SSC CGL formula sheet

Every formula and shortcut trick from the notes, on one page. Print it (Ctrl/Cmd + P) or save as PDF — it's formatted for paper.

Quantitative Aptitude

Number System

Number types, place value & counting

Multiples of k up to n

\left\lfloor \dfrac{n}{k} \right\rfloor

Multiples of k from a to b

\left\lfloor \dfrac{b}{k} \right\rfloor - \left\lfloor \dfrac{a-1}{k} \right\rfloor

Inclusion-exclusion

n(A \cup B) = n(A) + n(B) - n(A \cap B)

the overlap holds multiples of the LCM

Sum of first n natural numbers

\dfrac{n(n+1)}{2}

Sum of squares

1^2 + 2^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}

Sum of cubes

1^3 + 2^3 + \cdots + n^3 = \left[\dfrac{n(n+1)}{2}\right]^2

First n odd or even numbers

1 + 3 + \cdots + (2n-1) = n^2, \quad 2 + 4 + \cdots + 2n = n(n+1)

Any arithmetic series

S = \dfrac{n}{2}(a + l)

terms times average of first and last

Digit reversal

(10a + b) - (10b + a) = 9(a - b), \quad (10a + b) + (10b + a) = 11(a + b)

⚡ Floor-division counting. Count multiples of k up to the end of the range, then subtract the multiples before the start. No listing needed.

⚡ Place value in one product. Place value minus face value equals digit times (position value minus 1). The 9999 pattern does it in one line.

⚡ Middle term times count. For any evenly spaced list, the sum equals the number of terms times the middle term. The middle term is the average of first and last.

Divisibility rules

Divisibility by 11

\left(\sum \text{odd-place digits}\right) - \left(\sum \text{even-place digits}\right) \in \{0, \pm 11, \pm 22, \ldots\}

Composite divisor

pq \mid N \iff p \mid N \text{ and } q \mid N, \quad \gcd(p, q) = 1

split into co-prime parts only

Difference of powers

(a - b) \mid (a^n - b^n) \text{ for all } n

Difference of even powers

(a + b) \mid (a^n - b^n) \text{ when } n \text{ is even}

Sum of odd powers

(a + b) \mid (a^n + b^n) \text{ when } n \text{ is odd}

Repeated block

\overline{abcabc} = \overline{abc} \times 1001 = \overline{abc} \times 7 \times 11 \times 13

Divisibility by seven

N \to \left\lfloor \dfrac{N}{10} \right\rfloor - 2 \times (N \bmod 10)

repeat until small

⚡ Split into co-prime factors. Break the divisor into parts with no common factor and test each part separately. Order the tests from the most restrictive first.

⚡ The repeated-block family. Any six-digit number abcabc equals abc times 1001, so it is always divisible by 7, 11 and 13.

⚡ Sum of odd powers. For a to the n plus b to the n with odd n, check the options against a plus b first. No expansion is ever needed.

Remainders & remainder theorem

Division algorithm

N = d \times q + r, \quad 0 \le r < d

Product rule

\mathrm{rem}\left(\dfrac{a \times b}{d}\right) = \mathrm{rem}\left(\dfrac{R_a \times R_b}{d}\right)

same for sums

Fermat's little theorem

a^{p-1} \equiv 1 \pmod{p}, \quad p \text{ prime}, \ \gcd(a, p) = 1

Divisor-multiple rule

N \equiv r \pmod{D},\ d \mid D \ \Rightarrow\ N \equiv r \pmod{d}

one-way rule

Base one more than divisor

(ad + 1)^n \equiv 1 \pmod{d}

Base one less than divisor

(ad - 1)^n \equiv (-1)^n \pmod{d}

1 if n even, d - 1 if n odd

⚡ Make the base plus or minus one. Write the base as (multiple of divisor) plus or minus 1. The whole power then collapses to plus or minus 1.

⚡ Negative remainders for products. When factors sit just below the divisor, replace each by a small negative remainder and multiply those.

⚡ Divisor is a factor, just reduce. If the new divisor divides the old one, reduce the old remainder by the new divisor. If it does not divide it, this shortcut is not available.

Unit digit & cyclicity

Cyclicity rule

\text{unit}(a^n) = \text{unit}(a^r), \quad r = n \bmod 4 \ (r = 0 \Rightarrow r = 4)

Cycles of two, three, seven, eight

2: 2,4,8,6 \quad 3: 3,9,7,1 \quad 7: 7,9,3,1 \quad 8: 8,4,2,6

Factorials

n! \equiv 0 \pmod{10} \text{ for } n \ge 5

Last two digits, base ending in one

(10a + 1)^n \text{ ends in } \left[(a \cdot n) \bmod 10\right] 1

⚡ Last two digits of the exponent. For division by four, only the last two digits of the exponent matter, because 100 is a multiple of 4.

⚡ Even times five ends in zero. If a product contains an even factor and a factor ending in 5, the unit digit is 0 without any cycle work.

⚡ Factorial sums stop at four terms. From 5! on, every factorial ends in 0, so a factorial sum's unit digit comes from the first four terms alone.

Factors, prime factorisation & trailing zeros

Number of factors

d(N) = (a + 1)(b + 1)(c + 1)

Sum of factors

\sigma(N) = (1 + p + \cdots + p^a)(1 + q + \cdots + q^b) \cdots

Even factors

a \times (b + 1)(c + 1)

when 2 has exponent a

Product of all factors

N^{d(N)/2}

Trailing zeros in n factorial

\left\lfloor \dfrac{n}{5} \right\rfloor + \left\lfloor \dfrac{n}{25} \right\rfloor + \left\lfloor \dfrac{n}{125} \right\rfloor + \cdots

Highest power of a prime in n factorial

\sum_{k \ge 1} \left\lfloor \dfrac{n}{p^k} \right\rfloor

⚡ Even factors: force one two. If N has two to the power a in it, even factors equal a times the factor count of the odd part.

⚡ Successive division by five. For trailing zeros, keep dividing n by 5 and add the quotients until the quotient is 0.

⚡ Sum of factors as brackets. One bracket per prime, each running from 1 up to the full power. Multiply the brackets.

Fractions, decimals & recurring decimals

Pure repeating decimal

0.\overline{ab} = \dfrac{ab}{99}

Mixed repeating decimal

0.a\overline{bc} = \dfrac{abc - a}{990}

Terminating test

\dfrac{p}{q} \text{ terminates} \iff q = 2^m \times 5^n

q in lowest terms

⚡ Nines for pure repeats. One repeating digit gives 9 in the denominator, two give 99, three give 999.

⚡ Zeros after nines for mixed repeats. Denominator: one 9 per repeating digit, then one 0 per non-repeating digit. Numerator: all digits minus the non-repeating block.

⚡ Compare what is missing. For fractions close to 1, compare the shortfalls instead of the fractions.

Simplification

BODMAS, 'of', brackets & vinculum

'Of' before division

a \div b \text{ of } c = a \div (b \times c)

Division and multiplication left to right

a \div b \times c = \dfrac{a}{b} \times c

Minus before a bracket

a - (b - c) = a - b + c

Continued fraction

a + \dfrac{1}{b + \dfrac{1}{c}} = a + \dfrac{c}{bc + 1}

⚡ Bracket every 'of'. Rewrite each 'of' as a bracketed product before doing any division. The bracket makes the grouping visible and kills the trap.

⚡ Continued fractions from the bottom. Simplify the deepest fraction first, invert it, add the next layer, and repeat upward.

⚡ Digit-sum check. Casting out nines: digit sums must match on both sides of the equals sign. It kills wrong options without computing the product.

Algebraic identities in numerical simplification

Sum of cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Difference of cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Difference of squares

a^2 - b^2 = (a+b)(a-b)

Cube of sum

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

Squares combine

(a+b)^2 + (a-b)^2 = 2(a^2+b^2)

Squares subtract

(a+b)^2 - (a-b)^2 = 4ab

Zero-sum cubes

a+b+c = 0 \Rightarrow a^3+b^3+c^3 = 3abc

Square-sum from pair facts

a^2 + b^2 = (a+b)^2 - 2ab

⚡ Pattern-match the fraction. Cubes on top and three terms with a middle product below mean a sum or difference of cubes. The fraction is just a plus or minus of the bases.

⚡ Zero-sum check before cubing. Add the three bases. If they sum to zero, the cubes sum to three times their product, signs included.

⚡ Products around a round number. Write the pair as centre minus gap and centre plus gap. The product is centre squared minus gap squared.

Surds & indices

Same base

a^m \cdot a^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n}

Power of power

(a^m)^n = a^{mn},\quad (ab)^n = a^n b^n

Zero and negative index

a^0 = 1,\quad a^{-n} = \frac{1}{a^n}

Fractional index

a^{p/q} = \sqrt[q]{a^p}

Rationalisation

\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a - b}

Root of a surd

\sqrt{a \pm 2\sqrt{b}} = \sqrt{x} \pm \sqrt{y},\ x + y = a,\ xy = b

Reciprocal of a unit surd

x = a + \sqrt{b},\ a^2 - b = 1 \Rightarrow \tfrac{1}{x} = a - \sqrt{b}

⚡ Common base, then equate powers. Write both sides as powers of the same prime. The equation becomes a linear equation in the exponent.

⚡ Split a + 2 root b. Force the middle term into the 2 root b shape, then find two numbers with the given sum and product.

⚡ Reciprocal by the a squared minus b check. For x = a + sqrt(b), test a squared minus b. If it equals 1, the reciprocal is a minus sqrt(b) with no work.

Square roots & cube roots

Product rule

\sqrt{ab} = \sqrt{a}\,\sqrt{b},\quad \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b}

Infinite radical, plus

\sqrt{n + \sqrt{n + \cdots}} = \dfrac{1 + \sqrt{1 + 4n}}{2}

Infinite radical, minus

\sqrt{n - \sqrt{n - \cdots}} = \dfrac{-1 + \sqrt{1 + 4n}}{2}

Infinite nested product

\sqrt{x\sqrt{x\sqrt{x \cdots}}} = x

Least add or subtract

\text{subtract } N - k^2,\quad \text{add } (k+1)^2 - N

⚡ Cube root by unit digit and leading group. The last digit of the cube gives the last digit of the root; the leading group, compared with the small cubes, gives the first digit.

⚡ Square root by unit digit and size. Digit pairs fix the number of digits and the first digit; the last-digit map gives two candidates; one size check picks between them.

⚡ n equals k times k plus one radicals. Factor n into two consecutive integers. Plus signs give the larger one, minus signs the smaller.

Approximation

Percentage swap

x\% \text{ of } y = y\% \text{ of } x

Near-square root

\sqrt{n^2 + k} \approx n + \frac{k}{2n}

Percent-fraction anchors

12.5\% = \tfrac{1}{8},\quad 16.7\% = \tfrac{1}{6},\quad 33.3\% = \tfrac{1}{3},\quad 37.5\% = \tfrac{3}{8}

⚡ Swap the percentage. x percent of y equals y percent of x. Flip to whichever side has the nicer multiplier.

⚡ Balanced rounding for products. Round one factor up and the other down. The two errors cancel and the product stays close.

⚡ Anchors before bases. Turn the percent into its fraction anchor first, then apply it to the rounded base.

HCF & LCM

HCF & LCM: definitions and core relations

Product relation (two numbers)

\text{HCF} \times \text{LCM} = a \times b

Other number

b = \dfrac{\text{HCF} \times \text{LCM}}{a}

Co-prime case

\gcd(a, b) = 1 \Rightarrow \text{LCM}(a, b) = ab

HCF divides every difference

\gcd(a, b) \mid (a - b)

LCM is a multiple of the HCF

\gcd(a, b) \mid \text{LCM}(a, b)

Ratio pair

\text{numbers} = hm,\ hn; \quad \text{LCM} = hmn

⚡ Ratio split. Numbers in ratio a : b (co-prime parts) with HCF h are ha and hb. LCM = h times a times b; sum = h times (a plus b).

⚡ Product divided by HCF. Given any two of product, HCF, LCM, the product relation settles the third in one division.

⚡ Co-prime pairs from the quotient. For pair-count questions, divide the product by the square of the HCF, then count co-prime factor pairs of the result.

Finding HCF & LCM (incl. fractions and decimals)

HCF by factors

\text{HCF} = p_1^{\min} \times p_2^{\min} \times \cdots

common primes, lowest powers

LCM by factors

\text{LCM} = p_1^{\max} \times p_2^{\max} \times \cdots

all primes, highest powers

HCF of fractions

\text{HCF}\!\left(\dfrac{a}{b}, \dfrac{c}{d}\right) = \dfrac{\gcd(a, c)}{\text{LCM}(b, d)}

LCM of fractions

\text{LCM}\!\left(\dfrac{a}{b}, \dfrac{c}{d}\right) = \dfrac{\text{LCM}(a, c)}{\gcd(b, d)}

LCM from the HCF

\text{LCM}(a, b) = \dfrac{a \times b}{\gcd(a, b)}

⚡ Row of prime powers. Write each number as a row of prime powers in a small grid. LCM reads the column maxima, HCF the common minima. Three numbers take under thirty seconds.

⚡ Cross rule for fractions. HCF of fractions: HCF of tops over LCM of bottoms. LCM of fractions: LCM of tops over HCF of bottoms.

⚡ Shift the decimal point. Multiply by the power of 10 that clears every decimal, solve as integers, then shift the point back by the same number of places.

Standard word problems (tiles, bells, groups, divisible numbers)

Same remainder r

N = \text{LCM}(d_1, d_2, \ldots) \times k + r

Remainder is divisor minus c

r_i = d_i - c \Rightarrow N = \text{LCM} \times k - c

Divides with remainders

\text{answer} = \gcd(a - r_1,\ b - r_2,\ c - r_3)

Largest tile count

\text{tiles} = \dfrac{L \times W}{h^2}, \quad h = \gcd(L, W)

⚡ Subtract remainders, then HCF. For 'greatest number dividing a and b leaving remainders r1 and r2', the answer is the HCF of (a - r1) and (b - r2).

⚡ LCM plus r. For 'least number leaving remainder r with each divisor', add r to the LCM of the divisors.

⚡ Bells: add the LCM to the clock. Convert all intervals to one unit, take the LCM, and add it to the given start time.

Two-step LCM/HCF cases (extra condition, N-digit bounds)

Extra divisibility condition

N = Lk + r, \quad N \equiv 0 \pmod{p} \ \Rightarrow\ Lk \equiv -r \pmod{p}

Reconstruction from HCF and LCM

ab = \dfrac{\text{LCM}}{h}, \quad \gcd(a, b) = 1, \quad \text{numbers} = ha,\ hb

Pair count

\#\{(a, b): ab = M,\ \gcd(a, b) = 1,\ a \le b\}

Same unknown remainder

\text{answer} = \gcd(a - b,\ b - c,\ a - c)

⚡ Test k until it clicks. Once N = LCM x k + r is written, only k remains. Test k = 1, 2, 3 against the extra condition; small values click fast.

⚡ n-digit multiple scan. For the least n-digit multiple, divide the smallest n-digit number by the LCM and step up to the next multiple. For the greatest, subtract the remainder from the largest n-digit number.

⚡ Sum or difference picks the pair. With HCF, LCM and a sum or difference given, list co-prime pairs of LCM / HCF and match h times the parts to the sum.

Percentage

Percentage meaning & conversions

x% of a number

\frac{x}{100} \times N

Turn the percent into a fraction first; cancel where you can.

Whole from a part

\text{whole} = \frac{\text{part} \times 100}{x}

x is the percent that the part stands for.

A as a percent of B

\frac{A}{B} \times 100\%

B is the quantity that follows the word 'of'.

Remaining part

\text{rest} = 100\% - \text{given}\%

Only when both percentages are of the same whole.

⚡ Let the fraction table do the division. Replace the percent with its fraction from the table. 12.5\% becomes \dfrac{1}{8}, so the sum turns into a simple division.

⚡ Build the percent from 10% and 1% pieces. Move the decimal point for 10\% and 1\%, then add pieces. Works without pen-and-paper multiplication.

⚡ Swap x% of y into y% of x. x\% of y always equals y\% of x. Swap when one side turns into a friendly percent.

Percentage increase / decrease & successive change

Per cent change

\frac{\text{new} - \text{old}}{\text{old}} \times 100

Old value below the line.

Multiplier

\text{new} = \text{old} \times \frac{100 \pm x}{100}

Plus for a rise, minus for a fall.

Successive changes

a + b + \frac{ab}{100}

a and b carry their own signs.

Same x% up and down

\text{net loss} = \frac{x^2}{100}\%

x% rise followed by x% cut.

Fraction with both parts changed

\text{new} = \text{old} \times \frac{\text{top chip}}{\text{bottom chip}}

⚡ One multiplication per change. Replace every rise or fall with its multiplier fraction and multiply once. This replaces three lines of working.

⚡ a + b + ab/100 for two changes. Two changes in a row combine by this one formula. Keep the signs of both percents.

⚡ Round trip loses x squared over 100. A rise of x% followed by a cut of x% always ends below the start, by exactly \dfrac{x^2}{100} per cent.

'x% more than' ↔ 'x% less than' & chains

x% more, reversed

\frac{100x}{100+x}\% \text{ less}

x% more than becomes this much less than.

x% less, reversed

\frac{100x}{100-x}\% \text{ more}

x% less than becomes this much more than.

Per cent of a per cent

\frac{a}{100} \times \frac{b}{100} = \frac{ab}{10000}

Each 'of' multiplies.

⚡ Memorise the standard pairs. Four pairs cover most papers: 25/20, 20/16\dfrac{2}{3}, 50/33\dfrac{1}{3}, and 100/50. Read the question, write the partner, done.

⚡ Base-100 when memory fails. Put the base quantity at 100, write the other quantity, and divide by the new base. Slow but never wrong.

⚡ Awkward percents: use the fraction. 16\dfrac{2}{3}\% is 1/6, so 'more by 1/6' reverses to 'less by 1/7' of the bigger number.

Population growth, depreciation & elections

Growth for n years

P\left(1 + \frac{r}{100}\right)^n

r% added every year, compounding.

Depreciation for n years

P\left(1 - \frac{r}{100}\right)^n

r% of value lost every year.

Value n years ago

\frac{\text{now}}{\left(1 \pm \frac{r}{100}\right)^n}

Divide by the chip power to go back.

Election margin

\text{margin votes} = (\text{winner}\% - \text{rival}\%) \times \text{valid votes}

Percents on valid votes only.

⚡ Chip powers beat formulas. Write one multiplier per year and multiply. Squares like \left(\dfrac{5}{4}\right)^2 = \dfrac{25}{16} are worth remembering.

⚡ Remove invalid votes first. In elections, bring every statement onto valid votes before touching the margin.

⚡ Roll, cast, valid: walk the chain. Election numbers form a chain: roll, votes cast, valid votes, then candidates. Convert one link at a time with chips.

Marks, income–expenditure–savings & price–consumption

Maximum marks (two candidates)

M = \frac{(a+b) \times 100}{y - x}

a = shortfall below pass, b = marks above pass, x and y are the two percents.

Pass mark from failing

\text{pass} = \frac{x}{100}M + a

a is the shortfall.

Consumption cut after a price rise

\frac{100x}{100 + x}\%

Budget unchanged, price up x%.

Extra quantity after a price cut

\frac{100c}{100 - c}\%

Budget unchanged, price down c%.

⚡ Span of marks over span of percents. Two candidates give one clean fraction: marks apart over percents apart. Multiply by 100 for maximum marks.

⚡ Price rise to consumption cut is a flip. Reuse the more/less flip: price up x\% with the same budget means consumption down \dfrac{100x}{100+x}\%.

⚡ Extra kilograms reveal the price. After a price cut, extra kilograms equal money times the difference of the chips, divided by the new price. Solve for the price.

Ratio, Proportion, Partnership & Ages

Ratio basics & dividing amounts

Share of the total

\text{share} = \frac{\text{ratio term}}{\text{sum of terms}} \times N

Combining ratios

A:B = m:n,\ B:C = p:q \Rightarrow A:B:C = mp : np : nq

Cross-multiplication test

a:b > c:d \iff ad > bc

Multiplier method

\text{shares } ax, bx,\ (b-a)x = \text{given difference}

Duplicate / sub-duplicate

a:b \Rightarrow a^2:b^2 \text{ (duplicate)},\ \sqrt{a}:\sqrt{b} \text{ (sub-duplicate)}

⚡ x solves everything. Translate 'ratio a : b' into ax and bx. Every extra fact becomes one small equation in x.

⚡ LCM bridging for three terms. Scale A : B so its B-term matches the B-term of B : C, then read off A : B : C.

⚡ One share known, get the rest. Share = fraction × total. Recover the total from one share, then build any other share.

Proportion & proportional division of terms

Basic proportion

\frac{a}{b} = \frac{c}{d} \iff ad = bc

Fourth proportional

d = \frac{bc}{a}

Third proportional

c = \frac{b^2}{a}

Mean proportional

\text{mean} = \sqrt{ab}

Componendo & dividendo

\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a+b}{a-b} = \frac{c+d}{c-d}

Invertendo / alternando

\frac{b}{a} = \frac{d}{c},\quad \frac{a}{c} = \frac{b}{d}

⚡ Extremes times means. Set up the proportion in the exact order given, cross-multiply, done.

⚡ Mean proportional = geometric mean. Multiply the two numbers and take the square root. Exam numbers make it a perfect square.

⚡ Componendo-dividendo jump. When (x + y) and (x - y) both appear, jump straight to x/y by adding and subtracting the given ratio's terms.

Partnership

Simple partnership

P_1 : P_2 = C_1 : C_2 \quad (\text{same time})

Compound partnership

P_1 : P_2 : P_3 = C_1T_1 : C_2T_2 : C_3T_3

Working partner

\text{profit} = \text{manager's cut} + \text{residual split by } C_iT_i

Capital change mid-year

\text{effective capital} = C_a t_1 + C_b t_2 + \cdots

⚡ One row per partner. Write capital × months for each partner; the profit ratio is the row ratio.

⚡ Cut comes off the top. Salary or a per cent for the manager is removed from the whole profit before the capital-months split.

⚡ Equal shares, find the capital. Equal shares mean equal capital-months. Equate the products and solve.

Problems on ages

Present = ax, bx

\frac{ax \pm n}{bx \pm n} = \frac{p}{q}

Ratio after (or before) n years.

Constant difference

A - B \text{ is the same at every time}

Sum grows by 2 per year

(A + B)_{t+n} = (A + B)_t + 2n

Multiple of age

F = kS \Rightarrow F \pm n = k'(S \pm n)

k falls over time for elder-younger pairs.

⚡ Multiplier plus shift. Ages ax, bx now; shift both by n; equate the new ratio; solve for x.

⚡ Use the constant difference. The gap between two ages never changes — compute it once and reuse it at any time point.

⚡ Bracket the present with two ratios. With 'n years ago' and 'n years hence' ratios, both snapshots share one x — the 2n shift separates them.

Money ratios: income–expenditure, coins & mixed amounts

Savings equations

ax - py = s_1,\quad bx - qy = s_2

Two multipliers: x for incomes, y for expenditures.

Coin value

\text{total value} = k \sum_i (n_i \times d_i)

Price one set, then scale.

Wages together

\text{shares} \propto \frac{1}{\text{days alone}}

Difference of shares

\text{given excess} = (b - a)x

⚡ Subtract the savings equations. Write both savings equations and subtract — equal savings kill the constant and one variable.

⚡ Value per set for coins. Take one full set of coins in the given ratio, price it in paise, and scale.

⚡ Base the chain on the last person. For 'A gets half of B, B gets two-thirds of C', give the last-named person the unit and read the ratio off.

Average

Average & the sum bridge

Definition

\bar{x} = \frac{\sum x_i}{n} \iff \sum x_i = n\bar{x}

The sum form is the one you use.

Shift property

\overline{x_i + k} = \bar{x} + k,\quad \overline{k\, x_i} = k\bar{x}

Adding k shifts the average by k; multiplying by k scales it.

First n naturals / odd / even

\frac{n+1}{2},\quad n,\quad n+1

n = how many terms.

Squares / cubes

\frac{(n+1)(2n+1)}{6},\quad \frac{n(n+1)^2}{4}

Equal gaps

\text{average} = \frac{\text{first} + \text{last}}{2} = \text{middle term}

⚡ Flip to the sum first. Never push averages around. Change them to sums, adjust, divide back at the end.

⚡ Same operation on every value. Do to the average exactly what is done to each value, in the same order.

⚡ Equal gaps: jump from the middle. The average of evenly spaced numbers is the middle term. Step out from it instead of adding everything.

Members joining or leaving

Joining member

\text{value} = A' + n(A' - A)

n = old count; A' = new average.

Leaving member

\text{value} = A' + n(A - A')

A' = average of the remaining n members.

Replacement

\text{new} = \text{old} + n(A' - A)

The count does not change.

Count change both ways

\text{total of newcomers} = \text{new total} - \text{old total}

⚡ Joining: own share plus extras. New value = new average + (old count) × (average jump).

⚡ Replacement: the jump times n. Only the swapped item changes the sum, and it changes by n × (average jump).

⚡ Exclusion: subtract the totals. Old total − (new count × new average) = the removed value.

Weighted average & two-group problems

Weighted mean

\bar{x} = \frac{\sum n_i \bar{x}_i}{\sum n_i}

Missing group average

\bar{x}_2 = \frac{N\bar{x} - n_1\bar{x}_1}{n_2}

N = total count, overall average known.

Sizes from distances

\frac{n_1}{n_2} = \frac{\bar{x}_2 - \bar{x}}{\bar{x} - \bar{x}_1}

Reverse ratio of the distances.

⚡ Totals, not averages. Turn each group into a total, add, divide by the combined count.

⚡ Feel the balance point. The combined average splits the gap between the group averages in the ratio n₂ : n₁, the reverse of the sizes.

⚡ One newcomer is a tiny second group. A single new member changing a group average is just a weighted average with k = 1. Use value = B + n(B − A).

Batsman problems, overlapping sums & multi-step sets

Batsman score

\text{score} = A' + (n-1)d

A' = new average, d = rise.

Batsman new average

A' = \frac{(n-1)A + \text{score}}{n} = x - (n-1)d

Overlap subtraction

\text{Thu} - \text{Mon} = 3(b - a)

Three-day windows; multiply by the overlap length.

Shared middle item

\text{shared} = S_1 + S_2 - S_{\text{total}}

Split sums

n\bar{x} = \sum_{\text{chunks}} (\text{chunk total})

One equation per chunk.

⚡ Batsman: stay in totals. Old total + new score = new count × new average.

⚡ Overlaps: subtract the sums. Shared days cancel, leaving only the difference of the end days.

⚡ One variable for the unknown chunk. Name the smallest unknown x, express the rest through it, and close the equation with the leftover sum.

Interest (SI & CI)

Simple Interest

Simple interest

SI = \frac{PRT}{100}

Any one of the four recovers from the other three.

Amount

A = P + SI = P\left(1 + \frac{RT}{100}\right)

'Amounts to' includes the principal.

Recovering inputs

P = \frac{100\,SI}{RT},\quad R = \frac{100\,SI}{PT},\quad T = \frac{100\,SI}{PR}

n-times in T years

R = \frac{100(n-1)}{T}

Interest is only (n−1)P.

Equal yearly interest

SI_{\text{per year}} = \frac{SI_{\text{total}}}{T}

Same rupees every year.

⚡ Cover the unknown. Write SI = \dfrac{PRT}{100} and cover the letter you want. That covered formula is the whole equation.

⚡ n-times to rate. 'Becomes n times in T years' means the interest earned is (n-1)P. The P cancels.

⚡ One year at a time. Divide the total interest by the years; every year carries exactly that much.

Compound Interest

Compound amount

A = P\left(1 + \frac{R}{100}\right)^T

One chip per year.

Compound interest

CI = A - P = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]

Year-wise multipliers

A = P \times \frac{100 + r_1}{100} \times \frac{100 + r_2}{100} \times \cdots

For rates that change yearly.

Half-yearly compounding

A = P\left(1 + \frac{R}{200}\right)^{2T}

Rate halves, periods double.

Quarterly compounding

A = P\left(1 + \frac{R}{400}\right)^{4T}

Rate quarters, periods quadruple.

⚡ Multiply the chips. One chip per year, multiplied. Fraction chips cancel before you multiply.

⚡ Divide the chips to get P. An amount after T years walks back to the principal by dividing by the chip T times.

⚡ Half-yearly: halve rate, double count. Convert first, then run the chips on the periods.

CI vs SI: differences & doubling

2-year difference

CI - SI = P\left(\frac{R}{100}\right)^2

3-year difference

CI - SI = P\left(\frac{R}{100}\right)^2\left(3 + \frac{R}{100}\right)

CI doubling chain

2\times \text{ in } T \Rightarrow 2^k\times \text{ in } kT

SI multiple pace

n\times \text{ in } T \Rightarrow n'\times \text{ in } T',\ (n'-1) = (n-1)\frac{T'}{T}

Linear, not powers.

Rate from both figures

R = \frac{200 \times (CI_2 - SI_2)}{SI_2}

Two-year case.

⚡ The rate-fraction squared gap. Two equations — the SI and the difference — hand you the rate and the principal.

⚡ Count the doublings. Every T years at CI the money multiplies by the same factor, so count powers.

⚡ First-year interest lent again. For 2 years, think of the difference as the first year's interest earning R% once more.

Equal annual instalments

Present value (CI)

P = \sum_{k=1}^{n} \frac{x}{\left(1 + \frac{r}{100}\right)^k}

One term per instalment.

Simple interest instalment

P = nx - \frac{x\,r}{100} \cdot \frac{n(n-1)}{2}

Interest the early payments save.

Tabular step

\text{debt}_{k+1} = \text{debt}_k\left(1 + \frac{r}{100}\right) - x

Must end at zero.

⚡ Grow, subtract, repeat. Two or three rows of the debt table finish any small question.

⚡ Present-value one-liner. Discount each instalment by the chip power for its year and add.

⚡ SI instalment shortcut. Use P = nx - \dfrac{xr}{100} \cdot \dfrac{n(n-1)}{2} straight off for simple-interest loans.

Finding P, R, T from amount data

Principal from amount

P = \frac{A}{\left(1 + \frac{r}{100}\right)^T}

CI: divide by the chip power.

Rate from consecutive amounts

1 + \frac{r}{100} = \frac{A_{t+1}}{A_t} \quad (\text{CI})

Divide at CI.

Yearly SI from amounts

SI_{\text{year}} = A_{t+1} - A_t \quad (\text{SI})

Subtract at SI.

Two-amount system

\frac{A_2}{A_1} = 1 + \frac{r}{100} \Rightarrow P = \frac{A_1}{1 + r/100}

⚡ Ratio first, then walk back. The ratio of consecutive CI amounts exposes the chip; divide once more for P.

⚡ Equal yearly steps at SI. Check the amounts differ by a constant — that constant is the yearly interest.

⚡ Two amounts, two unknowns. Dividing the amounts kills P and hands you the chip.

Profit, Loss & Discount

Profit, Loss and CP–SP Basics

Profit / loss per cent

\text{P\%} = \frac{SP - CP}{CP} \times 100

Always on CP.

Selling price

SP = CP \times \frac{100 \pm x}{100}

Plus for profit, minus for loss.

Cost price from SP

CP = SP \times \frac{100}{100 \pm x}

Divide by the chip.

Equal profit and loss at two prices

CP = \frac{S_1 + S_2}{2}

When profit at S1 equals loss at S2.

⚡ Fraction swap for the per cent. Turn the profit into a fraction of CP and the per cent falls out. 170/850 is clearly 1/5.

⚡ Divide by the chip to get CP. A loss of 4\% means the chip 96/100. Divide the SP by the chip, nothing else.

⚡ Same per cent scales linearly. When the profit per cent is fixed, CP and SP scale together. Article counts often hide a clean ratio.

Successive Changes & Equivalent Single Change

Two successive changes

\text{net} = a + b + \frac{ab}{100}

Use negative b for a fall.

Same change twice (rise)

2x + \frac{x^2}{100}

Same change twice (fall)

2x - \frac{x^2}{100}

Equivalent single discount

D = d_1 + d_2 - \frac{d_1 d_2}{100}

For two discounts.

Kept fraction

\text{kept} = \prod \frac{100 - d_i}{100}

Any number of discounts; customer pays this share.

⚡ Multiply the chips. Chips handle any mix of rises and falls, in any order, and extend to three or more changes.

⚡ One line for two discounts. Two discounts collapse with d_1 + d_2 - \dfrac{d_1 d_2}{100}. One line, no chips.

⚡ Work backwards for the missing change. Net chip divided by the known chip gives the unknown chip.

Marked Price, Discount and the CP–MP–SP Chain

Discount per cent

\text{D\%} = \frac{MP - SP}{MP} \times 100

On the marked price.

Marked price for a target gain

MP = CP \times \frac{100 + g}{100 - d}

g = gain%, d = discount%.

SP through the chain

SP = CP\left(1 + \frac{m}{100}\right)\left(1 - \frac{d}{100}\right)

m = markup%.

Markup per cent

\text{markup\%} = \left(\frac{100+g}{100-d} - 1\right) \times 100

Multi-level markups

\text{final} = \text{cost} \times \prod \frac{100 + x_k}{100}

One chip per trader.

Free-item discount

\text{discount\%} = \frac{\text{free}}{\text{received}} \times 100

⚡ Build the MP-over-CP ratio. Gain g\% and discount d\% together fix MP as a multiple of CP: \dfrac{100+g}{100-d}.

⚡ Markup chip then discount chip. Two fractions give the whole story from CP to SP in one line.

⚡ Trade chains multiply. One chip per middleman. Work backwards from the final price by dividing in reverse order.

Dishonest Dealer, False Weights & Claims

False weight gain

\text{gain\%} = \frac{W - w}{w} \times 100

W = true weight, w = weight used; divide by w.

Weight from a given gain

w = W \times \frac{100}{100 + \text{gain\%}}

Claimed loss with short weight

\text{multiplier} = \frac{100 - L}{100 - c}

L = claimed loss%, c = short weight%.

Cheat at both ends

\text{multiplier} = \frac{100 + b}{100 - s}

b = extra taken while buying, s = shortfall while selling.

True gain from multiplier

\text{gain\%} = (\text{multiplier} - 1) \times 100

⚡ The W minus w over w rule. At cost price, the only question is which weight goes below the line. It is the weight he gives.

⚡ A claimed loss can still be a gain. Judge money per true gram, not per claimed gram.

⚡ Both-ends multiplier. Buy-side chip over sell-side chip. Two fractions, one division.

Same Selling Price: One Profit, One Loss

Net loss for equal plus-minus x%

\text{net loss\%} = \frac{x^2}{100}

Per cent of total CP.

Reconstructed costs

CP_1 = \frac{S}{1 + x/100}, \quad CP_2 = \frac{S}{1 - x/100}

S = common selling price.

Loss in rupees

\text{loss} = CP_1 + CP_2 - 2S

x from the loss per cent

x = 10\sqrt{\text{loss\%}}

⚡ The x squared over 100 reflex. Same SP, opposite equal percents: write the loss per cent straight away.

⚡ Rebuild both cost prices. Divide the common SP by each chip, add the CPs, compare with twice the SP.

Mixtures & Alligation

Mixture Concentration & Amounts

Amount from ratio

\text{part} = \text{total} \times \frac{\text{share}}{\text{sum of shares}}

Concentration

C = \frac{\text{ingredient}}{\text{mixture}} \times 100\%

Mean price

\text{mean} = \frac{q_1 p_1 + q_2 p_2}{q_1 + q_2}

Adding water

C' = \frac{A}{M + w}

A = amount of the other ingredient, unchanged.

Removing mixture

\text{each ingredient shrinks in its own share}

A uniform draw keeps the ratio.

⚡ Freeze the unchanged ingredient. Water added means milk unchanged. Hang the whole question on the ingredient that does not move.

⚡ Rebuild the per cent after dilution. Milk is fixed, the total grows — divide again.

⚡ One ratio change, one equation. Set the frozen ingredient against the wanted ratio and solve for the addition.

Rule of Alligation

Alligation ratio

\frac{\text{cheaper}}{\text{dearer}} = \frac{D - M}{M - C}

Weighted average

\bar{v} = \frac{n_1 v_1 + n_2 v_2}{n_1 + n_2}

Water as the free ingredient

\text{milk} : \text{water} = (m - 0) : (c - m),\ c > m

⚡ Cross the arms. Dearer-minus-mean and mean-minus-cheaper swap sides. Each quantity takes the other side's arm.

⚡ Water has price zero. Dilution is alligation with one strength at 0.

⚡ Averages alligate too. Salary, age, marks, weight — any average splits into an alligation.

Replacement & Repeated Operations

Repeated equal replacement

\text{left} = C\left(1 - \frac{x}{C}\right)^n

Milk : water after n rounds

\left(1-\frac{x}{C}\right)^n : \left[1-\left(1-\frac{x}{C}\right)^n\right]

Proportional removal

\text{lost} = \text{drawn volume} \times \frac{\text{ingredient share}}{\text{total}}

Unequal draws

C \prod_k \left(1 - \frac{x_k}{C}\right)

Multiply one factor per round when the draws differ.

⚡ One fraction per round. Each round multiplies the original liquid by (1 − x/C). Multiply the fractions; never subtract x twice.

⚡ Two rounds: square the kept fraction. The complement of the milk is the water — no separate calculation.

⚡ Draws keep the inside ratio. A uniform draw removes both liquids in proportion, so the ratio survives the withdrawal. Only the refill changes it.

Milk–Water Ratio & Profit by Adulteration

Adulteration gain

\text{gain\%} = \frac{\text{water}}{\text{milk}} \times 100

Target ratio

W : M = g : 100

For a gain of g% when sold at cost price.

Blend cost price

CP = \frac{q_1 c_1 + q_2 c_2}{q_1 + q_2}, \quad SP = CP\left(1 + \frac{g}{100}\right)

Alloy rebuild

\text{metal} = \text{alloy weight} \times \frac{\text{share}}{\text{sum of shares}}

⚡ Water over milk is the gain. Free litres per honest litre — that ratio, in per cent, is the profit.

⚡ Hit the target ratio by addition. Keep the bigger quantity constant and solve for the addition.

⚡ Rebuild the alloy, then change one metal. Ratio to grams, adjust one column, re-ratio.

Mixing Two Mixtures

Blend of two mixtures

f = \frac{V_1 f_1 + V_2 f_2}{V_1 + V_2}

Volumes for a target fraction

\frac{V_1}{V_2} = \frac{f_2 - f}{f - f_1}

Fraction from ratio

f_{\text{milk}} = \frac{m}{m + w}

Equal volumes

\bar{f} = \frac{f_1 + f_2 + f_3}{3}

Only when every vessel holds the same volume.

⚡ Fractions first, then average. Ratio → milk fraction per vessel → weighted average.

⚡ Alligate the fractions. The target sits between the two vessel fractions; the distances give the volumes.

⚡ Weight by volume when sizes differ. Multiply each vessel's fraction by its volume before adding.

Time & Work

Work Rates & the LCM Method

One-day work

\frac{1}{T}

Combined time (two workers)

T = \frac{ab}{a + b}

Combined time (three workers)

T = \frac{abc}{ab + bc + ca}

Work done and left

\text{done} = \frac{t}{T}, \quad \text{left} = 1 - \frac{t}{T}

⚡ LCM units, not fractions. Set the job to the LCM of the individual times; every rate becomes a small whole number.

⚡ The ab over a+b reflex. For exactly two workers, one formula — no fraction addition at all.

⚡ Three workers through the LCM. Same method with three rates added in one line.

Efficiency, 'Twice as Good' & Ratio Cases

Efficiency and time

\frac{E_A}{E_B} = \frac{T_B}{T_A}

k-times worker

T_B = (k+1)T, \quad T_A = \frac{(k+1)T}{k}

Times from an efficiency ratio

E_A : E_B = a : b \Rightarrow T_A : T_B = b : a

Per cent more efficient

a\% \text{ more} \Rightarrow E_A : E_B = (100 + a) : 100

⚡ Rate units from the efficiency ratio. Let B = 1 unit/day, A = k units/day; together (k+1) per day prices the job.

⚡ Invert, never scale. 'A is 3 times as good' means A's days = B's days ÷ 3.

⚡ Days-difference cases. 'B takes 24 days more than A' plus an efficiency ratio pins both times.

Pipes & Cisterns

Net speed

\text{net} = (\text{inlets}) - (\text{outlets})

Speeds in units per hour, with tank = LCM of the times.

Time to fill

T = \frac{\text{capacity}}{\text{net speed}}

Two inlets

T = \frac{a b}{a + b}

Both pipes fill.

One inlet, one outlet

T = \frac{a b}{b - a}

a = filling time, b = emptying time, b > a.

Leak time from two fill times

T_{\text{leak}} = \frac{t_1 t_2}{t_2 - t_1}

t₁ = normal time, t₂ = time with the leak.

Stage method

t_2 = \frac{\text{capacity} - \text{speed}_1 \times t_1}{\text{speed}_2}

⚡ One inlet, one outlet: multiply over subtract. For one filling pipe and one emptying pipe, time = product ÷ difference. No LCM needed.

⚡ Leak from two fill times. The leak is the only thing that changed, so the drop in speed belongs to the leak. Use product ÷ difference of the two fill times.

⚡ Check the options before finishing. Filling pipes together are always faster than the fastest pipe alone. Adding an outlet always makes it slower than the inlets alone. Use this to cut two options in a few seconds.

Men–Days–Hours Chain & Provisions

MDH chain

M_1 D_1 H_1 E_1 = M_2 D_2 H_2 E_2 \quad (W_1 = W_2)

Men and days constant

M_1 D_1 = M_2 D_2

Provisions remaining

\text{days} = \frac{M_1 (D_{total} - t)}{M_2}

Work scaling

W_2 = W_1 \times \frac{M_2}{M_1} \times \frac{D_2}{D_1}

⚡ Multiply resources, equate products. Everything that grows the work sits beside M; the missing quantity lands alone.

⚡ Provisions after reinforcement. Stock left = original men × days left; then divide by the new headcount.

⚡ Work left after a share is done. First find the remaining man-days, then apply them to the new team.

Joining / Leaving Mid-work, Alternate Days & Wages

Work done by A in t days

\frac{t}{T_A}

Leaves t days before the end

\frac{T - t}{T_A} + \frac{T}{T_B} = 1

Two-day cycle

\frac{1}{T_A} + \frac{1}{T_B} \text{ per 2 days}

Wage split

w_A = \text{total} \times \frac{1/T_A}{1/T_A + 1/T_B}

⚡ Subtract the finished part. Whoever continues inherits only the remainder.

⚡ Count full cycles, then the tail. Alternate days: measure progress per 2-day cycle.

⚡ Wages follow work, not days alone. Rate ratio × equal days = work ratio.

Time, Speed & Distance

Speed, Distance, Time & Unit Conversion

Basic relation

S = \frac{D}{T}, \quad D = S \times T

km/h to m/s

\text{km/h} \times \frac{5}{18} = \text{m/s}

Equal-distance average

\bar{S} = \frac{2ab}{a + b}

General average speed

\bar{S} = \frac{D_1 + D_2}{T_1 + T_2}

⚡ Convert first, always. Train lengths are in metres, times in seconds — reach m/s before anything else.

⚡ Harmonic mean for round trips. Same distance out and back → 2ab/(a+b) in one line.

⚡ Early and late gaps are time equations. Both runs cover the SAME distance — equate or subtract the two time expressions.

Relative Speed

Relative speed

S_{rel} = S_1 \pm S_2

Meeting time

t = \frac{\text{initial gap}}{S_1 + S_2}

Overtake time

t = \frac{\text{gap or combined length}}{S_1 - S_2}

Gap between two movers

d = (S_1 \pm S_2) \times t

⚡ Subtract for the same direction. Overtaking uses the difference; for two trains the distance is both lengths together.

⚡ Add for opposite directions. The closing speed is the sum even when one side is just a walking man.

⚡ Distance flown till meeting. Find the meeting time first; any third object's distance = its speed × that time.

Trains Crossing Poles, Platforms & Trains

Pole / man

L_{train} = S \times t

Platform / bridge

L_{train} + L_{platform} = S \times t

Train vs train

L_1 + L_2 = S_{rel} \times t

Two-equation extraction

S = \frac{P}{t_{platform} - t_{pole}}

⚡ Pole first: it is the train's own length. The pole crossing IS the train's length in motion.

⚡ Subtract the pole equation. Platform time minus pole time covers exactly the platform.

⚡ Platform crossing from rest. No pole given: just add both lengths and divide by the speed.

Boats & Streams

Effective speeds

u = b + s, \quad v = b - s

Boat and stream from legs

b = \frac{u + v}{2}, \quad s = \frac{u - v}{2}

Time for two legs

t = \frac{d_1}{b + s} + \frac{d_2}{b - s}

Round-trip average

\bar{S} = \frac{2uv}{u+v} = \frac{b^2 - s^2}{b}

Drift

\text{drift} = s \times \text{time}

⚡ Halve the sum, halve the difference. Down and up speeds give the boat and the stream in one line each.

⚡ Extract u and v from trip times. Each trip is one equation; the pair is linear in the reciprocals.

⚡ Two double-trip equations. Two journeys pin down both leg speeds exactly.

Races & Handicaps

Beating margin in metres

\frac{S_A}{S_B} = \frac{D}{D - x}

Beating margin in time

t_B = t_A + t; \quad S_B = \frac{x}{t}

Start handicap

B \text{ runs } D - \text{start} \ ( - \text{win margin})

Circular track meeting

t = \frac{L}{S_1 \mp S_2}

⚡ Translate beats by x m into a speed ratio. Same finishing time — distances are in the speed ratio.

⚡ By x metres or t seconds reveals both speeds. B's last x metres took t seconds — that is B's speed for free.

⚡ Handle the handicap. A start shortens one runner's distance — recompute the ratio.

Algebra

Basic algebraic identities

Square of sum or difference

(a \pm b)^2 = a^2 \pm 2ab + b^2

Difference of squares

a^2 - b^2 = (a+b)(a-b)

Sum of cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Difference of cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Cube of sum

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

Squares combine

(a+b)^2 + (a-b)^2 = 2(a^2+b^2)

Squares subtract

(a+b)^2 - (a-b)^2 = 4ab

Square of a trinomial

(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)

Fourth powers from the ladder

a^4 + b^4 = (a^2+b^2)^2 - 2a^2b^2

⚡ Build higher powers from a sum and a product. Chain the rungs: square-sum first, then cube-sum, then fourth powers. The individual letters are never needed.

⚡ Recognise the identity inside decimals. Cubed decimals over a trinomial mean a cube identity. The value is the sum or difference of the bases.

⚡ Value-putting among expression options. Put small numbers into the question and each option. Wrong options die in one round; ties die in a second.

$x+\frac{1}{x}$ type expressions

Square rung

x^2+\frac{1}{x^2} = \left(x+\frac1x\right)^2-2 = \left(x-\frac1x\right)^2+2

Cube rung, sum

x^3+\frac{1}{x^3} = \left(x+\frac1x\right)^3 - 3\left(x+\frac1x\right)

Cube rung, difference

x^3-\frac{1}{x^3} = \left(x-\frac1x\right)^3 + 3\left(x-\frac1x\right)

Fourth rung

x^4+\frac{1}{x^4} = \left(x^2+\frac{1}{x^2}\right)^2 - 2

Fifth rung

x^5+\frac{1}{x^5} = \left(x^2+\tfrac1{x^2}\right)\left(x^3+\tfrac1{x^3}\right)-\left(x+\tfrac1x\right)

Ladders link

\left(x+\frac1x\right)^2 - \left(x-\frac1x\right)^2 = 4

Equal-end quadratic

ax^2 - bx + a = 0 \Rightarrow x+\frac1x = \frac ba

Mixed form

\left(px+\frac{1}{qx}\right)^2 = p^2x^2+\frac{1}{q^2x^2}+\frac{2p}{q}

⚡ Run the ladder in order. Square rung first, then cube rung, then reuse them for higher rungs. Never restart from k for each question part.

⚡ Divide the quadratic by x. Equal first and last coefficients mean the quadratic is a k-value in disguise. Divide by x and read it off.

⚡ Switch ladders with one square. The two ladders differ by 4 under a square: k squared equals m squared plus 4. Convert once, then stay on the new ladder.

$a^3+b^3+c^3-3abc$ and conditional identities

Master identity

a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Sum form

a^3+b^3+c^3-3abc = s\left(s^2-3P\right)

Half form

a^3+b^3+c^3-3abc = \tfrac12\,s\left[(a-b)^2+(b-c)^2+(c-a)^2\right]

Zero-sum case

a+b+c = 0 \Rightarrow a^3+b^3+c^3 = 3abc

Equal case

a^2+b^2+c^2 = ab+bc+ca \Rightarrow a=b=c

Power-sum expansion

a^3+b^3+c^3 = s^3 - 3sP + 3R

Pair products

(a+b)(b+c)(c+a) = sP - R

Pairwise from squares

ab+bc+ca = \dfrac{(a+b+c)^2-(a^2+b^2+c^2)}{2}

⚡ Hunt for a hidden zero sum. Brackets like x minus y, y minus z, z minus x always sum to zero. The cube-sum is three times the product.

⚡ Close numbers: use the half form. When the three numbers differ by little, the squared differences are tiny, and half of s times their sum is quick arithmetic.

⚡ Value-putting under a condition. If a condition like a+b+c=0 is given, pick simple numbers that satisfy it and evaluate the expression.

Surds: rationalisation and square roots of surds

Rationalisation

\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a-b}

Conjugate product

(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b

Root of a surd, plus

\sqrt{a+2\sqrt{b}} = \sqrt{m}+\sqrt{n},\ m+n = a,\ mn = b

Root of a surd, minus

\sqrt{a-2\sqrt{b}} = \sqrt{m}-\sqrt{n}\ \ (m > n)

Product-one pair

x = p+\sqrt{q},\ p^2-q = 1 \Rightarrow \tfrac1x = p-\sqrt{q},\ x+\tfrac1x = 2p

Telescoping sum

\sum \frac{1}{\sqrt{n}+\sqrt{n+1}} = \sqrt{\text{last}} - \sqrt{\text{first}}

Difference of roots

\sqrt{a}-\sqrt{b} = \frac{a-b}{\sqrt{a}+\sqrt{b}}

⚡ Denest by sum and product. Rewrite the inner coefficient as 2 root something, then find two numbers with the given sum and product.

⚡ Spot the product-one conjugate. For x = p + sqrt(q), test p squared minus q. If it is 1, the reciprocal is the conjugate and the ladder runs.

⚡ Telescoping rationalisation. Each fraction over consecutive roots becomes a difference; middle terms cancel, leaving last root minus first.

Linear equations, graphs and polynomials

Unique solution

\frac{a_1}{a_2} \ne \frac{b_1}{b_2}

lines cross once

No solution

\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}

parallel lines

Infinite solutions

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

same line

Area with the axes

\text{Area} = \frac{1}{2}\cdot\left|\frac{c}{a}\right|\cdot\left|\frac{c}{b}\right|

Remainder theorem

p(x) \div (x-a) \Rightarrow R = p(a)

for px - q, substitute q/p

Roots of a quadratic

\alpha+\beta = -\frac ba,\quad \alpha\beta = \frac ca

Discriminant

b^2 - 4ac \gtrless 0

decides root nature

⚡ Area from intercepts. Put y = 0 for the x-intercept and x = 0 for the y-intercept, then halve the product of the absolute values.

⚡ Remainder equals substitution at the zero. No long division: substitute the zero of the divisor into the polynomial.

⚡ Test the options for the intersection point. For a meeting-point question, plug each option into both equations instead of solving the pair.

Maxima and minima (AM ≥ GM, quadratics)

AM-GM

\frac{x+y}{2} \ge \sqrt{xy}

equality when x = y

Plus-form floor

ax + \frac{b}{x} \ge 2\sqrt{ab}\ \ (x>0)

at x = sqrt(b/a)

Vertex location

x = -\frac{b}{2a}

Extreme value

\frac{4ac-b^2}{4a}

max for a < 0, min for a > 0

Fixed sum

x+y = S \Rightarrow xy \le \frac{S^2}{4}

Fixed product

xy = P \Rightarrow x+y \ge 2\sqrt{P}

Always positive

ax^2+bx+c > 0 \iff a > 0,\ b^2 < 4ac

strict inequality, strict discriminant

⚡ Equal split for weighted sums. To maximise a product under a weighted sum, set the weighted pieces equal, then square.

⚡ Complete the square, read the floor. For a > 0, the completed square shows the least value directly as the loose constant.

⚡ Equal roots sit on the boundary. Equal roots mean the discriminant is exactly zero; solve the resulting equation for the unknown.

Geometry

Lines and angles

Angles on a line / around a point

180^\circ,\ 360^\circ

A straight line totals 180 degrees; one full turn totals 360.

Co-interior angles

a+b=180^\circ

The two inside angles on one side of a transversal, between parallel lines.

Complement / supplement

\text{supplement}-\text{complement}=90^\circ

Complement = 90 − x, supplement = 180 − x.

Equal pairs at parallels

\text{corresponding}=\text{alternate}=\text{vertically opposite}

Each of these pairs is equal.

⚡ Say the shape, then the rule. Trace the two angles with your finger. An F or Z shape means they are equal. A C shape means they add to 180^\circ.

⚡ Supplement minus complement is always 90. Whatever the angle, its supplement and its complement differ by exactly 90^\circ. Use it as a free check, or to build the equation.

Triangles and their centres

Angle sum and exterior angle

A+B+C=180^\circ,\quad \text{ext at }A=B+C

Exterior angle = sum of the two remote (far) interior angles.

Centroid division

AG:GD=2:1

G is the centroid on median AD; the vertex piece is twice the base piece.

Incentre angle

\angle BIC=90^\circ+\dfrac{A}{2}

I = incentre, where the angle bisectors meet.

Circumcentre angle

\angle BOC=2A

O = circumcentre; the angle at O stands on the same arc BC as angle A.

Orthocentre angle

\angle BHC=180^\circ-A

H = orthocentre, where the altitudes meet.

Apollonius (median length)

m_a^2=\dfrac{2b^2+2c^2-a^2}{4}

Median to side a of a triangle with sides a, b, c.

Isosceles median to base

m=\sqrt{a^2-\left(\dfrac{b}{2}\right)^2}

a = equal side, b = base.

Heron's area

K=\sqrt{s(s-a)(s-b)(s-c)},\ s=\dfrac{a+b+c}{2}

s = semi-perimeter (half the perimeter).

⚡ Centre angles from one input. Only \angle A is needed. Read which centre the question names, then use its formula.

⚡ Centroid cut in the ratio 2 : 1. Call the median 3 parts. The centroid gives 2 parts on the vertex side and 1 part on the base side.

⚡ Median in an isosceles triangle. Skip Apollonius when two sides are equal. The median to the base is \sqrt{a^2-(b/2)^2} with a the equal side and b the base.

⚡ Heron families worth memorising. (3,4,5) gives 6, (5,12,13) gives 30, (13,14,15) gives 84, (7,24,25) gives 84, (9,12,15) gives 54, (10,24,26) gives 120.

Congruence, similarity and BPT

Similarity ratios

\frac{a_1}{a_2}=k,\quad \frac{P_1}{P_2}=k,\quad \frac{K_1}{K_2}=k^2

a = side, P = perimeter, K = area; k = scale factor.

Areas from perimeters

\frac{K_1}{K_2}=\left(\frac{P_1}{P_2}\right)^2

Square a length ratio to get the area ratio; take a root to go back.

BPT (Thales)

DE\parallel BC\Rightarrow\frac{AD}{DB}=\frac{AE}{EC}

A line parallel to one side cuts the other two sides in the same ratio.

Midpoint theorem

D,E\ \text{midpoints}\Rightarrow DE=\frac{BC}{2}

The join of two midpoints is half the third side and parallel to it.

Angle bisector theorem

\frac{BD}{DC}=\frac{AB}{AC}

The bisector of angle A splits BC in the ratio of the sides AB and AC.

⚡ Square the perimeter ratio for areas. Perimeters in ratio p:q mean areas in ratio p^2:q^2. Going back, take the square root.

⚡ Spot the word midpoints. 'Midpoints of two sides' means the joining segment is half the third side, and parallel to it.

⚡ BPT: write the ratio directly. A line parallel to a side cuts equal ratios on both other sides. Write the proportion, substitute, solve.

Pythagoras theorem and triplets

Pythagoras

h^2=p^2+b^2

h = hypotenuse (longest side, opposite the right angle).

Missing leg

p=\sqrt{h^2-b^2}

Subtract when a leg is missing.

Median to hypotenuse

m=\dfrac{h}{2}

The median drawn from the right-angle corner.

Rectangle diagonal

d=\sqrt{l^2+b^2}

The corner angles of a rectangle are right angles.

Triangle type test

a^2\lessgtr b^2+c^2

a = longest side; equal means right, greater means obtuse, smaller means acute.

⚡ See two numbers, recall the third. 12 with 13 gives 5. 24 with 25 gives 7. 15 with 17 gives 8. Multiples scale: 10-24-26 is 5-12-13 times 2.

⚡ Median to the hypotenuse is half of it. In any right triangle, the median from the right angle equals half the hypotenuse.

⚡ Classify: square the longest side only. To name the triangle type, compare the longest side's square with the sum of the other two squares.

Quadrilaterals and polygons

Quadrilateral angle sum

A+B+C+D=360^\circ

Any four-sided figure.

Parallelogram angles

A+B=180^\circ,\quad A=C

Adjacent angles supplement; opposite angles equal.

Cyclic quadrilateral

A+C=180^\circ,\quad B+D=180^\circ

Opposite corners on one circle.

Rhombus

K=\frac{1}{2}d_1d_2,\quad a=\sqrt{\left(\frac{d_1}{2}\right)^2+\left(\frac{d_2}{2}\right)^2}

d1, d2 = diagonals; they cross at right angles.

Trapezium

K=\frac{1}{2}(a+b)h

a and b are the two parallel sides; h is the gap between them.

Parallelogram

K=bh

Height is measured perpendicular to the base.

Regular polygon

\text{ext}=\frac{360^\circ}{n},\quad \text{int}=180^\circ-\text{ext},\quad \text{diagonals}=\frac{n(n-3)}{2}

n = number of sides.

⚡ Exterior angle to number of sides. Exterior angle =180^\circ- interior, and n=\dfrac{360^\circ}{\text{exterior}}.

⚡ Rhombus side from half-diagonals. Halve both diagonals. They are the legs of a right triangle whose hypotenuse is the side.

⚡ Count diagonals in one line. Each of the n corners joins n-3 others, and every diagonal is counted twice, giving \dfrac{n(n-3)}{2}.

Circles: chords, tangents, secants and cyclic angles

Chord from distance

\ell=2\sqrt{r^2-d^2}

d = distance from centre to chord.

Equal chords

\ell_1=\ell_2\Rightarrow d_1=d_2

Equal chords sit equally far from the centre.

Centre vs circumference angle

\angle BOC=2\angle BAC

Both angles stand on chord BC.

Tangent length

PT=\sqrt{d^2-r^2}

P is d from the centre of a circle of radius r.

Tangent-secant

PT^2=PA\cdot PB

Tangent squared = outside part times whole secant.

Intersecting chords

PA\cdot PB=PC\cdot PD

Two chords crossing inside the circle.

Alternate segment

\angle(\text{tangent},\ \text{chord})=\angle\text{ in alternate segment}

The angle between a tangent and a chord equals the angle the chord makes on the far side.

Common tangents (transverse / direct)

L_T=\sqrt{d^2-(r_1+r_2)^2},\quad L_D=\sqrt{d^2-(r_1-r_2)^2}

d = distance between centres.

⚡ The triplet inside the circle. Radius, distance and half-chord are the sides of a right triangle. Radius 13 and distance 5 give half-chord 12.

⚡ Tangent-secant: multiply the pieces. Tangent squared equals outside part times the whole secant. Whole = outside + inside.

⚡ Count common tangents from d. Compare the centre distance with r_1+r_2 and r_1-r_2 and read off 4, 3, 2, 1 or 0.

⚡ Chord of the outer circle touching the inner one. Concentric circles: a chord of the big circle that just touches the small one is 2\sqrt{R^2-r^2}. The small radius is its distance from the centre.

Mensuration (2D)

Areas of triangles

Triangle area

K=\frac{1}{2}bh

b = any side, h = perpendicular height on it.

Equilateral triangle

K=\frac{\sqrt3}{4}a^2,\quad h=\frac{\sqrt3}{2}a

a = side; height is root-three over two of the side.

Heron's formula

K=\sqrt{s(s-a)(s-b)(s-c)},\quad s=\frac{a+b+c}{2}

s = half the perimeter.

Altitude to hypotenuse

h=\frac{ab}{c}=\frac{2K}{c}

a, b legs, c hypotenuse; from equating two areas.

Inradius / circumradius

r=\frac{K}{s},\quad R=\frac{abc}{4K}

K = area, s = half-perimeter.

Median split

\text{median}\Rightarrow\text{two equal areas}

Each median halves the area of a triangle.

⚡ Triplet beats Heron. Before Heron, check for a triplet. A right triangle needs only half the product of its legs.

⚡ Isosceles: half it first. The median to the base is the height. Halve the base, then use Pythagoras with an equal side.

⚡ Two radii from Heron. After Heron gives the area, both radii are one division away: r divides by s, R uses abc over 4K.

Areas of quadrilaterals

Rectangle

K=lb,\quad P=2(l+b),\quad d=\sqrt{l^2+b^2}

Three linked facts; any two fix the third.

Square

K=a^2,\quad d=a\sqrt2

Diagonal is root-two times the side.

Parallelogram

K=bh=ab\sin\theta

theta = angle between the two given sides.

Rhombus

K=\frac{1}{2}d_1d_2

Diagonals cross at right angles and halve each other.

Trapezium

K=\frac{1}{2}(a+b)h

a, b = the two parallel sides.

Any quadrilateral

K=\frac{1}{2}d(h_1+h_2)

d = a diagonal; h1, h2 = perpendiculars onto it.

⚡ Rectangle identities, not quadratics. With perimeter and diagonal, jump straight to (l+b)^2=l^2+b^2+2lb and read off the area.

⚡ Rhombus ratio to perimeter. Ratio plus area fixes both diagonals. Halve them, spot the triplet, read the side.

⚡ Diagonal splits any quadrilateral. No special shape? Draw one diagonal. The area is half the diagonal times the sum of the two heights.

Circles, sectors and rings

Circle

K=\pi r^2,\quad C=2\pi r

Take pi as 22/7 unless stated.

Arc and sector

\text{arc}=\frac{\theta}{360}2\pi r,\quad \text{sector}=\frac{\theta}{360}\pi r^2

Same fraction of circumference and area.

Ring

K=\pi(R^2-r^2)

R = outer radius, r = inner radius.

Wheel revolutions

N=\frac{D}{C}=\frac{\text{distance}}{2\pi r}

Count turns by dividing distance by one circumference.

Quadrant / semicircle

\text{quad}=\frac{\pi r^2}{4},\quad \text{semi}=\frac{\pi r^2}{2}

Quarter and half of the circle area.

⚡ Radius first, always. Circumference, diameter or area given: convert to the radius before anything else. Every formula lives on r.

⚡ Simplify the sector fraction. Reduce theta over 360 first: 72/360 is 1/5, 90/360 is 1/4. Then one multiplication finishes.

⚡ Wheels: count the turns. One turn covers one circumference. Divide total distance by it, in the same units.

Regular polygons and inscribed figures

Regular hexagon

K=\frac{3\sqrt3}{2}a^2

Six equilateral triangles of side a.

Polygon from apothem

K=\frac{1}{2}\times P\times a

P = perimeter, a = apothem (centre to a side).

Exterior angle

\text{ext}=\frac{360^\circ}{n}

Equal turns around the boundary.

Interior angle

\text{int}=180^\circ-\text{ext},\quad \text{sum}=(n-2)180^\circ

One angle plus the total for all n.

Diagonals

d=\frac{n(n-3)}{2}

n sides give this many diagonals.

⚡ Hexagon = 6 equilateral triangles. Six triangles of side a. Use the equilateral area times six; the root-three factor is already familiar.

⚡ Exterior angle finds n. Divide 360 by the exterior angle to get the side count; the interior angle is its partner to 180.

⚡ Diagonal equation factors. Set n(n-3)/2 equal to the given count, then factor the quadratic. Exam answers are whole numbers.

Percentage change, similarity and re-bent shapes

Similar figures

\frac{a_1}{a_2}=k\Rightarrow\frac{K_1}{K_2}=k^2

Lengths take k once; areas take k squared.

Successive percentage change

\text{net}=a+b+\frac{ab}{100}

Works for two changes in a row, like both dimensions.

Reverse percentage area change

1+\frac{x}{100}=\left(1+\frac{y}{100}\right)^2

Area factor is the side factor squared.

Map areas

\text{true area}=\text{map area}\times(\text{scale})^2

Square the linear scale before converting units.

⚡ a + b + ab/100 in one line. Both dimensions change by the same percent: plug once, no quadratics, no decimals.

⚡ Reverse the square root. Area up 21%? Factor 1.21=1.1^2, so the side rose 10%. Recognise perfect squares of decimals.

⚡ Wire into shapes: circle wins. Same perimeter, compare areas. Circle beats square beats any rectangle, so guess before computing.

Mensuration (3D)

Cube and cuboid

Cuboid

V=lbh,\quad \text{LSA}=2h(l+b),\quad \text{TSA}=2(lb+bh+hl)

l, b, h are the three dimensions.

Cube

V=a^3,\quad \text{LSA}=4a^2,\quad \text{TSA}=6a^2

a = edge.

Diagonals

d=\sqrt{l^2+b^2+h^2},\quad d_{\text{cube}}=a\sqrt3

Corner to opposite corner.

Capacity

1\text{ m}^3=1000\text{ L},\quad 1\text{ L}=1000\text{ cm}^3

Convert once, at the end.

Painted cube counts

8,\ 12(n-2),\ 6(n-2)^2,\ (n-2)^3

3, 2, 1, 0 painted faces for n by n by n.

⚡ Cubes table, not cube roots. Know the cubes to 12 by heart. Reverse questions become look-ups: 343 is 7 cubed.

⚡ Capacity: metres to litres. Compute the volume in cubic metres, then multiply by 1000 for litres. Never by 100.

⚡ Edge scaling in powers. Edge times k: surface times k squared, volume times k cubed. Doubling is 4 and 8.

Cylinder

V=\pi r^2h,\quad \text{CSA}=2\pi rh,\quad \text{TSA}=2\pi r(h+r)

r = radius, h = height.

Missing height

h=\frac{V}{\pi r^2}

Reverse of the volume formula.

Capacity

\text{litres}=\text{m}^3\times1000

Same conversion as tanks.

Dimension change

V\propto r^2h,\quad \text{CSA}\propto rh

Scale each symbol by its own factor.

⚡ 154 and friends. pi r squared for r = 7, 14, 21 is 154, 616, 1386. Reverse questions then divide by a friendly number.

⚡ Scale the symbols, not the shape. Radius doubled, height halved: volume factor is 4 times 1/2 = 2. Track r squared and h separately.

⚡ Melt: equate volumes. Melting conserves volume. Write both volume formulas equal, cancel, solve for the new length.

Cone

Slant height

\ell=\sqrt{r^2+h^2}

Radius, height, slant: a right triangle.

Cone volume

V=\frac{1}{3}\pi r^2h

One-third of the same-base cylinder.

Cone surfaces

\text{CSA}=\pi r\ell,\quad \text{TSA}=\pi r(\ell+r)

Skirt alone, or skirt plus base.

Same base ratios

V_{\text{cone}}=\frac{V_{\text{cyl}}}{3}

Equal base and height.

⚡ Hunt the triplet. 7-24-25, 3-4-5 and their multiples cover nearly every cone. Two lengths known, read the third.

⚡ One-third both ways. Cone to cylinder: divide by 3. Cone volume given: multiply by 3 before dividing by the base area.

⚡ Tent = curved surface. Canvas touches only the slanted side, so use pi r l. Add the base circle only for a solid cone.

Sphere and hemisphere

Sphere

V=\frac{4}{3}\pi r^3,\quad S=4\pi r^2

One radius drives both.

Hemisphere

V=\frac{2}{3}\pi r^3,\quad \text{curved}=2\pi r^2,\quad \text{total}=3\pi r^2

Total adds the flat circle.

Melting into n parts

r_{\text{small}}^3=\frac{R^3}{n}

Divide the cubed length, then cube-root.

Radius scaling

V\to k^3V,\quad S\to k^2S

k = radius scale factor.

⚡ Total hemisphere = 3 circles. Curved shell 2 pi r squared plus flat face pi r squared equals 3 pi r squared. Never 2.

⚡ Cube-root the split. One sphere into 8 equal spheres: each cubed radius is one-eighth, so each radius is half.

⚡ Ratio: cube it or square it. Radii 3:4 mean volumes 27:64 and surfaces 9:16. Cancel pi, apply the right power.

Prisms, pyramids and painted cubes

Prism

V=\text{base area}\times\text{length}

Base can be any polygon.

Pyramid

V=\frac{1}{3}\times\text{base area}\times h

h = perpendicular height.

Frustum

V=\frac{\pi h}{3}(R^2+r^2+Rr)

R, r = the two end radii.

Lateral surfaces

\text{prism}=P\times L,\quad \text{pyramid}=\frac12 P\times\ell

P = base perimeter; L = length; slant for pyramid.

⚡ One-third rule of thumb. Same base and height: pyramid = one third of the prism. Use it to sanity-check any answer.

⚡ Add volumes, then root. Melting several solids: add the volumes, then take the cube root for a cube's edge.

⚡ Frustum: three terms. R squared, r squared, Rr. Radii 5 and 3 give 25 + 9 + 15 = 49, and the numbers turn friendly.

Trigonometry

Ratios and standard values

Primary ratios

\sin\theta=\frac{o}{h},\quad \cos\theta=\frac{a}{h},\quad \tan\theta=\frac{o}{a}

o = opposite, a = adjacent, h = hypotenuse.

Reciprocals

\text{cosec}=\frac{1}{\sin},\quad \sec=\frac{1}{\cos},\quad \cot=\frac{1}{\tan}

Flip the fraction.

Standard values

\sin\theta=\frac{\sqrt{k}}{2},\ k=0,1,2,3,4

For 0, 30, 45, 60, 90 degrees; cos runs the row backwards.

One ratio to all

\sin\theta=\frac{3}{5}\Rightarrow\cos=\frac45,\ \tan=\frac34

Draw the 3-4-5 triangle and read every ratio off it.

⚡ The root-k-over-2 row. sin at 0, 30, 45, 60, 90 is root-0, root-1, root-2, root-3, root-4, all over 2. Cos is the same row reversed.

⚡ Triplet finishes the ratios. Given sin = 3/5, place 3 and 5 in the triangle; the third side 4 completes 3-4-5 and every ratio follows.

⚡ Angle from a value. Isolate the ratio, then match the table entry. 2 sin = root 3 means sin = root-3 over 2, the 60-degree slot.

Fundamental identities

Pythagorean identities

\sin^2+\cos^2=1,\quad 1+\tan^2=\sec^2,\quad 1+\cot^2=\cosec^2

Three engines from one identity.

Conjugate pairs

(\sec+\tan)(\sec-\tan)=1,\quad (\cosec+\cot)(\cosec-\cot)=1

Sum and difference are reciprocals.

Squares of sums

(a+b)^2+(a-b)^2=2(a^2+b^2)

Cross terms cancel in pairs.

Reciprocal products

\sin\cdot\cosec=\cos\cdot\sec=\tan\cdot\cot=1

The constant that kills cross terms.

⚡ Sum given, difference taken. sec + tan = 5 means sec - tan = 1/5, because their product is 1. Then add or subtract the pair.

⚡ Divide through by cosine. A sin-cos fraction becomes a tan fraction when every term is divided by cosine. Substitute tan and finish.

⚡ Square and subtract two. tan + cot = 5: square it, use tan times cot = 1, and the squared sum is 25 - 2.

Complementary angles

Complementary swaps

\sin(90^\circ-\theta)=\cos\theta,\ \tan(90^\circ-\theta)=\cot\theta,\ \sec(90^\circ-\theta)=\cosec\theta

Drop the co- or add it.

Right triangle angles

A+B=90^\circ\Rightarrow\sin A=\cos B

The two acute angles are partners.

Pairing to one

\tan\theta\cdot\tan(90^\circ-\theta)=1

Complementary tangents multiply to 1.

⚡ Sum the angles first. Before computing anything, add the two angles. Ninety degrees means the terms are twins.

⚡ Chains pair from the ends. tan 5 with tan 85, tan 25 with tan 65: each pair multiplies to 1, and the lone tan 45 is 1.

⚡ Match partners to find the angle. sin of something = cos of something: the two somethings must add to 90. That gives a linear equation.

Value-putting and given-ratio questions

Square of sin+cos

(\sin\theta+\cos\theta)^2=1+2\sin\theta\cos\theta

The bridge from a sum to a product.

Divide by cosine

\frac{a\sin+b\cos}{c\sin+d\cos}=\frac{a\tan+b}{c\tan+d}

After dividing every term by cosine.

Cot fraction to triangle

\cot\theta=\frac{21}{20}\Rightarrow\text{hyp}=29

Two sides given, Pythagoras gives the third.

Reciprocal pair sum

x+\frac{1}{x}\ \text{from}\ x\cdot\frac{1}{x}=1

Conjugates of cosec plus cot.

⚡ Symmetry cancels first. cos squared 30 and sin squared 60 are the same number; scan for such twins before computing.

⚡ Square the sum. sin + cos given: square it to reach 1 + 2 sin cos, then read off the product.

⚡ Condition straight into the fraction. 5 tan = 4: divide the fraction by cosine, substitute, one line of arithmetic.

Maximum and minimum values

Amplitude of a sin + b cos

\max=\sqrt{a^2+b^2},\quad \min=-\sqrt{a^2+b^2}

General angles; on 0 to 90 check the endpoints too.

sin times cos

\sin\theta\cos\theta=\frac{\sin2\theta}{2}\le\frac12

Peak at 45 degrees.

AM-GM floor

x+\frac{1}{x}\ge2

For positive x; equality when x = 1.

Weighted squares

a\sin^2\theta+b\cos^2\theta\in[\min(a,b),\max(a,b)]

Rewrite as one constant plus one square.

⚡ Square, add, root. The maximum of a sin + b cos is the hypotenuse of the a-b right triangle. 4 and 3 give 5.

⚡ AM-GM floor of two. Anything plus its own reciprocal bottoms at 2: tan + cot, sec + cosec, all the same.

⚡ Weighted squares: constant plus square. Rewrite 5 sin squared + 12 cos squared as 5 + 7 cos squared. The range is then obvious.

Heights and Distances

Angles of elevation and depression

Tangent rule

\tan\theta = \frac{\text{height above eye}}{\text{horizontal distance}}

The angle sits at the observer. Height is opposite, distance is next to the angle.

Height and distance

h = d\tan\theta,\qquad d = h\cot\theta

Slanting length (thread, wire, ladder)

h = L\sin\theta,\qquad d = L\cos\theta

L is the slanting line of sight, the hypotenuse of the triangle.

Depression to elevation

\text{depression from top} = \text{elevation from bottom}

The two horizontal lines are parallel, so the angles are equal.

⚡ Swap depression for elevation first. Never work with a downward angle directly. Redraw it at the bottom of the tower and solve an ordinary elevation question.

⚡ Shadows without trigonometry. Same sun, same time, similar triangles. Set up the stick ratio and multiply; no tangent needed.

⚡ Forty-five degrees means equal legs. Whenever the angle is 45^\circ, the height above the eye equals the horizontal distance. Write the equal pair without any tangent.

Standard angles: 30°, 45°, 60°

Tangent values

\tan 30^\circ = \frac{1}{\sqrt{3}},\quad \tan 45^\circ = 1,\quad \tan 60^\circ = \sqrt{3}

Distance from height

d = h\cot\theta

cot 30 = sqrt3, cot 45 = 1, cot 60 = 1/sqrt3.

Ladder on a wall

h = L\sin\theta,\quad d = L\cos\theta

Theta is the ladder's angle with the ground.

Fifteen and seventy-five

\tan 15^\circ = 2-\sqrt{3},\qquad \tan 75^\circ = 2+\sqrt{3}

⚡ Read distance straight off the cot column. With a standard angle, distance = height \times the cot value. No division, no fraction juggling.

⚡ Fifteen and seventy-five multiply to one. Angles that add to 90^\circ have tangents that multiply to 1. Use this to check answers or flip a division into a multiplication.

Two observation points (two angles)

Same side (walk towards)

h = \frac{d}{\cot\alpha - \cot\beta}

d is the distance walked; beta is the nearer, bigger angle.

Opposite sides

h = \frac{d}{\cot\alpha + \cot\beta}

d is the full distance between the two observers.

Gap between two objects from a height

\text{gap} = h(\cot\alpha - \cot\beta)

Tower seen from foot and roof of a building

d = \frac{b}{\tan\beta - \tan\alpha},\quad H = d\tan\beta

b = building height; beta from the foot, alpha from the roof.

⚡ Thirty-sixty fast numbers. For the 30 and 60 pair: same side h = d\times\dfrac{\sqrt{3}}{2}; opposite sides h = d\times\dfrac{\sqrt{3}}{4}. Both come from \cot 30^\circ - \cot 60^\circ = \dfrac{2}{\sqrt{3}}.

⚡ Write the cot pair before the numbers. Always simplify \cot\alpha - \cot\beta (or the sum) first. Substituting numbers into unsimplified surds is where errors creep in.

Moving observers: speed and time

Moving observer chain

v\,t = h(\cot\alpha - \cot\beta)

alpha = first (farther) angle, beta = second (nearer) angle.

Time to reach the foot

t = \frac{h\cot\beta}{v}

Use the angle at the car's current position.

Vertical rise

\text{rise} = d(\tan\beta - \tan\alpha)

d = fixed horizontal distance; angles of depression shrink as the balloon rises.

Speed conversion

1\ \text{km/h} = \frac{5}{18}\ \text{m/s}

⚡ One chain, any unknown. v t = h(\cot\alpha - \cot\beta) contains every moving-observer question. Cover the unknown and solve.

⚡ Check the units before the triangle. Convert km/h to m/s with 5/18 before anything else. A speed in the wrong unit spoils an otherwise perfect triangle.

Compound figures: buildings, pedestals, broken objects

Stacked object (statue on pedestal)

s = d(\tan\beta - \tan\alpha)

d comes from the lower triangle: d = pedestal height x cot alpha.

Tower on a building

t = d(\tan\beta - \tan\alpha),\quad d = b\cot\alpha

From a roof: depression and elevation

d = b\cot\alpha,\qquad H = b + d\tan\beta

Broken tree

\text{stump} = x\tan\theta,\quad \text{broken} = \frac{x}{\cos\theta}

x = distance from the foot to where the top touches.

⚡ Subtract the tans, multiply once. For any stacked object, extra height = d(\tan\beta - \tan\alpha). Compute the tangent difference first, then one multiplication.

⚡ Broken tree: tan plus sec. Total height = x(\tan\theta + \sec\theta) where x is the ground distance to the touching point. At 30^\circ that is x\left(\dfrac{1}{\sqrt{3}} + \dfrac{2}{\sqrt{3}}\right) = x\sqrt{3}.

Statistics

Mean, weighted mean and combined mean

Average

\bar{x} = \frac{\text{sum of values}}{\text{count}}

Total

\text{total} = \bar{x} \times n

Combined average

\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}

Missing value

x = n\bar{x} - \sum(\text{known values})

New member

w = (n+1)\bar{x}_{new} - n\bar{x}_{old}

Use the same pattern for a leaving member with n-1.

⚡ Deviations from a round base. Pick a round number near the values. Add the small differences and divide by the count. The base can be anything.

⚡ Watch where the average is pulled. The combined average always lies between the two group averages, closer to the bigger group. Use this to reject impossible options in seconds.

Median and mode of raw data

Median (odd n)

\text{value at } \frac{n+1}{2}\text{th place}

Position in the sorted list.

Median (even n)

\frac{\text{(n/2)th} + \text{(n/2+1)th}}{2}

Empirical relation

\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Rough rule for mildly skewed data; rearrange for any missing one.

Median from mode and mean

\text{Median} = \frac{\text{Mode} + 2\,\text{Mean}}{3}

Transform

y = kx + c \Rightarrow \text{Med}_y = k\,\text{Med}_x + c

⚡ Count positions, do not hunt. For odd n, the median sits at position (n+1)/2 of the sorted list. Count to that position instead of scanning for the middle by eye.

⚡ Median without sorting everything. You only need the middle order statistics. In a long list, quickly bucket values as low or high; full sorting wastes time.

Median and mode of grouped data

Grouped mean

\bar{x} = \frac{\sum f x}{\sum f}

x is the midpoint of each class.

Grouped median

\text{Med} = L + \frac{\frac{n}{2} - c}{f} \times h

L: lower limit of median class; c: cf before it; f: its frequency; h: width.

Grouped mode

\text{Mode} = L + \frac{f_m - f_1}{2f_m - f_1 - f_2} \times h

f_m: modal class frequency; f_1, f_2: neighbouring frequencies.

⚡ Build the cf column once, use it thrice. Median, quartiles and 'how many below a value' questions all read the same cumulative column. Write it before touching any formula.

⚡ Cross-check the median class fast. Half of n must fall inside the median class. Glance: the cf before it is below n/2, the cf through it is at or above n/2.

Range, variance and standard deviation

Range

R = \text{max} - \text{min}

Variance

\sigma^2 = \frac{\sum (x-\bar{x})^2}{n}

Average of squared distances from the mean.

Standard deviation

\sigma = \sqrt{\sigma^2}

Shift and scale

\text{SD}(kx + c) = |k|\,\text{SD}(x)

Adding c changes nothing; multiplying scales the SD.

First n naturals

\sigma^2 = \frac{n^2-1}{12}

Two values

\text{SD} = \frac{|p-q|}{2}

Coefficient of variation

CV = \frac{\sigma}{\bar{x}} \times 100\%

Sum of squares

\sum x^2 = n(\bar{x}^2 + \sigma^2)

⚡ Skip the squares for symmetric lists. Values spaced evenly around their mean cancel in pairs: 2,4,6,8,10 gives squared deviations 16,4,0,4,16. Write only the distinct squares.

⚡ AP spread without listing. For k, 2k, 3k, ..., nk use variance k²(n²−1)/12. No squaring of long lists.

Averages of special series

Sum of first n naturals

\sum k = \frac{n(n+1)}{2}

Sum of squares

\sum k^2 = \frac{n(n+1)(2n+1)}{6}

Sum of cubes

\sum k^3 = \left[\frac{n(n+1)}{2}\right]^2

The square of the sum of the first n naturals.

Mean of first n naturals

\frac{n+1}{2}

Mean of first n odds

Mean of first n evens

n+1

Multiples of k

\text{sum} = \frac{kn(n+1)}{2},\; \text{mean} = \frac{k(n+1)}{2}

⚡ The middle is the mean. For any equally spaced list the mean is the middle term. Use it forwards (find the mean) and backwards (rebuild the list).

⚡ Odds add to squares. 1 + 3 + ... up to n odd numbers is exactly n². Use it to test counts fast.

Data Interpretation

Reading tables: totals, differences, ratios

Row total

\text{total} = \sum \text{row cells}

Row average

\bar{x} = \frac{\text{row total}}{\text{number of columns}}

Cell ratio

\text{ratio} = a : b \text{ (reduced)}

Rate from counts

\text{rate} = \frac{\text{passed}}{\text{appeared}} \times 100\%

⚡ Sweep the row with tick marks. For count-type questions (how many years above a value), tick the qualifying cells while reading once. No rewriting, no calculator.

⚡ Column-first for year questions. Any question about one year (best year, total of a year) lives in a single column. Add that column alone; ignore the rest of the table.

Percentage change and comparisons

Percentage change

\frac{\text{new}-\text{old}}{\text{old}} \times 100\%

Base = the old / original value.

\frac{\text{part}}{\text{whole}} \times 100\%

Points vs percent

\text{points} = a - b; \quad \%\text{age} = \frac{a-b}{b} \times 100

⚡ The 1% probe. Find 1% of the base first; then any percentage is that times n. 1% of 250 is 2.5, so 70 units is 28%.

⚡ Anchor on the double. A value that doubles is +100%; halves is −50%. Spot these before computing: they frame every other answer.

Pie charts: degrees, shares and totals

Slice value

\text{value} = \frac{\theta}{360} \times \text{total}

theta is the slice angle in degrees.

Angle to percent

\% = \frac{\theta}{3.6}

Percent to angle

\theta = \% \times 3.6

Part to angle

\theta = \frac{\text{part}}{\text{total}} \times 360

Per-degree value

1° = \frac{\text{total}}{360}

⚡ Divide by 36 for percent. Angle over 3.6 gives percent; angle over 36 gives percent over 10. 108° -> 30%, 54° -> 15%, 90° -> 25%.

⚡ Per-degree shortcut. Given one slice's value, find the value of 1 degree once, then price every other slice (and the whole pie) with one multiplication each.

Averages from data

Simple average

\bar{x} = \frac{\sum x}{n}

Weighted average

\bar{x} = \frac{\sum n_i x_i}{\sum n_i}

n_i is the count in each band or row.

Entry to hit a target average

x = (n+1)\bar{x}_{new} - n\bar{x}_{old}

⚡ Deviations from a round base. Table values near a round number: average = base + (sum of differences ÷ count). 480, 520, 460, 540, 500, 500 around 500: deviations sum to 0, average exactly 500.

⚡ Balance around the new average. If an added value lowers the average, each old entry gains (old avg − new avg); the new entry supplies all of it.

Growth rates and successive changes

Growth multiplier

\text{new} = \text{old}\left(1+\frac{r}{100}\right)

Successive growth

r_{total} = \left(1+\frac{a}{100}\right)\left(1+\frac{b}{100}\right)-1

Reverse to original

\text{old} = \frac{\text{new}}{1+\frac{r}{100}}

Total multiplier

\frac{\text{end}}{\text{start}} = 1 + \frac{r_{total}}{100}

⚡ Multipliers beat repeated percenting. Chain growth as multipliers: +10% then +10% is 1.1 x 1.1 = 1.21. One multiplication, no compounding errors.

⚡ End over start. The total growth multiplier over any span is just the last value divided by the first. 144/80 = 1.8 → +80%.

General Intelligence & Reasoning

Analogy

The idea of analogy

⚡ Say the link first, read the options second. Turn the full pair into a sentence before you look at the options. Then test each option with that same sentence. Options that are only from the same field fail the sentence at once.

Letter / letter-cluster analogy

Shift rule

\text{new position} = \text{old position} + k

k is the shift. Above 26, subtract 26; below 1, add 26.

Opposite letter

\text{opposite} = 27 - p

p is the position. A and Z, or M and N, always add to 27.

Rising shift

\text{shift of slot } i = i

Slot 1 moves +1, slot 2 moves +2 and so on, e.g. ABCD → BDFH.

⚡ Subtract once, then copy the pattern. Write one row of subtractions for the model pair. The pattern you get is the rule. Apply the same pattern to the third group and read the letters with EJOTY.

⚡ Spot opposite letters by the 27 total. If matching letters of the model pair add up to 27, the rule is opposite letters. Then write the opposite of each letter. No counting is needed.

⚡ Read the second group backwards. Before any counting, read the second group from right to left. If it gives the first group, the rule is plain reversal.

Word analogy

⚡ Two ticks on every option. Give each option a tick for "same kind of link" and a tick for "same exact link, same direction". The option with two ticks is the answer.

⚡ Tag animal pairs before reading options. For an animal pair, first tag it: adult : young, male : female, or animal : home, sound or group. Options often mix a young one with a female.

Number analogy

Multiply-and-add rule

b = k a + r

a is the first number, b the second. Find k and r from the model pair, then check.

Square family

b = a^2 \pm r,\ a(a+1),\ (a+1)^2

Try these when b is close to the square of a.

Digit rules

b = \text{digit sum},\ \text{digit product},\ (\text{digit sum})^2

Not allowed when the whole-number note is given.

⚡ The size test picks the family. Compare b with a before anything else. A few times a: multiply and add. Near a squared: square family. Very large: cubes. Smaller: roots or digit rules.

⚡ Check the rule on the model pair. Whatever rule you guess, make sure it gives b from a exactly. The check takes two seconds and prevents most wrong answers.

Number sets (triads) analogy

Chain rule

b = ka + r,\quad c = kb + r

a, b, c are the three numbers of a set. One step is used twice.

Third from first two

c = ab,\ a^2 + b^2,\ k(a+b),\ k(a-b)

Test on every model set.

Power set

(a,\ a^2 \pm r,\ a^3 \pm r)

Near-squares and near-cubes of the first number.

⚡ Compute only the last number. Once you have the rule, work out what the last number of each option should be. Traps nearly always change only that number, so one pass finds the answer.

Mixed clusters and same-relation selection

Word to number

\text{code} = k \times (\text{sum of positions}) + r

Often k = 1 and r = 0: the plain sum of letter positions.

Reverse position

\text{reverse value} = 27 - p

A = 26, B = 25 ... Z = 1.

Reverse sum shortcut

\text{reverse sum} = 27n - \text{plain sum}

n is the number of letters in the word.

⚡ Two columns: letters and numbers. Make one column for the letters and one for the numbers. Solve each column on its own. An error then stays inside one column.

⚡ One sentence for all four pairs. In same-relation questions, make the link sentence once from the model pair. Test every option against that same sentence, in the same order.

Classification (Odd One Out)

The idea of odd one out

⚡ Write the majority rule, not the exception. Instead of hunting the odd item, write the rule the other three follow. Confirm each of the three obeys it and the last one breaks it. This stops you marking an item that is merely unusual.

Word odd one out

⚡ Category first, then a binary property. First find the broad category shared by three words. If the fourth also fits that category, switch to a binary property: salt or fresh, input or output, element or alloy, male or female. One word will stand alone on the …

Number odd one out

Divisibility by 11

|S_{odd} - S_{even}| \equiv 0 \pmod{11}

S = sums of alternate digits from the left; e.g. 2728: (2+2) − (7+8) = −11, divisible.

Perfect square endings

n^2 \in \{0,1,4,5,6,9\}

A square never ends in 2, 3, 7 or 8 — instant elimination.

Digit sum rule for 9

9 \mid n \iff S(n) \equiv 0 \pmod 9

S(n) is the sum of the digits of n.

⚡ Last-digit scan for squares. Squares end only in 0, 1, 4, 5, 6 or 9. An option ending in 2, 3, 7 or 8 cannot be a square. For the rest, place the number between two known squares.

⚡ Prime check by small divisors. To test a number under 200 for prime, divide only by 2, 3, 5, 7, 11 and 13. Memorise the fake-prime list: 51, 57, 87, 91, 119, 133, 143, 161.

Letter-cluster odd one out

Step of a cluster

s_i = Q(L_{i+1}) - Q(L_i) \pmod{26}

Q is the alphabet position. Compute steps for every option; three must match as tuples.

Opposite-pair sum

Q(X) + Q(Y) = 27

Letter pairs whose positions add to 27: AZ, BY, ..., MN.

⚡ Step tuples in one line. Under each cluster write its steps: ACE → (+2, +2), BDF → (+2, +2), GIK → (+2, +2), MOP → (+2, +1). The mismatched tuple is the answer. No vowel counting is needed.

⚡ Vowel count as tiebreaker. If the step tuples agree, or the clusters look irregular, count the vowels (A, E, I, O, U). Three clusters with no vowel and one with a vowel (or the reverse) is a complete exam pattern.

Pair odd one out

⚡ Relation sentence, then four ticks. Form the relation sentence from the three pairs that clearly agree, then tick or cross all four options. In number pairs, compute the rule for each pair; three will match.

Number pairs and sets odd one out

Pair rule

b = f(a),\; f(a) \in \{a^2,\ a^3,\ a^2 \pm k,\ ka \pm r\}

Find f from one pair, then check the other three pairs.

Chain triad

(a,\ f(a),\ f(f(a)))

The same operation applied twice, e.g. ×3 gives (3, 9, 27).

Outer-to-middle triad

(a,\ g(a, c),\ c),\; g = ac \text{ or } a^2 + c^2

The middle number is made from the two outer numbers.

First-two-to-third triad

(a,\ b,\ h(a, b)),\; h = k(a+b) \text{ or } ab \pm r

The third number is made from the first two.

⚡ Size tells the operation. Compare sizes first. If the second number is near the square of the first, think square. If the middle number is large and the outer ones small, think product of the outer numbers.

⚡ Test the rule on two options before trusting it. A rule found from one option can be a coincidence. Confirm it on a second option, then check the rest. The option that fails is the answer.

Series (Number & Letter)

How every series is cracked

Gap between terms

d_n = a_{n+1} - a_n

Write these gaps under the series. If they are not clear, take gaps of the gaps.

Same number added

a_n = a_1 + (n - 1)d

d is the equal gap. Use it when the first row of gaps is constant.

⚡ Two rows of gaps in ten seconds. Write the gaps under the series. If they are not equal, write the gaps of the gaps. Stop when a row is clear. Then work back up.

⚡ Say the rule aloud in words. Before you calculate, say the rule in one plain sentence, like 'add 3, then add 6, then add 9'. If you cannot say it in a sentence, the rule is wrong.

Number series

Same gap (AP)

a_n = a_1 + (n - 1)d

d is the equal gap from the ladder.

Multiply then adjust

a_{n+1} = k \cdot a_n \pm r

Try k = 2 and 3 first, with r = 1 or 2.

Squares family

a_n = n^2 \pm k \ \text{ or } \ n^2 + n

If the second gaps are all 2, the series is built on n squared.

Same ratio (GP)

a_n = a_1 \cdot r^{\,n-1}

r is the fixed ratio. Test ×2, ×3 and ×1.5.

⚡ Ratio test before anything fancy. Divide each term by the one before it. If the result is close to 2 or 3, try ×k ± r. Find r from what is left over.

⚡ Split a jumpy series. If the terms go up and down, write the odd places on one line and the even places on another. Solve each line on its own. Then find which line the asked place belongs to.

⚡ Cubes hiding in the gaps. If the gaps grow very fast, check for cubes (1, 8, 27, 64, 125) or powers of 2 (1, 2, 4, 8, 16).

Letter series

Opposite letter

\text{place of opposite} = 27 - \text{place}

A and Z, B and Y, and so on. Each pair adds up to 27.

Wrap around

\text{place} > 26 \Rightarrow \text{place} - 26,\ \ \text{place} < 1 \Rightarrow \text{place} + 26

Use this whenever the rule takes you past Z or before A.

⚡ A step row under the letters. Write the place under every letter, then the gap between places. Work on the numbers only. Convert to a letter once, at the end.

⚡ Wrap around with the number 26. Never count letters on your fingers past Z. Add or subtract 26 on the number, then convert.

Alphanumeric series

⚡ Two columns, two answers, one join. Write letters in one column and numbers in the other. Solve each column on its own. Join only at the end, in the format of the question.

⚡ Test the letter-place link first. Check whether the number equals the place of the letter, or its square. If yes, you only need the next letter.

Wrong number in the series

⚡ Ladder with one broken rung. Write the gaps. Find two bad gaps side by side. The term between them is the wrong term.

Repeating letter series — fill the blanks

⚡ Write the blocks one below another. Cut the line into equal rows. Each column must hold one letter. Blanks take the letter of their column.

⚡ A double letter marks a mirror turn. If the line has a doubled letter like rr or nn, the block probably reads forward and then backward. The block is twice the length of the first half.

Coding - Decoding

The coding-decoding family

⚡ Find the difference list first. Write the sample word above its code. Subtract the positions slot by slot. The list of differences is the rule. Apply the same list to the new word.

Letter shift coding

Shift list

\text{code}_i = \text{letter}_i + d_i \pmod{26}

d is the shift for place i, found from the sample. Above 26, subtract 26.

Decoding

\text{letter}_i = \text{code}_i - d_i \pmod{26}

To decode, subtract the same shifts. At 0 or below, add 26.

⚡ Hop from an anchor, do not count. To shift a letter, jump from the nearest EJOTY anchor. W is Y − 2, so W + 4 = Y + 2 = A. Two small hops are faster than counting W, X, Y, Z, A.

⚡ Decoding: run the shifts backwards. When the code is given and the word is asked, subtract the shift instead of adding it.

Reversal and reverse-plus-shift coding

⚡ Read the code backwards first. Read the code from right to left. If you see the word, just reverse new words. If every letter is off by the same k, reverse and shift by k.

Number coding

Position sum

\text{code} = \sum p_i

p = position of a letter, A = 1 … Z = 26.

Sum with an extra step

\text{code} = k \cdot \sum p_i + r

Find k (multiply) or r (add) from the sample. Check with a second sample if one is given.

Weighted sum

\text{code} = \sum i \cdot p_i

i = place of the letter in the word (1st, 2nd, 3rd …).

Reverse positions

p^{\text{rev}} = 27 - p

Z counts as 1 and A as 26.

⚡ Plain sum first. Work out the position sum of the sample. If it equals the code, you are done. If not, compare: code − sum, then code ÷ sum. Only then try weighted or side-by-side codes.

⚡ Anchors make sums quick. Letters near E, J, O, T and Y are small hops from 5, 10, 15, 20 and 25. Group them to add fast.

Symbol and digit coding

⚡ Common letters match common symbols. When two coded words share letters, circle the shared letters and the shared symbols. They belong together. The letters left over take the symbols left over.

⚡ Conditions before the table. Read the conditions and test them on the word before you touch the table. The table gives the symbols; the condition only changes their order.

Substitution and sentence coding

⚡ Line up the sentences. Write each sentence above its code. Tick the words that repeat and the code words that repeat. The word you need is either matched directly or is the one item left over.

Language (sentence) coding with common words

⚡ Intersect, then subtract. For a word, take the code words shared by all sentences that contain it. Remove codes already fixed for other words. One left is the answer; two left means cannot be determined.

Mathematical Operations

Operator puzzles and BODMAS

BODMAS order

B \to O \to D/M \to A/S

D and M share a rank and go left to right; so do A and S.

Missing number

? = \dfrac{\text{RHS} - \text{loose terms}}{\text{factor}}

Undo + and − first, then the multiply or divide around the ?.

⚡ Circle the × and ÷ first. Read the line once and circle every × and ÷. Work those pieces first. Then read the line again for + and −. Your eye lands on the high-rank work before any arithmetic.

Sign substitution

⚡ Compute the trap value too. After the correct value, evaluate the original expression once. If that number sits among the options, the examiner planted it. Seeing it confirms your substitution is the different one.

Interchanging signs and numbers

⚡ One rewrite for every swap. Write the swaps on the left, like + ↔ × and 2 ↔ 8. Then rewrite the expression once with all swaps applied. Evaluate. Piececemeal swapping invites double swaps.

⚡ Fixed-order sweep for fix-the-equation. Sweep number pairs left to right: (n1,n2), (n1,n3), (n1,n4), (n2,n3), (n2,n4), (n3,n4). One written value per row. The row matching the right side is the answer.

Balancing and operator-filling

⚡ Value column for pick-the-equation. List the four options in a column. Evaluate each left side in one written line. Exactly one row matches its right side. The column doubles as your final check.

Hidden-rule and defined operations

Sum of squares

a \# b = a^2 + b^2

A very common hidden rule: 5 # 3 = 34.

Difference of squares

a @ b = a^2 - b^2

Also equals (a+b)(a−b): 8 @ 2 = 60.

Product plus sum

a \diamond b = ab + a + b

Multiply, then add both numbers: 4 ◇ 3 = 19.

⚡ Rule ladder for hidden operations. Test in this order: a + b, a × b, a² + b², a² − b², (a+b)², (a−b)², ab + a + b. Stop at the first rule that fits ALL examples.

Missing Number

How missing-number puzzles work

Rule check

f(a_1, b_1) = c_1 \text{ and } f(a_2, b_2) = c_2 \Rightarrow \text{use } f

One row fitting proves nothing. Two rows fitting is strong proof.

Common families

a + b,\ a - b,\ ab,\ ab \pm k,\ (a + b)k,\ a^2 \pm b

Sums and products cover most puzzles. Try them first.

⚡ Test the spare row. Write your guessed rule as a formula. Test it on the second complete row. Only then use it on the row with the ?. This takes ten seconds and stops the most common wrong answer.

⚡ Build the biggest number first. Pick the biggest number in the row. Try to make it from the smaller ones. This finds the rule faster than reading left to right.

Grids with row (or column) rules

Sum times a number

c = (a + b) \times k

Rows (4, 7, 22) and (6, 3, 18) give k = 2.

Square minus the second

c = a^2 - b

Rows (8, 15, 49) and (7, 10, 39) fit.

Sum of two squares

c = a^2 + b^2

Rows (3, 4, 25) and (6, 8, 100) fit.

Average of two

c = \dfrac{a + b}{2}

Rows (12, 8, 10) and (20, 6, 13) fit.

⚡ Build the biggest cell first. Write the biggest cell of a row. Rebuild it from the other two. This shows the rule faster than reading left to right.

⚡ Copy the grid as three rows. Write the grid as three lines in your rough space, with the ? in place. Reading numbers off the figure while you calculate leads to mistakes.

Digit-sum and digit-reversal rules

Digit sum

S(47) = 4 + 7 = 11

Add the digits of the number.

Reversal

R(43) = 34

Write the digits in the opposite order.

Digit product

P(47) = 4 \times 7 = 28

Multiply the digits of the number.

⚡ Switch to digits when standard rules fail. Give ordinary rules about 30 seconds. If both complete rows refuse them, test these in order: digit sums, reverse of the sum, digit sums plus a number.

⚡ Add first, then flip. A reversal answer must read backwards cleanly. Add the two numbers first, then reverse the sum. Confirm on two rows.

Blood Relations

How to solve blood-relation questions

Mirror pairs

\text{father} \leftrightarrow \text{son},\ \text{uncle} \leftrightarrow \text{nephew}

Every relation has a mirror. If A is B's uncle, B is A's nephew or niece.

Marriage words

\text{child's spouse} \to \text{son/daughter-in-law}

Spouse's parent = father/mother-in-law. Sibling's spouse or spouse's sibling = brother/sister-in-law.

Cousin

\text{cousin} = \text{child of a parent's sibling}

An uncle's or aunt's child is always a cousin, never a nephew.

⚡ Write each sentence as an arrow. In the margin, turn every sentence into an arrow: 'Suresh's daughter is Pooja' becomes Suresh to Pooja (F). When every sentence is an arrow, the tree draws itself.

⚡ Flip the word, not the tree. One tree answers both directions. Keep the drawing, swap to the mirror word: father goes to son, uncle to nephew, grandfather to grandson.

Multi-statement tree puzzles

⚡ Start from the root. The root is the person nobody calls a child. Build downward from there. Every statement then clicks into an empty slot.

⚡ Count arrows, then gender. The arrow count picks the family of the word (father, grandfather, great-grandfather). Only then does gender pick inside it. Never guess the word first and draw later.

Pointing and introducing

⚡ One layer per line. Underline the innermost 'my' and work outward. Each line of rough work resolves one 'of'. Then place the speaker and the target in a two-person sketch.

⚡ Scan for the fold-back. Before peeling anything, scan the phrase for 'my mother's only child', 'my father's only son', 'my mother's only daughter'. If one is there, test the speaker's gender against it first.

Coded (symbol) relations

⚡ Write the sentences under the string. For each symbol, write its one-line sentence under the string, then draw. Six symbols become three short sentences, and the drawing finishes the job.

⚡ Audit genders before answering. Before marking, list every letter with the gender its symbols fixed. If the asked word needs a gender that stayed blank, the answer is the neutral word or 'Cannot be determined'.

Direction & Distance

Direction conventions and displacement

Shortest distance

d = \sqrt{x^2 + y^2}

x = net East-West move, y = net North-South move, after cancelling.

Number triples

3\text{-}4\text{-}5,\ 5\text{-}12\text{-}13,\ 8\text{-}15\text{-}17,\ 7\text{-}24\text{-}25

Doubles and triples of these also work: 6-8-10, 9-12-15.

Diagonal leg

k\sqrt{2}\ \text{along NE} = k\ \text{North} + k\ \text{East}

The same rule holds for NW, SE and SW.

⚡ Two-line tally. Keep one running total for East-West and one for North-South. Write West and South as minus. The two totals are the whole answer, with no drawing needed.

⚡ Spot the triple. If the two net moves are 3 and 4, or 5 and 12, or 8 and 15, or 7 and 24 (or their multiples), the answer is the third number. No square root is needed.

Direction faced and net direction

Clockwise cycle

N \to E \to S \to W \to N

Every right turn moves one step along this circle.

Anticlockwise cycle

N \to W \to S \to E \to N

Every left turn moves one step along this circle.

Reading the end point

(x>0,\ y>0) \Rightarrow \text{North-East}

x is the net East move and y is the net North move. Their signs give the compass word.

⚡ Write the direction after every turn. After each turn, write one letter: N, E, S or W. Overwrite the old letter. Three written letters are safer than one long chain in your head.

⚡ Reverse the pair, flip the word. If B is North-East of A, then A is South-West of B. The mixed words flip together.

Shadow questions

⚡ Change the shadow into a facing first. Before any other step, rewrite the sentence as a facing sentence. Morning and shadow behind him means he faces East. Evening and shadow in front means he faces East.

Position of one point from another

Distance between two points

d = \sqrt{(\Delta E)^2 + (\Delta N)^2}

Delta E is the East gap and Delta N is the North gap between the two people.

Distance from speed and time

\text{distance} = \text{speed} \times \text{time}

Use it first when two people walk at different speeds.

⚡ Start from the person named after 'from'. For "direction of C from B", draw B first at the middle. The chain then hangs off B. The answer is simply where C lands.

Order & Ranking

How ranking questions work

Rank from the other end

p' = n + 1 - p

n = total people, p = rank from one end.

Total from two ranks

n = a + b - 1

The same person is a-th from one end and b-th from the other end.

Between count (same end)

\text{between} = |a - b| - 1

Both ranks are from the same end.

Ahead and behind

\text{ahead} = r - 1,\quad \text{behind} = n - r

r = the person's rank from the front.

⚡ Check that the two ranks add to total plus 1. For one person, the two ranks from opposite ends always add up to the total plus 1. If they do not, you have misread a number.

⚡ Draw the end zones. Draw a short row of boxes. 'a-th from the left' means a - 1 people stand before the person. Counting people at each end is often faster than any formula.

Ranks and positions

Rank from the other end

p' = n + 1 - p

n = class size or row size.

Total from two ranks

n = a + b - 1

The same person, ranked from both ends.

People joining or leaving ahead

r_{\text{new}} = r_{\text{old}} \pm k

Plus for k joiners ahead. Minus for k leavers who were ahead.

⚡ Check that the sum of the two ranks is total plus 1. For one person, the two ranks add up to the total plus 1. Test it before you mark the answer.

People between two positions

Between, same end

\text{between} = |a - b| - 1

Both ranks are counted from the same end.

Between, opposite ends

\text{between} = n - a - b

n = total people, a and b = ranks from opposite ends.

Middle person

m = \dfrac{a + b}{2}

a and b from the same end. The result must be a whole number.

Rank from the gap

b = a + k + 1

k people are between a and b, counted from the same end.

⚡ Convert first, then count. Never subtract ranks from different ends. Convert one rank with n + 1 - p. Then use the same-end formula.

Ordering and comparison chains

⚡ Write two symbols for each clue. Turn each sentence into two letters with a sign, such as S < V. Keep the sign pointing the same way all through. The reversed words then become easy to see.

⚡ Start from the person named most often. The person who appears in the most clues is the best starting point. Build the chain outwards from that person and it forms in one pass.

Seating Arrangement

How seating puzzles are set

Circular neighbour rule

L(i) = (i+1) \bmod n,\quad R(i) = (i-1) \bmod n

Seats numbered clockwise, everyone facing the centre. Both swap if facing outward.

Opposite seat

i \leftrightarrow (i + n/2) \bmod n

Only when n is even. An odd table has no opposite seat.

People between two seats

\text{between} = |a - b| - 1

a and b are seat numbers in the same row.

Viewer versus person

\text{person faces south} \Rightarrow \text{their left} = \text{your right}

Applies to the person's own left and right, never to the row ends.

⚡ Pin the absolutes first. Ends, the middle and opposite seats fix a person outright. Fill them before you touch any chain. The remaining people usually fall into one or two slots.

⚡ Test one clue both ways. If you are unsure of the facing, place one clue for north and for south. The two seats differ, and this shows you which one you must use.

Single-row arrangements

⚡ Chain links, do not guess gaps. 'Second to the right of A' pins B exactly two seats from A. Mark it with an arrow as soon as you read it. Most puzzles have four such arrows, and the picture then completes itself.

⚡ Turn the row to face up. If the row faces south, redraw it from east to west. Now every person 'faces up' and left is your left. Use this drawing for the whole puzzle.

Round-table arrangements

⚡ Walk the circle in one direction. Take the longest chain of 'immediate left' clues and lay it out clockwise. Close the loop with the last clue.

⚡ Convert left to right around a circle. In a circle of n, the k-th to the left is the same person as the (n - k)-th to the right. Use the smaller number to save steps.

Exam strategy for puzzles

⚡ Two-seating bail-out. When exactly two seatings survive, check what the question asks. If both give the same answer, mark it and move on.

⚡ Verify with one closed loop. In a circle, walk the left-neighbour links around your final drawing. If the string does not close, one arrow was drawn backwards.

Syllogism

What a syllogism is and how to test it

Transitivity of All

A \subset B \wedge B \subset C \Rightarrow A \subset C

'All A are B + All B are C → All A are C' is the only freely chained rule.

Conversion of All

\text{All } A \subset B \Rightarrow \text{Some } B \subset A

Valid (classes are assumed non-empty): 'Some B are A' follows.

Conversion of No / Some

\text{No } A \cap B \Leftrightarrow \text{No } B \cap A;\quad \text{Some } A \cap B \Leftrightarrow \text{Some } B \cap A

Both convert symmetrically.

⚡ Find the shared term. The conclusion must join the two outer terms. Find the term that appears in both statements, and ask what travels through it.

⚡ One counter-picture kills it. You do not need to prove a conclusion. You need only one picture that fits the statements and breaks it.

Possibility conclusions

⚡ Defence lawyer thinking. For a possibility, flip your mindset. Build one picture where it holds. Done.

Either-or (complementary pairs)

⚡ Spot the opposite pair. Look for (All, Some-not) or (Some, No) on the same two terms. Then test each alone.

Only-a-few and definite-case rulings

⚡ Split on sight. Rewrite every 'Only a few A are B' as two lines: Some A are B, and Some A are not B. Then solve as normal.

Venn Diagrams

What a Venn diagram encodes

Two-set union

|A \cup B| = |A| + |B| - |A \cap B|

The overlap is counted twice on the right, so subtract it once.

Only A

|A \setminus B| = |A| - |A \cap B|

Only A is A minus the overlap.

Number of parts

2^n

n circles make 2 to the power n parts, including the outside.

⚡ Write numbers into the regions. Do not keep numbers in your head. Write each one in its region. The answer is then a single sum.

⚡ Shade and compare. For a 'which expression' question, shade the phrase first. Then shade each option. Pick the match.

Two-circle counting

Two-set union

|A \cup B| = |A| + |B| - |A \cap B|

The overlap is counted twice on the right, so subtract it once.

Only A

|A \setminus B| = |A| - |A \cap B|

Only A is A minus the overlap.

Neither

\text{neither} = \text{total} - |A \cup B|

Everyone outside both circles.

Exactly one

|A| + |B| - 2|A \cap B|

Only A plus only B.

⚡ Add the four parts to check. After you fill only A, both, only B and neither, add them. If the sum is not the total, one part is wrong.

⚡ Overlap limits. The overlap can never be more than the smaller group. It can never be less than A + B − total.

Three-circle counting

Three-set union

Add singles, subtract pairs, add the triple back.

Exactly two

\Sigma|A \cap B| - 3|A \cap B \cap C|

Sum of the pair figures minus three times the triple.

At least two

\Sigma|A \cap B| - 2|A \cap B \cap C|

Exactly two plus the triple.

Exactly one

|A \cup B \cup C| - \text{exactly two} - \text{all three}

Remove the multi-group members from the union.

⚡ Strip the triple. For 'exactly two', take the pair numbers and remove the triple three times. It works because each pair holds the triple once.

⚡ Fill the centre first. Start with the all-three part. Then fill the pairs, then the singles. Each step needs only the numbers you already have.

Choosing the correct diagram

⚡ One pair at a time. Never judge the whole picture at once. Decide each pair, then pick the option that agrees with all three pairs.

⚡ Big class first. Find the widest word first. Draw it as the big circle. Then place the other words inside, across, or outside it.

Dictionary Order & Alphabet

Dictionary order rules

⚡ Find the first split point. Write the words one under another and scan the columns from the left. The first column with different letters decides the whole comparison. Ignore every letter after it.

Arranging words and picking slots

⚡ Sort once, count slots. Fix one numbered order for the whole question, then answer any slot lookup from it. Re-sorting for each option is where errors and lost time come from.

Letter positions and shifts

Position from the right

p_{\text{right}} = 27 - p_{\text{left}}

The two positions of the same letter add to 27.

Shifted position

p' = p \pm k

Move right (+) or left (−) by k letters, staying inside 1 to 26.

⚡ EJOTY plus a tiny shift. Anchor at the nearest of E, J, O, T, Y and step from there. Position 18 is T (20) minus 2, which is R. Two seconds, no counting.

⚡ The 27 mirror. Any from-the-right count becomes from-the-left by 27 − k. Convert first, then do everything else from the left.

Dictionary rank and word surgery

Rank of a word

\text{rank} = 1 + \sum_i c_i \times (r_i)!

c_i = unused letters smaller than the letter at position i; r_i = letters remaining after position i.

⚡ Count smaller unused letters. Freeze the sorted letter list. For each letter of the word, left to right, count how many still-unused letters are smaller. Multiply by the factorial of what remains, sum, add one.

⚡ Three letters: write all six. For a three-letter word the whole list has just 6 entries. Writing them beats any formula and never miscounts.

Statement & Conclusion

Statement and conclusions

⚡ The extreme-word filter. Circle words like only, all, always, never, surely, best, entire and must. If the conclusion has one and the statement does not, it almost always fails.

⚡ Needed is not the same as enough. 'Only X can do Y' means X is required. It does not mean X is enough. Check which way the conclusion runs.

Statement and assumptions

⚡ The negation test. Put 'not' into the assumption. Does the speaker still have a reason to speak? If not, the assumption is implicit.

Courses of action

⚡ The proportion check. Ask: would a sensible official do this tomorrow? Inspecting, repairing, warning, supplying and treating usually pass. Banning, shutting forever and blaming everyone usually fail.

Counting Figures

Counting triangles

Apex lines

T = \frac{(k+1)(k+2)}{2}

k extra lines drawn from the apex to the base.

Apex lines + horizontal cuts

T = \frac{(k+1)(k+2)}{2}\times(h+1)

h lines parallel to the base, each crossing every apex line.

⚡ Square with diagonals: 8 or 16. A rectangle with both diagonals always has 8 triangles. Add both midlines and it becomes 16. Memorise both.

⚡ Count the lines, then choose two. In an apex figure, count the lines through the apex (extra lines plus both sides) and choose any two of them.

Counting squares and rectangles

Rectangles in an m × n grid

R = \binom{m+1}{2}\binom{n+1}{2}

Pick 2 of the m + 1 vertical lines and 2 of the n + 1 horizontal lines.

Squares in an m × n grid

S = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1)

An n × n grid gives 1 squared + 2 squared + ... + n squared.

⚡ Line-pair trick. A rectangle is decided by its two vertical sides and two horizontal sides. Count line pairs, never shapes.

⚡ Memorise the small grids. 2 × 2: 5 squares, 9 rectangles. 3 × 3: 14 squares, 36 rectangles. 4 × 4: 30 squares, 100 rectangles. They appear inside bigger questions.

Counting straight lines

Lines in a grid

L=(m+1)+(n+1)

m columns, n rows; add extra slanting lines separately.

⚡ Direction sweep. One pass for horizontals, one for verticals, one per slant direction. The running total is the answer.

⚡ Merge before counting. When cell diagonals sit end to end on one path, they are one line. Join them in your head first.

Mirror & Water Images

Mirror images (vertical mirror)

⚡ Check the ends first. The image starts with the last character of the word, flipped. Reject options by looking at the first character only.

⚡ Symmetric-letter shortcut. A word made only of A H I M O T U V W X Y that reads the same backwards looks unchanged in a vertical mirror.

Water images (horizontal mirror)

⚡ Order stays, letters go upside down. In the options, look for the one that keeps the original order and has only the unsymmetric letters turned upside down.

⚡ Name the image from two clues. Same order and upside down: water. Reversed order and flipped left to right: mirror. Reversed and upside down: half turn.

Mirror image of a clock

Mirror time

T_{\text{image}} = 11{:}60 - T_{\text{actual}}

Works in both directions. For 12:xx, use 23:60 minus the time.

⚡ 11:60 rule. Subtract the hours from 11 and the minutes from 60.

⚡ Check by adding. Add the given time and your answer. The sum must be 12:00.

Paper Folding & Cutting

Folding and punching: unfolding the pattern

⚡ Reflect, do not rotate. Each unfold is a mirror. The copy is at the same distance from the fold line, on the other side. A half turn of the flap is wrong.

⚡ Count first, then check positions. Holes = punches times layers under each punch. Use the count to remove wrong options, then check positions.

Counting layers and holes

Holes after full half-folds

H = p \times 2^{n}

p punches through n full half-folds.

Layers after full half-folds

L = 2^{n}

Each full fold doubles the layers.

⚡ Doubling chain. Write 1, 2, 4, 8 as you read each full fold. Multiply by the number of punches at the end.

⚡ Multiply the steps. Each step multiplies the layers: a half fold by 2, a three-way fold by 3.

Embedded Figures

Finding the hidden part in a figure

⚡ Reject by a missing direction. Look at which slants the big figure has. An option with a slant the figure lacks cannot be hidden in it.

⚡ Count strokes by type. Count the flat, upright and slanting strokes in the part. The big figure must have at least as many of each type.

⚡ Two finalists: compare at the anchor. If two options look alike, compare only the strokes that touch the anchor. The wrong one usually breaks there.

Which figure contains the given part

⚡ Reject by a missing direction. List the directions in the part: flat, upright, / and \. An option that lacks one of them is out at once.

⚡ Count the strokes first. The option must have at least as many strokes as the part. Fewer strokes means it cannot contain the part.

⚡ Two finalists: compare at the anchor. If two options survive, compare only the strokes that touch the anchor. The wrong one usually breaks a stroke there.

Cube & Dice

Dice: finding opposite faces from positions

Standard die

1+6 = 2+5 = 3+4 = 7

Opposite faces of a standard die add to 7.

Hidden faces of a standard die

\text{hidden} = 21 - \text{shown}

All six faces add to 21.

⚡ Strike out the neighbours. List every face seen with the asked face and cross them out. If one face is left, it is the answer.

⚡ Two common faces. Two pictures with two faces in common: the two other faces are opposite.

⚡ Hidden faces of a standard die. All six faces add to 21. Subtract the three faces you can see.

Cube nets: folding a sheet into a cube

⚡ Skip one. Along a row or column, skip one square. The squares on both sides of the skipped one are opposite.

⚡ Reject first. In a which-cube question, throw out every option that shows an opposite pair. Often only one option is left.

Painted cube cut into small cubes

Two faces painted

12(n-2)

Cubes on the edges, not the corners.

One face painted

6(n-2)^2

Cubes in the middle of each face.

No face painted

(n-2)^3

The hidden inner block.

Total check

8 + 12(n-2) + 6(n-2)^2 + (n-2)^3 = n^3

The four counts add up to all the small cubes.

⚡ Check the total. The four counts must add up to n cubed. Use it to catch mistakes.

⚡ At least one painted. Take all the small cubes and remove the core.

Figure Series

Movement and rotation

Position after k steps

p_{k} = (p_{0} + k\,s) \bmod 8

s = step (positive = clockwise) on the 8 border places.

⚡ One element, one line. Write one short row for each element: the places in every frame, then the step, then the next place. Any option that fails a row is out.

⚡ Arrows as clock hours. Think of the arrow as the hand of a clock. A 90-degree clockwise turn goes 12, 3, 6, 9.

⚡ Cross out by one element. Test the easiest element first, such as the arrow. Cross out the options that fail it before you look at anything else.

Adding, removing and changing elements

⚡ Count first. Count the elements in each frame. Only options with the right count survive. Then check positions or details.

⚡ Differences of differences. If the differences are not equal, write them as a list. A list like 1, 2, 3 grows by 1 each time.

⚡ Cross out by the detail. Test the small detail first, such as a shape or an arrow. It removes options quickly.

English Comprehension

Reading Comprehension

How to attempt an RC passage in SSC exams

⚡ Two-minute structure read. On the first read, do not hunt for facts. After each paragraph, say its job in three words. Facts are easy to find again. Structure is what saves time.

⚡ Vocabulary first, main idea last. Answer the replace-the-word question and the line-specific details before the main-idea question. By then you have touched half the passage and the theme is clear.

⚡ Line anchoring. Every detail question has a home line. Match a distinctive word from the question (a name, a number, an odd noun) to the passage. Read two lines around it. Choose the option that paraphrases those lines.

Main idea, central theme and best-title questions

⚡ First and last sentence sketch. Read the opening and closing sentences of the passage, plus the first sentence of each middle paragraph. This skeleton usually states or implies the theme. Build your ten-word summary from it before you read the options.

⚡ Count the mentions. The idea the author repeats in different words across paragraphs is the theme. If an option's core idea appears in only one paragraph, it is a detail, not the theme.

⚡ Title versus theme wording. Titles are noun phrases. Theme options are full claims. Judge both by scope, not by grammar.

Factual detail questions ('According to the passage…')

⚡ Paraphrase before you look. After you locate the line, cover the options and answer in your own words. Then uncover and match.

⚡ Reversal scan. Before you mark a detail option, check its direction words: only, always, all, may, often, some, can, cannot. Wrong options flip these while keeping the same topic words.

⚡ Option pairing. When two options say the same thing in different words, both are wrong. A fact has one meaning. Remove the pair, then judge the other two.

Inference and extrapolation questions

⚡ One-step rule. Correct inference = stated idea plus one logical step. Two steps, extra facts or a change of scale means reject.

⚡ Verbatim is a verdict. In an inference question, an option that repeats passage wording without adding a step is wrong by definition. Cross it out first.

⚡ Extremity filter. Inferences with only, never, must or inevitably are almost always one step too far. Moderate options (may, tends to, is likely) fit the must-be-true test better.

Vocabulary in context (replace-the-word / meaning questions)

⚡ Own words first. Cover the options and replace the word with your own simple word. Then find the option that matches your word. This removes the dictionary-meaning bait.

⚡ Tone match. The substitute must keep the sentence's attitude. In a critical sentence, choose the critical option. A neutral synonym that drains the criticism is wrong.

⚡ Two-line rule for phrases. For phrase-meaning questions, read one line before and one line after. Figurative phrases get their meaning from the surrounding argument, not from the words.

Tone, attitude and author's-opinion questions

⚡ Adjective audit. Sweep the passage for adjectives and adverbs that carry feeling, such as quietly, badly, remarkable, merely, confident. Three of them in the same direction fix the tone.

⚡ However counts double. The clause after but, however or yet carries the author's real position. Tone and opinion questions key off the second half.

⚡ Degree matching. Match intensity as well as direction. Pick the option at the same temperature as the passage.

Cloze Test

What a cloze test is and how to attack it

⚡ Tense anchor first. Before you touch any blank, underline the time words: every morning, last month, by evening. Verb blanks obey these anchors.

⚡ Predict, then match. Cover the options with your hand. Read the sentence and think of your own word. Options are written around the right meaning, so a far-off option is usually wrong.

⚡ The read-aloud finish. After you answer all five, read the passage as if someone else wrote it. Grammar errors and logic breaks become easy to hear. Re-check any blank where you stumble.

Grammar inside the passage: tense, agreement, articles

⚡ Strip-to-subject. Delete every phrase between the subject and the verb blank. Then apply agreement to the short skeleton.

⚡ Only-one-past trick. In a past story, one option may be the only past form among present and -ing forms. The anchor already decided it. Mark it and move on.

⚡ Say the article aloud. For an a/an blank, pronounce the first sound of the next word. It is *an hour* but *a university*. Sound decides it, not the first letter.

Vocabulary blanks: meaning, register and collocation

⚡ Name the missing idea. Before you look at the options, say what the sentence lacks, for example: it needs the idea of consistency. Then choose the option that says it.

⚡ Fixed pairs to bank. Learn common pairs: gain momentum, pay attention, take measures, reach a decision, bear fruit, meet a deadline, break the news, strike a balance. Cloze verbs and nouns often come from such pairs.

⚡ Polarity check. Mark the sentence + or - before you choose. A praising sentence rejects negative words, even when they are grammatically perfect.

Connector blanks: contrast, cause, result, addition

⚡ Opposite-direction test. Give each clause a + or - sign. Same signs point to addition or result. Opposite signs point to contrast.

⚡ Partner hunt. Scan the sentence for the other half of a pair. If you see *not only*, the blank must be *but also*. No further thinking is needed.

⚡ Comma tell. A pattern like *clause, _ clause* with a one-word slot wants yet, but, so, for or and. Options like however and therefore fit the pattern *clause. _, clause* instead.

Preposition and phrasal-verb blanks

⚡ Governor chant. Underline the word that governs the blank and say its partner: depend on, afraid of, good at, solution to. About ten pairs cover most exam prepositions.

⚡ Particle direction sense. Ask what the sentence does to the object. Does it cancel it (call off), reject it (turn down) or build it (set up)? The particle shows the direction of the action.

⚡ Substitute the simple verb. Replace the phrasal verb with a one-word verb in your head. The sentence must keep its meaning. If it does not, that option is wrong.

Error Spotting

The master checklist: SVA → tense → article → preposition

⚡ The sweep in one line. Verb first: find the subject and check its number. Then tense words. Then a / an / the by sound. Then the preposition partner. Only then pronouns, pairs and word choice.

⚡ Strip the middle. Delete the phrase between the subject and the verb, such as 'of my friends' or 'as well as the players'. Read the bare skeleton. Agreement errors become loud.

⚡ Earn the No error answer. Choose 'No error' only after the full sweep. About one in five of these items hides a quiet agreement or preposition fault.

Subject–verb agreement errors

⚡ Hunt the subject backwards. Start at the verb and walk backwards past every 'of ...' phrase. The first noun or pronoun with no preposition in front is the subject.

⚡ Nearness rule flash. See neither ... nor or either ... or with two subjects? The verb matches the second one. The item usually solves at once.

⚡ A quantity is singular. Money, distance, time and weight taken as one amount use a singular verb, even in plural form: 'Ten kilometres is a long walk.'

Tense and sequence-of-tenses errors

⚡ Circle every anchor. Circle yesterday, ago, since, for, by the time, when and next week before you judge any verb. Most tense items solve themselves.

⚡ The will check. Scan for will inside a when / if / until clause with future meaning. SSC plants this error again and again.

⚡ Since and for sorting. Since + starting point (since Monday, since 2019, since childhood). For + length (for ten years, for a week). The wrong partner is the answer.

Article and preposition errors

⚡ Say it aloud. Articles are sound decisions. Say the phrase. 'An university' fails the ear because of the 'yu' sound. 'A honest man' fails too.

⚡ Governor chant. Underline the word the preposition depends on. Recite its partner: insist on, angry with (person), prefer to, married to. A wrong partner is the answer.

⚡ The with superlatives. Superlatives and ordinals need the: 'the most intelligent girl', 'the first attempt'. A bare superlative hides an article error.

Pronoun and correlative-conjunction errors

⚡ Preposition plus I? Kill it. Between you and I, for he and I: all wrong. After a preposition, use the object form every time.

⚡ Did plus base form. After did, does or do, the verb keeps its base form: 'No sooner did the bell ring'. A past form after did is always an error.

⚡ Cover and match the pair. Cover everything except the pair words. If one half stands with the wrong partner (hardly ... than), that part is the error.

Comparison, redundancy and confusable-word errors

⚡ Scan for more + er. Look for more before a comparative (more bigger) or most before a superlative. That is an error at once.

⚡ Flinch at echo pairs. In pairs like return back, repeat again and revert back, the second word repeats the first. These are planted answers.

⚡ The that-of test. When two nouns' qualities are compared, the second noun needs that of or those of: 'The roads of Delhi are wider than those of Patna.'

Verb forms, question tags, conditionals and structure

⚡ Flip the tag. Hear the end of the sentence. A positive statement needs a negative tag. A negative word (never, rarely, hardly) needs a positive tag.

⚡ Unreal past conditional. After 'if' about the unreal past, use had + participle. 'Would have' in the if-part is the planted error.

⚡ Copy the first item. In a list, the first item sets the shape (-ing, to + verb or noun). Every other item must copy it.

Sentence Improvement

The elimination ladder: how to choose in 20 seconds

⚡ Find the anchor before you read the options. Look for the time word or helper verb near the bold part. Decide what the verb must look like. Then pick the option that matches your form.

⚡ Tag every option with its flaw. In your head, label each option: 'plural verb', 'wrong partner', 'breaks inversion'. If no option has a flaw, the answer is No substitution required.

⚡ Meaning guard. Two options are both grammatical? Choose the one that keeps the original meaning word for word. An option that changes who did what is wrong.

Tense and sequence fixes

⚡ The yesterday test. See a finished-time word? Cross out every perfect and continuous option at once. Usually one option is left.

⚡ Backshift plus straight order. In a reported question, two things change: the tense moves back and the order becomes a statement. Each wrong option usually breaks one of these. Check both.

⚡ Since and for health check. 'Since 2015' or 'for ten years' with a continuing action needs 'has / have been'. Options without it die at once.

Agreement and verb-form fixes

⚡ Cover the phrase and read the skeleton. Cover 'of the students' type phrases with a finger. Read what is left: 'One _ won'. The right helper is now easy.

⚡ Check the helper chain. did takes the base verb, have takes the participle, be takes -ing or the participle. If the bold part has a helper, check its partner form first.

⚡ Two subjects with and. Two subjects joined by 'and' take a plural verb. The exception is a pair that names one thing, like 'bread and butter is'.

Comparison and structural fixes

⚡ Insert that of or those of. Two 'the X of A' phrases are compared, but the second is bare. Add 'that of' for a singular noun or 'those of' for a plural noun.

⚡ Check for a double marker. Look for more + a word that already has -er, or most + a word that already has -est. If you find one, the repair is a subtraction.

⚡ Count first. Find the count phrase in the sentence. 'Of the two' means the comparative with 'the'. 'Of all' means the superlative.

Precision fixes: articles, prepositions, pronouns, word choice

⚡ Sweep the small words. Read only the small words in the bold part: a, an, the, in, on, at, me, myself. A wrong one usually stands alone. Fix it and finish.

⚡ The myself firewall. A reflexive like 'myself' never replaces a plain object pronoun. Any option that offers 'myself' where 'me' belongs is a decoy.

⚡ Redundancy flinch. Learn the fatal pairs: return back, repeat again, revert back, final conclusion. When one is bold, the shorter option is the key.

Structural fixes: inversion, dangling modifiers, parallelism

⚡ Opener triggers a flip. Negative word first? Expect helper before subject: had I reached, did he arrive. Then check the partner: than or when.

⚡ Who does the -ing?. Name the doer of the opening action. If the main subject is not that doer, pick the option that makes the doer the subject.

⚡ Shape match across pairs. Cover the pair words. Compare the shapes on both sides. If one side is a verb and the other a noun, that is the fault.

Fill in the Blanks

Collocation blanks: word partnerships that decide the answer

⚡ Answer before you look at the options. Read up to the blank and stop. Ask what the word before it usually takes. Say your answer, then find it among the options.

⚡ Cut options by topic. Collocation options often share an ending (-tion, -ment). Remove the ones that do not suit the topic first. A border dispute invites talks or deliberations, never medications.

⚡ Finish the phrase in your head. For a verb blank, turn the phrase around: 'attention is _' gives 'paid'. If your word is an option, mark it and move on.

Phrasal-verb blanks: verb + particle as one unit

⚡ Say the meaning in one word, then match. Put the sentence's action in one plain word: cancelled, tolerated, abolished. Then pick the phrasal verb that means that word.

⚡ Swap in a one-word verb. Replace each option with a one-word verb (call off = cancel, turn down = reject). The option whose swap keeps the meaning is right.

⚡ Use the direction of the small word. Ask what happens to the object: stopped or removed (off, out), raised (up), lowered or refused (down), accepted (in). Match that to the particle.

Verb-form and tense blanks

⚡ Who owns the 'to'?. If 'to' belongs to a phrase like look forward to or object to, it is a preposition, so use -ing. If it belongs to a verb like decide or hope, use the base verb.

⚡ The by-the-time ladder. With 'by the time' and two past events, the earlier event takes had + V3 and the later one takes simple past. Put the blank on its step of the ladder.

⚡ Three short verb lists. Learn three lists. -ing after: avoid, enjoy, mind, suggest, finish. to + verb after: decide, hope, refuse, plan. Plain verb after: had better, would rather, let, make.

Connective and relative-pronoun blanks

⚡ Find the other half of the pair. Scan the sentence for the first half of a pair. 'Hardly had' fixes the blank as 'when'. 'No sooner had' fixes it as 'than'.

⚡ He / him test for who / whom. Rewrite the clause as a sentence. If 'he' fits, use who. If 'him' fits, use whom.

⚡ Clause or noun after the blank?. A full clause (subject + verb) after the blank needs although / though. A noun or -ing form needs despite / in spite of.

Confusable and homonym blanks

⚡ Word type first, meaning second. Decide noun, verb or adjective from the sentence. Remove every option of the wrong type. Usually only two options are left for the meaning check.

⚡ Picture the sentence. Make a quick mental picture. A shop full of paper and pens → stationery. A parked van → stationary.

⚡ Learn the groups of three. Make flash cards for groups: assure / ensure / insure, site / sight / cite, allusion / illusion / elusion. These appear together in options.

Double blanks: two gaps, one sentence

⚡ Spot the structure. Learn the skeletons: too + adjective + to + V1; so + adjective + that + clause; such + noun + that + clause; hardly ... when; no sooner ... than. Once you see the skeleton, both blanks fill at once.

⚡ Let the verb decide. Plural verb with two subjects → Both ... and. Singular verb → Either ... or / Neither ... nor, agreeing with the nearer noun.

⚡ Learn fixed expressions in pairs. Revise safe and sound, part and parcel, null and void, first and foremost, ways and means together. Options often join a real first half to a wrong second half.

Synonyms & Antonyms

The root-word method: decode words you have never seen

⚡ Split, then eliminate. Cut the unknown word at its joins, translate the parts you know, and remove options that clash. One sure root can remove two options.

⚡ Learn roots as families. Learn words in root groups, not in alphabet order. The dict family: dictate, verdict, predict, contradict. The loqu family: eloquent, loquacious, soliloquy.

⚡ Warm or cold. Roots carry a feeling. Bene-, eu- and am- are warm. Mal-, dys-, phob- and mis- are cold. When stuck, pick an option with the same feeling.

Choosing the nearest meaning: elimination by class, degree, charge

⚡ The replace test. The right synonym can replace the word in most sentences. If the sentence changes meaning, the option is only related.

⚡ Tone sorts half the options. Mark the given word + or - (notorious is -, celebrated is +). Options with the other mark can go without more thought.

⚡ The strength ruler. Draw a line from mild to extreme. Put the word and the two surviving options on it. The option at the same mark wins.

Antonyms: prefix flips and real opposites

⚡ Pair-of-sentences test. Put the word and each option in the same short sentence. The option that gives a real contradiction is the antonym.

⚡ Guess the flip first. Before reading the options, guess the opposite yourself (conformity, then nonconformity). If your guess is in the options, mark it.

⚡ Absence is not opposition. Cross out options that only lack the quality. Careful does not fight brave. Cowardly does.

Exam strategy for vocabulary questions

⚡ Two passes. Answer every sure vocabulary item in the first pass at 15 seconds each. Come back to the unknown words with the time left.

⚡ Options show the scale. If the options are kind, cruel, wealthy, talkative, the scale is temperament. Now guess the tone of the given word from its parts and pick a matching option.

⚡ Words repeat across years. Solve the last five years of synonym and antonym questions. Words come back within two to four years.

One Word Substitution

Suffix families: decode the word by its ending

⚡ Family first, root second. Read the phrase and name the family (killing, fear, study). Then match the root. Two steps remove most options.

⚡ Stack two families. Rare words are often two parts joined. Nycto + phobia is fear of darkness. Entomo + logy is study of insects.

⚡ Make look-alike cards. Write each look-alike pair on one card with the meaning of each word. Revise the cards every week.

One word for people: traits, trades and types

⚡ Object first. Underline the object or habit in the stem before you read the options. It names the family at once.

⚡ Learn words in pairs. Take each word with its partner. Widow goes with widower. Miser goes with spendthrift. The pair is one memory item.

⚡ Speak the parts. Say the parts aloud: somn, ambul, ist. Hearing the parts locks the meaning.

One word for places and groups

⚡ Animal decides the group. Name the animal in the phrase first. Then recall its one group word.

⚡ Sound hooks. Link each look-alike to a sound: api sounds like ape-bee, avi is in aviation. Hooks beat plain rote.

⚡ Twenty and twenty. About twenty place words and twenty group words cover nearly every paper. Learn them in one sitting and revise once a week.

Government systems, studies and word-craft

⚡ Ruler first. Circle the ruler in the stem before the options. It gives the word.

⚡ Three dead words. Learn elegy, eulogy and epitaph as one set. One question can ask any of them.

⚡ Link the verb. Pair each cannot-be word with its verb: read illegible, see invisible, hear inaudible.

Exam strategy for one-word substitution

⚡ Specific beats general. If both a general and a specific word fit, choose the specific one. It matches the detail the examiner gave.

⚡ Weekly weak list. Any word missed twice goes on a short list. Read only that list twice a week.

⚡ Syllable check for spelling. For spelling items say the word slowly in syllables and match each syllable to the option.

Idioms & Phrases

Why literal meaning is always wrong — and how to find the picture

⚡ Photo test. If you could take a photo of what an option says, it is almost always wrong. This removes one option in nearly every question.

⚡ Plug-back check. Put your chosen meaning into a sentence in place of the idiom. If the sentence sounds natural, keep it.

⚡ Tone from the key word. Cold, dark and hot often signal trouble. Cloud nine and red-letter signal joy. Use this when you do not know an idiom.

Animal and body-part idioms

⚡ Character first. Name the animal or body part, then recall its character or job. The meaning usually follows.

⚡ Body parts in sets. Learn hand idioms together: lend a hand, have your hands full, wash your hands of. The job is help or work.

⚡ One card for opaque idioms. Write a card for each idiom whose picture is hard to see. Read those cards first in revision.

Colour and food idioms

⚡ Colour to feeling. Say the colour and its feeling before you read the options. Blue means sudden or rare, and the options then sort themselves.

⚡ Food gives a verdict. Ask what verdict the food gives: easy, hard, flattery or trouble. That is the meaning.

⚡ Pair cards. Write the twins on one card: red tape and red-letter day, apple of the eye and apple of discord.

Idioms for situations and types of people

⚡ Family first. Name the family of the situation before you read the options. Effort, trouble and suddenness cover a large share.

⚡ One line per story. Store each story idiom as one line: Achilles, heel, weak point. Do not learn the whole story.

⚡ Read the whole sentence. In a blank question, say what the person does. Escape means took to his heels. Hard work means burnt the midnight oil.

Phrasal verbs (verb + particle phrases)

⚡ Swap for one word. Replace the phrasal verb with one plain word, such as cancel, refuse or tolerate. Test it in the sentence.

⚡ Subject shows the particle. A car breaks down. A fire breaks out. A thief breaks in. The subject picks the particle.

⚡ Learn by verb. Learn one verb with all its particles in one sitting: give up, give in, give away, give out.

Exam strategy: banking 300 idioms in 30 days

⚡ Own sentence. Write one sentence of your own for each idiom. The sentence is what you remember under pressure.

⚡ Format first. Name the format before you start. Meaning, replace, complete or opposite. Each has its own first step.

⚡ Same-meaning trap. In an opposite question, the option with the same meaning as the idiom is always wrong.

Spelling

i before e — and the exceptions that fill the paper

⚡ The c-test. Is there a c right before the pair? Then choose ei (receive). No c? Usually choose ie (believe). Ten exceptions override this: weird, seize, height, foreign, sovereign, leisure, either, neither, their, protein.

⚡ Rhyme-anchor pairs. Some words match each other. Height and weight match. Believe and achieve match. Receive and deceive match. Fix one word of a pair and you get its partner free.

⚡ Say it in syllables. Say the word slowly in your head: be-lieve, re-ceive. Your ear often confirms what your eye doubts.

Doubling consonants and one-letter betrayals

⚡ Stress decides doubling. Say the word. If the last syllable is stressed, double the consonant before a vowel suffix (begin, beginning; occur, occurred). If not, do not double (benefit, benefited).

⚡ Double-double checklist. Five words need two sets of double letters: accommodate (c, m), committee (m, t), embarrass (r, s), occurrence (c, r), millennium (l, n). Count them.

⚡ Letter census. Count the repeated letters in each option. Accommodate must show two c's and two m's. Embarrass must show two r's and two s's. Cut any option that fails the count.

-able / -ible, -ance / -ence, -ant / -ent

⚡ Root trace. Strip the suffix and look at the root. Exist gives existence. Maintain gives maintenance. Resist gives resistance. Persist gives persistence. The root's own spelling helps decide the ending.

⚡ -fer means -ence. Nouns from *fer* take -ence: reference, preference, difference, inference. The verb doubles by stress: refer, referring.

⚡ The -cient family. Sufficient, efficient and ancient all end in -cient. Science is sci + ence, and conscience is con + science.

The trap-word bank: silent letters and inherited mistakes

⚡ The n-test for -ment words. Government is govern + ment. Environment is environ + ment. The root's final n must survive. Ask what the root is, and the letter reappears.

⚡ Three -ceed, one -sede. Proceed, succeed and exceed are the only -ceed words. Supersede is the only -sede word. Every other word with this sound ends in -cede.

⚡ Plurals already plural. Criteria, phenomena and stimuli are already plurals. An option that adds -s to them (criterias) is wrong.

Exam strategy: how to judge a spelling you have never seen

⚡ Difference-first reading. Do not read four full words. Find the one position where they differ. That spot is the whole question.

⚡ Syllable count. Count the syllables. Mischievous has three. Rhythm has two. A wrong spelling often forces a syllable that cannot be said.

⚡ Write-from-memory drill. Seeing a word is not the same as spelling it. Cover the list, write the words, mark the misses and repeat tomorrow. The misses shrink each round.

Active & Passive Voice

Voice basics: the three moves

⚡ Tense handover. The helping verb must carry the original tense: wrote becomes was, writes becomes is, will write becomes will be, has written becomes has been. A changed tense in an option is a planted error.

⚡ V3-only law. After be, been or being, the main verb is always V3 (written, given, built). Options like *is wrote* and *was build* are wrong on sight.

⚡ Pronoun mirror. The subject pronoun becomes the object pronoun behind by (he becomes him). The object pronoun becomes the subject pronoun in front (me becomes I).

Tense-by-tense conversions in action

⚡ Row recall. Name the active tense. Then say its passive row: present continuous is is/are being + V3, past perfect is had been + V3, future is will be + V3. Each wrong option breaks one cell of the row.

⚡ Agreement sweep. After converting, check is/are, was/were and has/have against the NEW subject before anything else.

⚡ Being or been in one look. Continuous in the active gives being. Perfect in the active gives been. A modal gives plain be.

Modals, imperatives, infinitives and questions

⚡ Let anchor. An order (no subject, base verb) points to Let + object + be + V3. An option with no Let and no modal cannot be right.

⚡ By whom first. Who-questions become *By whom + was/were/is + new subject + V3*. Check the word order first, because options scramble it.

⚡ Frame words. Please gives *You are requested to*. Advice gives *You are advised to*. A warning gives *You are warned not to*.

Reverse conversions: passive to active

⚡ First helping verb. Read only the first helping verb. Was means past. Is means present. Has or had means perfect. Will means future. Is being and was being mean continuous.

⚡ Doer hunt. The by-phrase gives the new subject. No by-phrase means *they*, *people* or *someone*.

⚡ Object-case check. The last pronoun must be in the object case: *praised him*, not *praised he*. Many options fail exactly here.

Exam strategy: the 20-second conversion protocol

⚡ Name the fault. Every wrong option breaks one rule: V-form, being/been, tense, agreement or pronoun. Name the fault and the option is out.

⚡ Five-row priority. Learn simple present, simple past, present perfect, present continuous and modals first. They cover most of the questions.

⚡ Write it, do not just say it. Voice errors hide on paper. Convert sentences in writing every day. The hand learns the pattern faster than the ear.

Direct & Indirect Speech

Reporting verbs and the mechanics of the shift

⚡ Read the tone first. Before you touch tense or pronouns, name the tone of the quote: order, advice, request, question, joy or sorrow. The matching verb alone often removes two options.

⚡ said to or told. Use *said to* + person, or *told* + person. Never mix them. Any option with *said me* or *told to me* is wrong.

⚡ Connector test. Statement: that. Yes/No question: if or whether. Wh-question: the wh-word alone. Order, request, advice: to + V1.

Backshift: the tense conversion table

⚡ Say the ladder aloud. Present goes to past. Present continuous goes to past continuous. Perfect goes to past perfect. Past goes to past perfect. Will goes to would. Can goes to could. May goes to might. Must (duty) goes to had to.

⚡ Frozen five. Could, would, should, might and ought to never change. If the quote has one of these, its tense is already right. Any shift is an error.

⚡ Truth test. Is the quote a law of nature, a habit or a proverb? Then do not backshift. Write: said that the earth revolves.

Pronouns, and the time/place word table

⚡ SONA sweep. Mark the three anchors before you change anything. First person goes to the reporter. Second person goes to the listener. Third person stays. Most wrong options break exactly one of these.

⚡ Yesterday and tomorrow pair. Yesterday becomes the previous day. Tomorrow becomes the next day. If the option shifts the verb but keeps the old time word, it is wrong.

⚡ Check the reporting verb first. If the reporting verb is 'says' (present), nothing shifts: not the tense, not the time words. Only the pronouns follow SONA.

Questions, commands and exclamations

⚡ Skeleton first. Name the type of the quote. Then write its skeleton: asked if ... / told ... to ... / exclaimed with joy that ... Pour the tense, pronoun and time-word changes into the skeleton.

⚡ Remove the helper do. In a direct question, do / does / did is only a helper. It vanishes in indirect speech. 'Do you know' becomes 'if I knew', never 'if I did know'.

⚡ Exclamation map. Hurrah: exclaimed with joy. Alas: exclaimed with sorrow. Bravo: applauded. Good morning: wished. Thank you: thanked.

Exam strategy: the four-check protocol

⚡ Name the fault. For each wrong option, name its fault: verb, tense, pronoun or time word. If you cannot name it, look again. If two options share no fault, you may have misread the tone verb.

⚡ Reverse-mode clues. In indirect to direct, 'the previous day' means the quote had 'yesterday'. 'Asked if' means the quote was a yes/no question with the helper verb first.

⚡ Write ten a day. Convert ten sentences a day by hand. Mix statements, questions and commands. In a week the four checks run together.

Sentence Rearrangement

The PQRS method: label, anchor, chain, eliminate

⚡ Pairs before strings. Do not read the four strings first. Build two or three pairs from P, Q, R, S. Then see which strings keep them.

⚡ Frame test. After you chain the parts, read S1, your chain and S6 as one paragraph. If any *it*, *this* or *they* has no noun, the chain is wrong.

⚡ Adjacent-swap check. Wrong strings usually swap two neighbours. Find the one swap in each wrong string and name the broken pair. That proves your answer.

Opening and closing sentence clues

⚡ Pronoun orphan test. A part that starts with he, she, it, they, this or these needs its noun earlier. If a string gives no earlier noun, cross it out.

⚡ Connector ban. But, However, Therefore, Thus and So cannot open a free jumble. Two of four options often start with such parts. Cross them out. In a PQRS item, check whether S1 is the claim first.

⚡ Ask what comes before S6. Read S6 and ask which sentence must come just before it. It is often the part with *Thus*, *That is why* or a summing-up judgement. Pin it, then build backwards.

Mandatory pairs: the links that never break

⚡ Noun-first scan. List the nouns and pronouns in P, Q, R, S. Circle each pronoun and each *the* noun. Draw an arrow to the part that brings in its noun. The arrows are the answer skeleton.

⚡ This + noun hunt. For every *this + abstract noun*, find the part that holds that idea. The order is then forced.

⚡ A-to-the cascade. The part with *a + noun* comes before every part with *the + same noun*. This orders many food and story passages at once.

Time-sequence chains

⚡ Timeline first. Write the four to six events as a numbered list. Then map the numbers to the letters. The string writes itself.

⚡ Tense radar. Past perfect is earlier. Past continuous is background. Simple past is the main story line. Use tense when time words are missing.

⚡ By-time capping. *By midnight* and *By dawn* close the phase they describe. They follow the event chain and never open it.

Exam strategy: elimination, budget and Tier 2 jumbles

⚡ Layered elimination. Layer 1: the opener ban. Layer 2: the mandatory pair. Layer 3: the S6 check. Each layer is one question. Two layers usually finish the item.

⚡ Local-difference test. The two surviving strings differ in one place. Isolate that part and test only its two neighbours.

⚡ Odd-sentence radar. For odd-one-out items, mark each sentence's key noun. Four share it. The odd one only orbits it.

Grammar Essentials (Handbook)

Parts of speech: nouns, pronouns, adjectives, adverbs

⚡ Cover the other name. When a pronoun sits next to a noun (*Ravi and I / Ravi and me*), cover the noun. Read the sentence with the pronoun alone. The form that sounds right is the answer.

⚡ Scan every noun. In an error-spotting sentence, circle each noun. Ask two questions. Can I count it? Is it one of the special groups (no plural, always plural, cattle-type)? Words like informations, furnitures and advices are favourite e…

⚡ Few or a few: empty or half full. *Few* and *little* mean the glass is nearly empty (a negative feeling). *A few* and *a little* mean there is something (a positive feeling). Let the rest of the sentence show the feeling.

⚡ Adjective order and word families. Adjectives follow the order OSASCOMP: Opinion, Size, Age, Shape, Colour, Origin, Material, Purpose. Example: a beautiful small old round brown Indian wooden table. Some adjectives have no degrees: unique, perfect, ideal,…

⚡ Numbers with nouns. After a definite number, *dozen, hundred, thousand, million, score, lakh, crore* and *pair* take no -s: *three dozen eggs, five hundred rupees*. But *dozens of eggs* and *hundreds of people* keep the -s. A number + noun …

⚡ Where adverbs go. Frequency words go before the main verb but after *be*: *He always comes late. He is always late.* Use *very* with the plain adjective and with -ing adjectives (very tall, very interesting). Use *much* with comparatives …

Tenses and sequence of tenses

⚡ Scan for the signal word. Underline the time signal before reading the options. Since or for gives a perfect tense. Yesterday, ago or last gives the simple past. By + future time gives the future perfect. When, if or as soon as with future meanin…

⚡ No will after when or if. If a clause starts with when, if, unless, until or as soon as and talks about the future, remove will from that clause. Will stays in the main clause only.

Subject-verb agreement (all rules and exceptions)

⚡ Strike-through method. Draw a line through every phrase that starts with a preposition (of the boys, in the box, with his friends). Also strike every add-on phrase (as well as, along with, together with). What stays before the verb is the true…

⚡ Nearest subject for or / nor pairs. For either ... or, neither ... nor, not only ... but also, or and nor, look only at the subject next to the verb. Put the plural subject second so the plural verb sounds natural.

⚡ The number and a number. *The number* is one figure, so it is singular. *A number* means many, so it is plural. Memory hook: A = Abundant.

Articles (a, an, the, zero article) and determiners

⚡ Say it aloud. For a or an, whisper the next word. A vowel sound (on-est, em-el-ay, ex-ray) needs an. A y or w sound (you-niversity, won-day) needs a.

⚡ The geography rule of thumb. Plural names, water and chains take the: the Alps, the Ganga, the Indian Ocean, the Andamans. A single peak, a single lake, a city or most countries take no article: Mount Abu, Lake Dal, Delhi, Nepal.

Prepositions and conjunctions

⚡ Find the partner. When you see hardly, scarcely, no sooner, not only, neither, either, both, whether, lest or though, look ahead for its fixed partner. A wrong or missing partner is very often the error.

⚡ Group prepositions by meaning. Stopping: prevent, refrain, abstain, desist from. Dependence: depend, rely, count on. Skill: good, expert, adept at; proficient in. Latin comparatives: senior, junior, superior, inferior, prior, prefer …

Modals, conditionals and the subjunctive

⚡ Match the type by the if-clause. Read only the if-clause. Present simple: use will or can in the result. Past simple: use would in the result. Past perfect: use would have + V3.

⚡ Modal + base form. After any modal the next verb is bare. If you see a modal followed by to, -s, -ing or -ed, that is the error. The only exceptions are ought to and used to.

Degrees of comparison and question tags

⚡ Flip the polarity. Read the statement. Note whether it is positive or negative. Then use the helper verb with the opposite sign and a pronoun. Command? Use will you. Let's? Use shall we.

⚡ Watch than, that of and other. When a comparison feels odd, check two things. First: are two like things compared (use *that of* or *those of*)? Second: does the comparison come from the same group (add *other*)?

Non-finites (infinitive, gerund, participle), inversion and parallelism

⚡ Preposition then -ing. If the word before the blank is a preposition (of, in, at, without, after, before, by, about), the verb must end in -ing. Watch for to in look forward to and used to: it is a preposition there.

⚡ Negative start means flip. When a sentence opens with never, rarely, hardly, no sooner, not only or not until, flip the first helper before the subject. If there is no helper, add do, does or did.

General Awareness

Ancient Indian History

Indus Valley (Harappan) Civilisation

⚡ Four corners: Many Donkeys Are Slow. Manda is north. Daimabad is south. Alamgirpur is east. Sutkagendor is west.

⚡ One site, one find. Lothal is the dockyard. Kalibangan is the ploughed field. Mohenjo-daro is the Great Bath. Chanhudaro is the bead workshop. Dholavira is the water reservoirs.

Vedic Age & Vedic Literature

⚡ Punjab rivers: Very Angry Parrots Visit Sutlej. Vitasta is Jhelum, Asikni is Chenab, Parushni is Ravi, Vipas is Beas, Sutudri is Sutlej. Modern order: J-C-R-B-S.

⚡ Veda and priest: HUAB. Hotri with Rig, Udgatri with Sama, Adhvaryu with Yajur, Brahma with Atharva.

⚡ Mandalas 3-7-9-10. 3 Gayatri. 7 Ten Kings. 9 Soma. 10 Purusha Sukta.

Mahajanapadas & Rise of Magadha

⚡ Capital pairs by first letter. Vatsa-Kaushambi and Vajji-Vaishali. Avanti-Ujjain and Anga-Champa. Gandhara-Taxila. Kuru-Indraprastha.

⚡ Magadha order: Hari Shishu Nanda. Haryanka, then Shishunaga, then Nanda, then Maurya. Council 1 is Ajatashatru (Haryanka). Council 2 is Kalashoka (Shishunaga).

Buddhism & Jainism

⚡ Councils: places, kings, presidents. Places: Rajagriha, Vaishali, Pataliputra, Kashmir. Kings: Ajatashatru, Kalashoka, Ashoka, Kanishka. Presidents: Maha-Saba-Moggali-Vasu.

⚡ Buddha symbols in order. Birth lotus, renunciation horse, enlightenment tree, first sermon wheel, death stupa.

⚡ Tirthankara symbols. Bull begins and lion ends. The 22nd has a conch. The 23rd has a snake.

Mauryan Empire

⚡ Edict numbers: 2, 5, 12, 13. 2 hospital ward. 5 officer rank. 12 faiths tolerated. 13 the unlucky war.

⚡ Dynasty start dates: 322, 185, 73. Maurya begins in 322 BCE. Shunga begins in 185 BCE. Kanva begins in 73 BCE.

Post-Mauryan Age: Shungas to Kushanas

⚡ 57 minus, 78 plus. Vikram Samvat is 57 BCE, so subtract. Shaka is 78 CE, so add.

⚡ Satavahana: S-P-G-H. Simuka founder. Pratishthana capital. Gautamiputra greatest. Hala poet.

Gupta Age & Harshavardhana

⚡ Gupta order. Sri Gupta, Ghatotkacha, then Chandragupta I, Samudragupta, Chandragupta II, Kumaragupta I, Skandagupta.

⚡ Inscription authors: H, R, B. Harishena wrote the Prayag Prashasti of Samudragupta. Ravikirti wrote the Aihole inscription of Pulakeshin II. Banabhatta wrote Harsha's biography.

⚡ Pilgrims: F-H-I. Fa-Hien (Chandragupta II), then Hiuen Tsang (Harsha), then I-Tsing.

Sangam Age & South Indian Dynasties

⚡ B-T-F. Chera Bow, Chola Tiger, Pandya Fish.

⚡ Imperial Cholas: V-P-R-R. Vijayalaya, Parantaka, Rajaraja, Rajendra.

Ancient Literature & Authors

⚡ Kalidasa's three K-words. Kalidasa wrote Meghaduta, Raghuvamsha, Kumarasambhava, Shakuntalam. Shakuntalam is his most famous play.

⚡ Two chroniclers. Kalhana wrote the history of Kashmir. Banabhatta wrote Harsha's life.

Medieval Indian History

Early Medieval India (c. 750–1206)

⚡ Tripartite: PPR fight for K. Pala, Pratihara, Rashtrakuta fight for Kannauj. East, west and south all wanted the centre.

⚡ Tarain: 1 to Prithviraj, 2 to Ghori. 1191, the first battle: Prithviraj wins. 1192, the second battle: Ghori wins. The second number goes to the invader.

Delhi Sultanate (1206–1526)

⚡ Who did what: first-letter hooks. Iltutmish = Iqta. Balban = Bow down (sijda, paibos). Alauddin = All prices fixed. Muhammad bin Tughlaq = Move capital and Money tokens. Firoz = Fields and canals.

⚡ Dynasty order: Sabka Khana Tum Sab Lo. Slave, Khalji, Tughlaq, Sayyid, Lodi. "Tum Sab Lo": Sayyid comes just before Lodi.

Vijayanagara & Bahmani Kingdoms

⚡ Vijayanagara dynasties: Some Silly Tigers Attack. Sangama, Saluva, Tuluva, Aravidu. Krishnadevaraya sits third, in the Tuluva line.

⚡ Deccan five: BiG AB B. Bijapur (Adil), Golconda (Qutb), Ahmadnagar (Nizam), Bidar (Barid), Berar (Imad).

Mughal Empire & Sher Shah Suri

⚡ Mughal order: Bahut Hi Accha Jalebi Shahi Aur. Babur, Humayun, Akbar, Jahangir, Shah Jahan, Aurangzeb. Sher Shah sits between Humayun's two reigns (1540-55).

⚡ Three Panipats: 26, 56, 61. 1526 Babur beats Ibrahim Lodi. 1556 Akbar's side (Bairam Khan) beats Hemu. 1761 Ahmad Shah Abdali beats the Marathas.

Marathas & Sikh Gurus

⚡ Guru order: Nana Aur Amar Ram Arjun, Har-Har-Har Tegh Gobind. Nanak, Angad, Amar Das, Ram Das, Arjan, then the three Hars (Hargobind, Har Rai, Har Krishan), then Tegh Bahadur, then Gobind Singh.

⚡ Chauth vs Sardeshmukhi. Chauth = Chaar ana in the rupee (1/4). Sardeshmukhi = 10%, the tenth claimed by the hereditary head.

Bhakti & Sufi Movements

⚡ Philosophy trio: RaVi MaD. Ramanuja gives Vishishtadvaita. Madhva gives Dvaita. Shankara gives Advaita.

⚡ Language of the epic. Tulsidas = Awadhi (A for Ayodhya's Ram). Surdas = Braj (Bal-Krishna).

Medieval Books & Foreign Travellers

⚡ Mughal chroniclers: Abul praises, Badauni bites. Abul Fazl wrote the official, praising *Akbarnama* and *Ain*. Badauni wrote the bitter, critical *Muntakhab-ut-Tawarikh*.

⚡ Travellers by century. 11th Al-Biruni, 13th Marco Polo, 14th Ibn Battuta, 15th Conti and Abdur Razzaq, 16th Paes and Nuniz, 17th Hawkins, Roe, Bernier and Tavernier.

Modern Indian History

Europeans in India & British Expansion

⚡ Plassey then Buxar: "57 buys, 64 seals". 1757 Plassey gave the British a foothold (Clive beat Siraj). 1764 Buxar sealed control (three rulers beaten). The Diwani followed in 1765.

⚡ Governor-General reforms in order. Cornwallis Settles (1793) → Wellesley Subsidises (1798) → Bentinck bans Sati (1829) → Ripon starts Local self-government (1882).

Revolt of 1857 and Peasant & Tribal Uprisings

⚡ 1857 pairs: "Kanpur Nana, Lucknow Lady, Jhansi Rani, Bihar Kunwar". Kanpur–Nana Saheb · Lucknow–Begum Hazrat Mahal (the Lady) · Jhansi–Rani Lakshmibai · Bihar (Jagdishpur)–Kunwar Singh, the old hero.

⚡ Tribal movements by first letter. Santhal = Sidhu (1855) · Munda = Birsa, at the millennium's end (1899–1900).

Socio-Religious Reform Movements

⚡ 1875: the year of three. Arya Samaj (Dayanand), Theosophical Society (Blavatsky and Olcott) and the Aligarh MAO College (Sir Syed) all date to 1875.

⚡ Founder hooks by first letter. Ram Mohan = Reform of sati (Brahmo, 1828) · Dayanand = Discover the Vedas (Arya) · Phule = People of lower castes (Satyashodhak) · Vivekananda = Vedanta and service (Ramakrishna Mission, 1…

Congress, Moderates & Extremists (1885–1916)

⚡ Five-six-seven-nine: "Split, League, Split, Separate". 1905 Bengal split → 1906 Muslim League → 1907 Congress split (Surat) → 1909 separate electorates (Morley–Minto).

⚡ Home Rule twins: "TAB then BSM". Tilak first (April, Belgaum) → Besant next (September, Madras).

Gandhian Era & Road to Independence (1915–1947)

⚡ Gandhi's first three: "C-A-K, 17-18-18". Champaran 1917 (indigo, Bihar) → Ahmedabad 1918 (mill workers, hunger strike) → Kheda 1918 (farmers, revenue).

⚡ Session presidents: "J-P-B" from 29 to 39. Lahore 1929 = Jawaharlal (Purna Swaraj) · Karachi 1931 = Patel (Fundamental Rights) · Tripuri 1939 = Bose.

⚡ Dandi numbers. 12 March start · 6 April salt law broken · 78 marchers · about 385 km (240 miles) · from Sabarmati Ashram.

Newspapers, Journals & Books of the Freedom Era

⚡ Gandhi's papers: "YNH". Young India · Navajivan · Harijan (plus *Indian Opinion* from South Africa). Any other paper in the options belongs to someone else.

⚡ Books written in jail. Tilak → *Gita Rahasya* in Mandalay. Nehru → *The Discovery of India* in Ahmednagar Fort.

Art & Culture

Classical Dances & Their Exponents

⚡ Kerala has two, the rest one each. Kerala gives Kathakali and Mohiniyattam. Every other dance has one home state: TN Bharatanatyam, AP Kuchipudi, Odisha Odissi, Manipur Manipuri, Assam Sattriya, UP Kathak.

⚡ Exponent hooks. Birju is Kathak. Kelucharan is Odissi (Konark poses). Rukmini Devi revived Bharatanatyam (Kalakshetra). Vempati is Kuchipudi (village). Vallathol is Kerala Kalamandalam (Kathakali).

Folk Dances by State

⚡ North-East quick five. Cheraw is Mizo. Hojagiri is Tripura. Nongkrem is Khasi (Meghalaya). Wangala is Garo (Meghalaya). Bagurumba is Bodo (Assam).

⚡ Men and women in Punjab. Bhangra goes with Bhaiya (men). Giddha goes with Girls (women).

Music, Instruments & Maestros

⚡ Carnatic Trinity. T for Tyagaraja, M for Muthuswami Dikshitar, S for Syama Sastri. Purandara Dasa is the father, not part of the Trinity.

⚡ Instrument classes. Tata is strings. Sushira is air, so wind. Avanaddha is skin drums. Ghana is solid, such as bells, cymbals and ghatam.

Paintings, Textiles & Handicrafts

⚡ Kalam, Patta, Phad. Kalamkari is Andhra (the pen, kalam). Pattachitra is Odisha (cloth, patta). Phad is Rajasthan (Pabuji scroll). Madhubani is Bihar (Mithila).

⚡ Bidri is Bidar. The craft carries its town's name. Bidriware comes from Bidar (Karnataka). Patola comes from Patan. Chanderi comes from Chanderi.

Temple & Rock-cut Architecture

⚡ Shikhara North, Gopuram South. A gopuram means Dravida. A curved shikhara means Nagara. A star-shaped Hoysala plan means Vesara (Deccan).

⚡ Odisha pagodas. Black Pagoda is Konark (dark stone, Sun temple). White Pagoda is Puri (whitewashed Jagannath).

Festivals & Fairs

⚡ Hornbill closes the year. The Hornbill Festival runs 1-10 December. Nagaland Statehood Day is 1 December.

⚡ Two cattle fairs. Pushkar is in Rajasthan, the Pink City state, for camels. Sonepur is in Bihar, for cattle and elephants.

UNESCO Heritage & Cultural Institutions

⚡ WHS count ladder. 2021 gives 40 (Dholavira). 2023 gives 42 (Hoysala). 2024 gives 43 (Moidams). 2025 gives 44 (Maratha forts). 2026 gives 45 (Sarnath).

⚡ Three Akademis. Sangeet Natak comes first (1953). Sahitya and Lalit Kala come together in 1954.

Indian Polity & Constitution

Making of the Constitution & borrowed features

⚡ Committee to chairman. Nehru took Union Powers, Union Constitution and States. Patel took Provinces and Rights. Prasad took Rules, Steering and Flag. Ambedkar took Drafting.

⚡ Borrowed features speed-pair. UK: Parliament. USA: Rights. Ireland: DPSP. Canada: strong Centre. Australia: Concurrent List. USSR: Duties. France: Republic. Germany: Emergency. Japan: law procedure.

Preamble, Parts and the 12 Schedules

⚡ Schedule story-line. 1 Names, 2 Salaries, 3 Oaths, 4 RS seats, 5 SC/ST areas, 6 Tribal north-east, 7 Three Lists, 8 Languages, 9 Land laws, 10 Defection, 11 Panchayat (29), 12 Municipality (18).

⚡ Preamble keyword count. Sovereign, Socialist, Secular, Democratic, Republic make five descriptors. Only Justice splits into three (social, economic, political).

Fundamental Rights, DPSP and Fundamental Duties

⚡ FR article blocks. 14-18 Equality, 19-22 Freedom, 23-24 Exploitation, 25-28 Religion, 29-30 Culture and Education, 32 Remedies. The blocks hold 5, 4, 2, 4, 2 and 1 articles.

⚡ Duty amendment pair. The 42nd Amendment gave ten duties. The 86th gave the eleventh and also inserted Article 21A.

Union and State Executive

⚡ Electoral college contrast. The President is elected by elected MPs and elected MLAs. The Vice-President is elected by all MPs of both Houses, including the nominated ones.

⚡ Pardon ladder. The President's Article 72 covers court-martial and death sentences. The Governor's Article 161 covers neither.

Parliament and the Judiciary

⚡ Committee trio numbers. PAC has 22, Public Undertakings has 22, Estimates has 30. Estimates is drawn only from the Lok Sabha.

⚡ Writ first letters. H-M-P-C-Q stands for Habeas, Mandamus, Prohibition, Certiorari, Quo warranto. Prohibition stops a case early. Certiorari quashes after.

Important Constitutional Amendments

⚡ Decade clusters. 1950s structural (1st, 7th). 1970s power (24, 42, 44). 1980s politics (52, 61). 1990s grass roots (73, 74). 2010s and 2020s economy and equality (101 GST, 103 EWS, 106 women).

⚡ 42 versus 44. The 44th undid parts of the 42nd. Lok Sabha term went from 6 back to 5 years. 'Internal disturbance' became 'armed rebellion'. Property moved from a right to Article 300A.

Panchayati Raj, Municipalities and Emergency

⚡ Committee tier rhyme. Balwant-3 (1957), Ashok-2 (1977), Rao-district (1985), Singhvi-Constitution (1986). Remember the years as 57, 77, 85, 86.

⚡ Emergency digits. 352 is National, with three proclamations (1962, 1971, 1975). 356 is President's Rule, first used in Kerala in 1959. 360 is Financial, used zero times.

Constitutional & Statutory Bodies (with current heads)

⚡ Article blocks for bodies. 76 AG, 148 CAG, 280 FC, 315 UPSC, 324 EC. Reading upward gives the order AG, CAG, FC, UPSC, EC.

⚡ Statutory versus constitutional filter. A dated Act (NHRC 1993, CVC 2003, CIC 2005, Lokpal 2013) means statutory. An Article number means constitutional. NITI Aayog is neither.

Indian Geography

Location, extent, neighbours and islands

⚡ Tropic of Cancer states: GRM CJ WTM. West to east: Gujarat, Rajasthan, Madhya Pradesh, Chhattisgarh, Jharkhand, West Bengal, Tripura, Mizoram. Odisha and Bihar are the usual wrong options. The Tropic misses both.

⚡ Border order: Big Cats Prowl Near My Big Area. Longest to shortest: Bangladesh, China, Pakistan, Nepal, Myanmar, Bhutan, Afghanistan.

⚡ Channel numbers fall as you go south-west. Ten Degree splits Andaman and Nicobar. Nine Degree splits Minicoy and Lakshadweep. Eight Degree splits Minicoy and the Maldives. The Andaman group has the biggest number.

Himalayas, peaks, passes, plateau, coasts and lakes

⚡ Pass to state by first letters. Zoji, Khardung, Karakoram: Ladakh. Rohtang, Shipki, Baralacha: Himachal. Lipulekh, Mana: Uttarakhand. Nathu, Jelep: Sikkim. Bomdi: Arunachal. Palghat: the Kerala-TN gap.

⚡ Highest-peak ladder. Claimed by India: K2. Administered by India: Kangchenjunga. South India: Anamudi. Nilgiris: Doddabetta. Aravallis: Guru Shikhar. Satpura: Dhupgarh.

⚡ Lake adjectives: Fresh Wular, Brackish Chilika, Salty Sambhar, Floating Loktak. Say it as one chain. Examiners swap the adjectives between the four lakes.

Rivers, dams and waterfalls

⚡ Panch Prayag: Very Nice Kids Read Daily. Vishnuprayag, Nandprayag, Karnaprayag, Rudraprayag, Devprayag, from upstream to downstream. At Devprayag the river becomes the Ganga.

⚡ West-flowers: Nobody Takes Mahi Seriously. Narmada, Tapi, Mahi and Sabarmati flow into the Arabian Sea and form estuaries. Narmada and Tapi flow through rift valleys.

⚡ Waterfall and river pairs. Jog with Sharavati. Chitrakote with Indravati. Dudhsagar with Mandovi. Athirappilly with Chalakudy. Shivanasamudra and Hogenakkal with Kaveri. Dhuandhar with Narmada.

Monsoon, local winds and soils

⚡ Black soil: LIMP is rich, NPH is poor. Rich in Lime, Iron, Magnesium, Potash. Poor in Nitrogen, Phosphorus, Humus.

⚡ Shower names follow the crop. Mango showers: mango, Kerala. Blossom showers: coffee, Karnataka. Kalbaisakhi: tea, jute, rice in Bengal and Assam. Western disturbances: rabi wheat in the north-west.

Agriculture, revolutions, minerals and ports

⚡ Revolution colours follow the product. White is milk. Blue is water and fish. Yellow is oilseeds (mustard flower). Silver is eggs. Round is potato. Green is grains.

⚡ Kharif crops need water. Water-loving rice, jute and cotton are kharif. Cool-weather wheat, gram and mustard are rabi. Summer melons are zaid.

National parks, tiger reserves and wildlife

⚡ Animal-first recall. Rhino: Kaziranga. Lion: Gir. Cheetah: Kuno. Snow leopard: Hemis. Hangul: Dachigam. Sangai: Keibul Lamjao. Nilgiri tahr: Eravikulam. Lion-tailed macaque: Silent Valley. Saltwater crocodile: Bhitarkanika. Olive Ridley: Gah…

⚡ Year chain 36-73-86-92. 1936 first NP (Hailey/Corbett). 1973 Project Tiger. 1986 first biosphere reserve (Nilgiri). 1992 Project Elephant.

World Geography

Solar system, Earth's motions, latitudes and time

Time difference from longitude

\Delta t = \Delta\lambda \times 4\ \text{minutes}

1° of longitude = 4 min; 15° = 1 hour

Local time

T_{\text{local}} = T_{\text{GMT}} \pm \frac{\lambda}{15}\ \text{h}

+ for east longitudes, − for west

⚡ Planet superlatives in one line. Mercury smallest and nearest. Venus hottest, brightest, spins backward. Mars red. Jupiter largest, with Ganymede. Saturn rings and lightest. Uranus on its side. Neptune farthest and windiest.

⚡ IST from the meridian. India's meridian is 82.5 E. 82.5 x 4 = 330 minutes, which is 5 h 30 min ahead of GMT.

Earth's interior, earthquakes, rocks and volcanoes

⚡ Discontinuity order: Come Make Good Lunch. Conrad (inside crust), Moho (crust and mantle), Gutenberg (mantle and core), Lehmann (outer and inner core). Repetti sits in the mantle between Moho and Gutenberg.

⚡ Metamorphic pairs. Lime to Marble, Sand to Quartz(ite), Shale to Slate, Granite to Gneiss. Marble from limestone is the most asked, for example the marble of the Taj Mahal.

Atmosphere, pressure belts, planetary and local winds

⚡ Layer order: The Strong Man Throws Eggs. Troposphere (weather), Stratosphere (ozone, jets), Mesosphere (coldest, meteors), Thermosphere (ionosphere, radio), Exosphere.

⚡ Local wind by country. Chinook: USA and Canada. Foehn: Alps. Sirocco: Sahara to Italy. Harmattan: West Africa. Mistral: France. Khamsin: Egypt. Santa Ana: California. Loo: India.

Oceans, currents, straits and canals

⚡ Cold current and desert pairs. Humboldt gives Atacama. Benguela gives Namib. West coasts with a cold current have dry air, so deserts form.

⚡ Strait to two countries. Gibraltar: Spain and Morocco. Hormuz: Iran and Oman. Bering: Russia and USA. Malacca: Malaysia and Indonesia. Palk: India and Sri Lanka. Dover: UK and France.

Continents, superlatives, grasslands and tribes

⚡ Grasslands: Pretty Pam Visits Steppe Down. Prairies (North America), Pampas (Argentina), Veld (South Africa), Steppes (Eurasia), Downs (Australia).

⚡ Lake superlatives. Caspian is largest (salt). Superior is largest freshwater. Baikal is deepest. Titicaca is highest navigable. The Dead Sea shore is the lowest land.

Nicknames and boundary lines

⚡ Sun pair. Rising Sun is Japan, in the east of Asia, where the Sun comes up first. Midnight Sun is Norway, in the Arctic summer.

⚡ Line and neighbours. Radcliffe split India and Pakistan. McMahon runs along the Himalayas with China. Durand divides Pakistan and Afghanistan.

Indian Economy

National income, growth and base years

NNP

NNP = GNP - \text{Depreciation}

GNP already includes net factor income from abroad

GDP deflator

\text{Deflator} = \frac{\text{Nominal GDP}}{\text{Real GDP}} \times 100

divide nominal by (deflator/100) to get real

Market price bridge

MP = FC + \text{Indirect taxes} - \text{Subsidies}

moving between factor cost and market price

⚡ G-D-N ladder. GDP (territory) + NFIA = GNP; GNP − depreciation = NNP. 'Territory → Nation → Net'.

⚡ New base-year pairs. Production takes financial years: GDP and IIP = 2022-23; prices of consumers take a calendar year: CPI = 2024; wholesale moved to 2022-23 with PPI twins in June 2026.

Five Year Plans and NITI Aayog

⚡ Plan-era story line. 1-farm (1951) → 2-machines (Mahalanobis) → 3-wars → holiday (66-69) → 4-stability + bank nationalisation → 5-Garibi Hatao → rolling (Janata) → 6-7 growth → 8-reforms (1992) → 9-12 inclusive → NITI (2015).

⚡ Plan-break tags. Two gaps: Plan Holiday 1966-69 (wars/drought) and Annual Plans 1990-92 (transition after the 8th was delayed).

Money, banking and the RBI

⚡ Corridor arithmetic. Repo is the middle: SDF = repo − 0.25 (floor), MSF = Bank Rate = repo + 0.25 (ceiling). With repo 5.25 → 5.00 / 5.25 / 5.50.

⚡ Nationalisation anchors. 1935 born, 1949 nationalised (RBI), 1955 SBI, 1969 fourteen, 1975 RRBs, 1980 six.

⚡ MPC fixed points. 6 members; 4 meetings a year minimum; target CPI 4% ± 2%; Governor's casting vote.

Inflation and price indices

⚡ Index-compiler pairs. CPI-IIP-GDP = MoSPI/NSO; WPI-PPI = DPIIT's Office of Economic Adviser. 'M for MoSPI, W for DPIIT(OEA)'.

⚡ Core = strip the volatile. Core inflation = headline − food − fuel. Stagflation = stagnation + inflation together.

Budget and fiscal policy

Fiscal deficit

FD = \text{Total Expenditure} - \text{Total Receipts (excl. borrowings)}

equals government borrowing requirement

Revenue deficit

RD = \text{Revenue Expenditure} - \text{Revenue Receipts}

revenue items only

Primary deficit

PD = FD - \text{Interest Payments}

strips out past borrowing costs

⚡ Budget 2026-27 number sheet. Spend 53.5 • Receipts 36.5 • Borrow 16.9 (₹ lakh crore); FD 4.3% • RD 1.5% • PD 0.7%; Capex 12.2; Debt 55.6%.

⚡ Article trio for money matters. 112-Budget • 266-Consolidated Fund • 267-Contingency Fund; 265-no tax without law.

Taxation and GST

⚡ GST article trio. 246A power • 269A inter-state • 279A Council. 101st Amendment, 1 July 2017.

⚡ GST 2.0 slab story. Two working slabs — 5 (merit) and 18 (standard) — with 40 kept aside for sin/luxury; 12% and 28% abolished on 22 Sep 2025.

⚡ Tax-year switch. Income-tax Act, 2025 → effective 1 April 2026; one 'tax year' replaces previous/assessment year pair; 536 sections, 23 chapters.

Government schemes and missions

⚡ Launch-year clusters. 2014: Jan Dhan, Make in India • 2015: BBBP, APY, MUDRA, PMAY • 2016: Ujjwala • 2018: Ayushman • 2019: KISAN • 2020: SVANidhi, PLI • 2023: Vishwakarma, Drone Didi • 2024: Surya Ghar • 2026: VB-G RAM G live (125 days).

⚡ MGNREGA → RAM G swap. 100 → 125 days; 15-day → weekly wages; Act of 2025, in force 1 July 2026.

Regulators, exchanges and international organisations

⚡ Regulator HQ pairs. RBI-SEBI-NABARD = Mumbai; IRDAI = Hyderabad; SIDBI = Lucknow; PFRDA = New Delhi.

⚡ Bretton-Woods twins. IMF + World Bank both born at Bretton Woods (1944), both in Washington DC; WTO (1995, Geneva) replaced GATT (1947).

⚡ Asian banks map. ADB 1966 → Manila; AIIB 2016 → Beijing; NDB (BRICS) 2015 → Shanghai. India founding member in all three.

Physics

SI units, conversions and measuring instruments

Kilowatt-hour

1\ \text{kWh} = 3.6 \times 10^{6}\ \text{J}

1 'unit' of electricity

Horsepower

1\ \text{hp} \approx 746\ \text{W}

power unit

Light year

1\ \text{ly} \approx 9.46 \times 10^{15}\ \text{m}

a unit of distance

⚡ Base units in one line. Say it as Metre, Kilogram, Second, Ampere, Kelvin, Mole, Candela. Anything not on this list is derived.

⚡ Read the root of the instrument name. Hygro is humidity. Hydro is liquid density. Lacto is milk. Anemo is wind. Sphygmo is pulse and blood pressure. Pyro is fire, so high temperature. Seismo is earthquake.

Motion, gravitation, work-energy, fluids and levers

First equation of motion

v = u + at

v is final speed, u is initial speed, a is acceleration, t is time

Second equation of motion

s = ut + \tfrac{1}{2}at^{2}

s is the distance covered

Third equation of motion

v^{2} = u^{2} + 2as

no time needed

Newton's second law

F = ma = \frac{\Delta p}{\Delta t}

p = mv

Law of gravitation

F = \frac{G m_1 m_2}{r^{2}}

G = 6.67 × 10⁻¹¹ N m² kg⁻²

Kinetic and potential energy

KE = \tfrac{1}{2}mv^{2},\quad PE = mgh

m is mass, v is speed, h is height

Work and power

W = Fs\cos\theta,\quad P = \frac{W}{t}

1 W = 1 J/s

Pressure

P = \frac{F}{A},\quad P_{\text{liquid}} = h\rho g

unit pascal

⚡ Lever class from the middle item. Whatever sits in the middle decides the class. F is 1, L is 2, E is 3.

⚡ Squares in energy. KE depends on v^2. Double the speed and KE is 4 times. Triple it and KE is 9 times. Momentum only doubles when speed doubles.

Heat, temperature and sound

Temperature scales

\frac{C}{100} = \frac{F-32}{180} = \frac{K-273}{100}

−40 °C = −40 °F

Heat absorbed

Q = mc\,\Delta T

c = specific heat

Latent heat

Q = mL

no temperature change during phase change

Wave speed

v = f\lambda

speed = frequency × wavelength

Echo distance

d = \frac{v\,t}{2}

sound travels to the wall and back

⚡ Sound speed order. Solids are fastest, then liquids, then gases. A vacuum has no sound at all.

⚡ Pitch and loudness. Pitch goes with Frequency. Loudness goes with Amplitude.

Light — mirrors, lenses, eye, dispersion and scattering

Power of a lens

P = \frac{1}{f\,(\text{m})} = \frac{100}{f\,(\text{cm})}

unit dioptre

Lens formula

\frac{1}{f} = \frac{1}{v} - \frac{1}{u}

Cartesian sign convention

Mirror formula

\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

f = R/2

Refractive index

n = \frac{c}{v}

c = 3 × 10⁸ m/s

⚡ Eye defect and its lens. Myopia goes with concave. Hypermetropia goes with convex. A short-sighted eye needs a diverging lens to push the image back onto the retina.

⚡ Mirror by job. Need a wide view? Use convex (rear-view). Need an enlarged image or a focused beam? Use concave (shaving, dentist, headlight).

Electricity, magnetism and the EM spectrum

Ohm's law

V = IR

V is voltage, I is current, R is resistance

Series resistance

R_s = R_1 + R_2 + \cdots

the same current flows through each

Parallel resistance

\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots

two resistors: R₁R₂/(R₁+R₂)

Electric power

P = VI = I^{2}R = \frac{V^{2}}{R}

P is power in watt

Joule heating

H = I^{2}Rt

t is time in seconds

Resistance of a wire

R = \rho\frac{L}{A}

ρ = resistivity

⚡ Fleming's hands. L comes before M, so Left goes with Motor. G is for generator, so Right goes with Generator.

⚡ Spectrum order. Good X-rays Use Visible Infra-Micro Radios. The wavelength rises along the list. Frequency and energy fall.

Nuclear physics, inventions and scientists

Mass–energy equivalence

E = mc^{2}

Einstein

⚡ Fission or fusion. Fission means split: heavy nucleus to lighter ones (reactors, atom bomb). Fusion means join: light nuclei to a heavier one (Sun, hydrogen bomb).

⚡ Particle discoverers. Electron: Thomson (1897). Nucleus: Rutherford. Neutron: Chadwick (1932). Proton rays: Goldstein.

Chemistry

Matter, separation methods and atomic structure

Moles from mass

n = \frac{m}{M}

M = molar mass in g/mol

Number of particles

N = n \times N_A,\quad N_A = 6.022 \times 10^{23}

Avogadro number

Boyle's law

P_1V_1 = P_2V_2

temperature constant

Charles's law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

pressure constant, T in kelvin

Ideal gas equation

PV = nRT

R = 8.314 J mol⁻¹ K⁻¹

Maximum electrons in a shell

2n^{2}

K = 2, L = 8, M = 18

⚡ Iso-words by the letter. Isotopes share the atomic number (P for protons). Isobars share A, the mass number. Isotones share neutrons.

⚡ Sublimation list: CNIAD. Camphor, Naphthalene, Iodine, Ammonium chloride, Dry ice. If an option is on this list, it sublimes.

Periodic table, elements and record-holders

⚡ Crust order: OSAIC. Oxygen, Silicon, Aluminium, Iron, Calcium. So the top element is O and the top metal is Al.

⚡ Two liquids at room temperature. Mercury is the liquid metal. Bromine is the liquid non-metal. Gallium and caesium melt just above room temperature.

Acids, bases, pH, common compounds and reactions

\text{pH} = -\log_{10}[\text{H}^{+}]

pH + pOH = 14 at 25 °C

Neutralisation

\text{Acid} + \text{Base} \rightarrow \text{Salt} + \text{H}_2\text{O}

Setting of Plaster of Paris

\text{CaSO}_4\cdot\tfrac{1}{2}\text{H}_2\text{O} + 1\tfrac{1}{2}\,\text{H}_2\text{O} \rightarrow \text{CaSO}_4\cdot 2\text{H}_2\text{O}

PoP → gypsum

⚡ Kitchen acid map. Vinegar-Acetic, Lemon-Citric, Tamarind-Tartaric, Tomato-Oxalic, Curd-Lactic, Apple-Malic, Ant-Formic. Matching first letters helps: Tamarind and Tartaric, Formica (Latin for ant) and Formic.

⚡ Vitriol colours by metal. Blue vitriol is copper sulphate. Green is iron (ferrous) sulphate. White is zinc sulphate. Oil of vitriol is H₂SO₄.

Metals, ores, metallurgy and alloys

Gold purity

\text{Purity}\,(\%) = \frac{\text{carat}}{24} \times 100

22 ct ≈ 91.67%

⚡ Reactivity series sentence. Please Stop Calling Me A Zebra, I Like Her Calling Me Smart Goat gives Potassium, Sodium, Calcium, Magnesium, Aluminium, Zinc, Iron, Lead, Hydrogen, Copper, Mercury, Silver, Gold.

⚡ Brass or bronze. Brass has zinc. Bronze has tin. Both have copper.

Carbon compounds, fuels, polymers, industry and pollution

Alkane / alkene / alkyne

C_nH_{2n+2},\quad C_nH_{2n},\quad C_nH_{2n-2}

Haber process

N_2 + 3H_2 \rightleftharpoons 2NH_3

iron catalyst

Complete combustion of methane

CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O

⚡ Process to product. Haber gives ammonia. Contact gives sulphuric acid. Ostwald gives nitric acid. Solvay gives soda. Catalysts: Haber Fe, Contact V₂O₅, Ostwald Pt.

⚡ Japanese diseases. Minamata goes with mercury. Itai-itai (bone pain) goes with cadmium.

Biology

Cell, organelles and genetics

⚡ Organelle nicknames: Power-Mito, Suicide-Lyso, Protein-Ribo, Packing-Golgi. These four nicknames answer most organelle questions.

⚡ Cell discoverers in order. Hooke saw the cell (1665, dead cork). Leeuwenhoek saw it alive. Brown saw the nucleus. Schleiden and Schwann made the theory. Virchow said cells come from cells.

Human body — organs, blood, digestion, glands

Body mass index

\text{BMI} = \frac{\text{mass (kg)}}{\text{height (m)}^{2}}

Mass in kilograms divided by height in metres squared. 18.5–24.9 is the normal range (WHO)

⚡ Blood group giving: O gives, AB takes. O has no A or B antigens, so it can give to all. AB has no anti-A or anti-B antibodies, so it can take from all.

⚡ Brain parts: Cerebrum thinks, Cerebellum balances, Medulla keeps you alive. Thinking and memory: cerebrum. Balance and posture: cerebellum. Heartbeat and breathing: medulla oblongata.

Vitamins, minerals and nutrition

⚡ Fat-soluble vitamins: KADE (or ADEK). A, D, E and K dissolve in fat and are stored in the body. B and C dissolve in water.

⚡ Deficiency pairs: A-Andha (night blind), B1-Beriberi, C-sCurvy, D-Deformed bones (rickets), K-Klotting. Add B3 with Pellagra ('3 D's) and B12 with Pernicious anaemia (cobalt).

Diseases, pathogens, vectors and vaccines

⚡ Bacterial diseases: TB, Cholera, Typhoid, Diphtheria, Tetanus, Leprosy, Plague, Pertussis. A disease in this list is bacterial. Polio, measles, AIDS, dengue, rabies, chickenpox and hepatitis are viral. Malaria and kala-azar are protozoan.

⚡ Mosquito map: Anopheles-Malaria, Aedes-Dengue, Culex-Filaria. Aedes also spreads chikungunya, yellow fever and Zika (day-biting). Only the female mosquito bites for blood.

Classification, plants, hormones and animal facts

Photosynthesis

6CO_2 + 6H_2O \xrightarrow[\text{chlorophyll}]{\text{sunlight}} C_6H_{12}O_6 + 6O_2

O₂ comes from water. Inputs are carbon dioxide and water. Outputs are glucose and oxygen

Aerobic respiration

C_6H_{12}O_6 + 6O_2 \rightarrow 6CO_2 + 6H_2O + \text{energy (ATP)}

The reverse of photosynthesis in terms of matter. It releases energy as ATP

⚡ Underground but not a root. Potato, ginger, turmeric and onion grow underground but are stems. They have nodes or buds (eyes). Carrot, radish, sweet potato and beetroot are true roots.

⚡ Five kingdoms: My Pet Fish Plays Alone. Monera, Protista, Fungi, Plantae, Animalia. Whittaker, 1969.

Branches of biology and ecology

10% law of energy transfer

E_{n} = E_{1} \times (0.1)^{\,n-1}

E₁ = energy at the producer level, n = trophic level. Each step keeps one tenth

⚡ Culture words by root. Api is bee (apiary). Seri is silk. Pisci is fish (Pisces). Viti is vine or grapes. Pomo is fruit (pomegranate). Olera is vegetables.

⚡ Study-of roots. Ornitho is bird. Ento is insect. Ichthyo is fish. Herpeto is creeping animals. Myco is fungus. Phyco is seaweed or algae. Onco is tumour. Nephro is kidney.

Static GK

Books & Authors

⚡ Booker ladder. Say it in order: Rushdie 81, Roy 97, Desai 06, Adiga 08. The books run Midnight's Children, God of Small Things, Inheritance of Loss, White Tiger. Geetanjali Shree (2022) won the International Booker, which is for transl…

⚡ Sports book hooks. Sunny Days is Sunil Gavaskar. Race of My Life is Milkha Singh, who ran races. Unbreakable is Mary Kom, the boxer. Ace Against Odds is Sania Mirza, the tennis player. Test of My Life is Yuvraj Singh, who fought cancer.

Sports: Trophies, Terms, Players & Venues

⚡ Badminton cups. Thomas Cup is the men's team event. Uber Cup is the women's team event. Sudirman Cup is mixed. In tennis, Davis is for men and Billie Jean King is for women.

⚡ Team size ladder. Count up from five: basketball 5, volleyball 6, kabaddi 7. Cricket, football and hockey are 11. Only polo (4) and rugby union (15) sit far from this band.

Awards & Honours

⚡ RRR 1954. The first Bharat Ratnas were Rajagopalachari, Radhakrishnan and Raman. Three Rs, one year: 1954.

⚡ Award decade ladder. 1950s: Bharat Ratna (1954). 1960s: Arjuna (1961), Jnanpith (1965), Phalke (1969). 1980s: Dronacharya (1985). 1990s: Khel Ratna (1991-92).

Important Days

⚡ Birthday days. Pair the day with the person. 12 Jan Vivekananda, 29 Aug Dhyan Chand, 5 Sep Radhakrishnan, 15 Sep Visvesvaraya, 14 Nov Nehru, 22 Dec Ramanujan.

⚡ Two constitution anchors. 26 Nov 1949 is adoption, so it is Constitution Day. 26 Jan 1950 is enforcement, so it is Republic Day.

National Symbols & Firsts

⚡ Women firsts hook. Sarojini Governs, Sucheta Chiefs. Governor is Sarojini Naidu (1947). Chief Minister is Sucheta Kripalani (1963). Both from UP.

⚡ Flag numbers. Length to width is 3 : 2. The chakra has 24 spokes. The designer is Pingali Venkayya.

Organisations, Capitals, Currencies & Parliaments

⚡ Geneva, Rome, Nairobi. Geneva holds health, trade and labour bodies: WHO, WTO, ILO. Rome holds food bodies: FAO, WFP, IFAD. Nairobi holds UNEP, the environment body.

⚡ Neighbour currencies. East of India: Bangladesh uses the taka, Bhutan the ngultrum and Myanmar the kyat. Nepal, Sri Lanka and Pakistan use the rupee.

Current Affairs — How to Prepare

How current-affairs questions are framed

⚡ Pairing drill. Read any news item and write its two halves — e.g. 'X award → field', 'Y summit → host'. Exams ask the halves, not the story.

Award categories to track — structure, not winners

⚡ Ladder memory. Civilian: Ratna > Vibhushan > Bhushan > Shri. Sports: Khel Ratna > Arjuna (players) > Dronacharya (coaches) — teachers below students is the joke that fixes the order.

Sports events and how tournament news is asked

⚡ Cycle split. Four-yearly: Olympics, Winter, CWG, Asian Games, FIFA, ODI WC. Two-yearly: T20 WC, Thomas/Uber. Annual: IPL, Ranji, Khelo India.

Summits, groupings and organisations to track

⚡ HQ trio. SCO — Beijing; SAARC — Kathmandu; BIMSTEC — Dhaka — the three most-swapped headquarters. ASEAN sits at Jakarta as the fourth.

Important days — national and UN framework

⚡ Person-day strings. Chain them by date: 12 Jan Vivekananda → 1 Jul B.C. Roy → 29 Aug Dhyan Chand → 5 Sep Radhakrishnan → 31 Oct Patel → 14 Nov Nehru → 22 Dec Ramanujan.

Indexes and reports — index ↔ publisher

⚡ UN vs NGO split. UN bodies: UNDP (HDI), SDSN (Happiness), WIPO (Innovation), IMF (WEO). NGOs: Hunger = Concern + Welthungerhilfe, Peace = IEP, Press = RSF, ASER = Pratham, Corruption = TI.

Computer Knowledge

Computer Fundamentals

Five generations of computers

⚡ Generation ladder mnemonic. Vacuum tube, Transistor, IC, Microprocessor, AI. Read the first letters as "Very Tiny ICs Made AI". Five words match five generations in order.

⚡ Firsts trio. ENIAC 1946 was built. UNIVAC-I 1951 was sold. Intel 4004 1971 shrank the CPU onto a chip. Remember 1946, 1951, 1971.

Types of computers

⚡ Size ladder. "Some Ministers Must Promise" gives Supercomputer, Mainframe, Minicomputer, Microcomputer. It runs from largest to smallest.

⚡ PARAM pin. PARAM is India's supercomputer series, built by C-DAC, Pune. PARAM 8000 came in 1991. Pair it with Vijay Bhatkar.

Basic organisation: CPU, memory, I/O

⚡ FDES chant. "Fat Dogs Eat Snacks" gives Fetch, Decode, Execute, Store. This is the order for every instruction.

⚡ ALU or CU. If the question says performs calculations or comparisons, answer ALU. If it says controls or coordinates, answer CU.

Pioneers and programming-language levels

⚡ Compiler or interpreter. Compiler means Complete first: whole program, fast run, all errors together. Interpreter means Instant: line by line, stops at the first error.

⚡ Who did what. Babbage Built the idea. Ada Added the first program. Turing Thought up the theory. Von Neumann Noted the stored program. Berners-Lee Browsed the Web.

Core abbreviations and full forms

⚡ ROM ladder. PROM, then EPROM, then EEPROM. Each step is easier to erase. UV light for EPROM, Electricity for EEPROM.

⚡ OCR, OMR, MICR trio. OCR reads Characters. OMR reads Marks. MICR reads Ink on cheques. Say Characters, Marks, Ink.

Hardware, Memory & Number Systems

Memory hierarchy, RAM/ROM and cache

⚡ Volatile means Vanishes. Volatile memories vanish when power goes: registers, cache, RAM. Everything you keep files on (ROM, SSD, HDD, pen drive) is non-volatile.

⚡ ROM family eraser. PROM is written once. EPROM is erased by UV light. EEPROM is erased electrically. More E's, more electricity.

⚡ Cache position. Any question that says 'memory between the CPU and main memory' points to cache. L1 is inside the core and fastest.

Memory units and conversions

Step multiplier

1\ \text{unit}_{next} = 2^{10} = 1024\ \text{units}_{prev}

KB->MB->GB->TB->PB all multiply by 1024

Byte identity

1\ \text{Byte} = 8\ \text{bits},\quad 1\ \text{Nibble} = 4\ \text{bits}

half a byte = nibble

Common ladders

1\ \text{GB} = 1024\ \text{MB} = 2^{20}\ \text{KB} = 2^{30}\ \text{B}

4 GB = 4096 MB = 2^22 KB

bits vs bytes

1\ \text{MB} = 8\ \text{Mb}

capital B = byte, small b = bit (speeds are usually bits)

⚡ The 1024 ladder. Every step up multiplies by 1024 (2^{10}), never 1000. Half a KB is 512 B.

⚡ Bit or byte speed check. Line speeds are in bits (Mbps). File sizes are in bytes (MB). To get MB per second, divide Mbps by 8.

Number systems: binary, octal, decimal, hexadecimal

Positional value

N = \sum d_i \cdot b^{i}

digit d_i at position i (from 0, rightmost), base b

Decimal to base b

N = (\ldots r_2 r_1 r_0)_b\ \text{from repeated division by } b

read remainders bottom to top

Grouping shortcuts

1\ \text{octal digit} \leftrightarrow 3\ \text{bits},\quad 1\ \text{hex digit} \leftrightarrow 4\ \text{bits}

group binary from the RIGHT; pad with leading zeros

⚡ 3 and 4 grouping. Octal uses 3-bit groups. Hex uses 4-bit groups. Both start from the right. Pad the left with zeros. No division is needed.

⚡ Hex letter wheel. A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. So FF = 15 × 16 + 15 = 255, which is 11111111.

⚡ Power-of-2 positions. Memorise 1, 2, 4, 8, 16, 32, 64, 128. Then binary to decimal means adding the places that hold a 1.

Input and output devices

⚡ Marks, Characters, Ink. OMR is Marks (bubbles). OCR is Characters (text). MICR is Ink on cheques (banks). Match the noun in the question.

⚡ Impact means hit. If a printer physically hits the paper (dot matrix, daisy wheel), it is impact. It is noisy but can make copies. Inkjet, laser and thermal never strike the paper.

Ports and connectors

⚡ Serial is a single lane. Serial sends one bit at a time over few wires. It suits long distance. Parallel sends a whole byte across 8 wires. It suits short distance, like old printers.

⚡ VGA or HDMI. VGA carries video only and is analog with 15 pins. HDMI is digital and carries audio and video together. 'Sound and picture together' means HDMI.

Storage and backup media

⚡ Optical capacity ladder. CD is 700 MB. DVD is 4.7 GB. Blu-ray is 25 GB (single layer). Say it as '700, 4.7, 25'.

⚡ SSD or HDD in one line. SSD has no moving parts. It is silent, shock-proof, faster and costlier per GB. HDD has spinning platters and gives cheap bulk storage.

Software & Operating Systems

Types of software: system, application, utility

⚡ SUApp test. Ask who the program serves. The machine means system. Cleaning or protecting the machine means utility. The user's own job means application. Windows is system, antivirus is a utility, Excel is an application.

⚡ Licence four-word key. Freeware is free forever but closed. Shareware is a sample first. Open source has open code. Proprietary means pay.

Operating system functions, types and booting

⚡ PMF-SUIE services. The OS gives Process, Memory, File, Security, User interface and Error handling. A 'manage the machine' answer is OS work. Computing payroll is application work.

⚡ Boot words. Cold boot starts from a cold, switched-off machine. Warm boot restarts without a power cut. POST always comes first.

Windows OS features

⚡ Plug and Play. Plug and Play means the OS detects and sets up new hardware by itself. The phrase 'automatically detected' in a question points to PnP.

⚡ Task Manager shortcut. The direct route to Task Manager is Ctrl+Shift+Esc. Ctrl+Alt+Del opens a security menu first. Exams ask for the direct one.

File systems and file management

⚡ FAT32 four-GB wall. FAT32 cannot hold a single file bigger than 4 GB. This is the classic exam line. NTFS and exFAT do not have this limit.

⚡ Wildcards. The question mark masks exactly one character. The asterisk masks any number. So *.docx finds all Word files.

MS Word

Word interface, views and basics

⚡ Views ladder. Default view is Print Layout: what you print is what you see. Read Mode is for reading. Web Layout is the browser look. Outline shows headings. Draft is bare text.

⚡ Backstage tab. New, Open, Save, Print and Options live under the File tab. They are not on the Home ribbon.

Formatting text, paragraphs and pages

⚡ Alignment letters. Each shortcut letter starts the word: Left is Ctrl+L, Right is Ctrl+R, Justify is Ctrl+J. Centre breaks the pattern and uses E, because Ctrl+C is Copy.

⚡ Spacing numbers. The number in the shortcut is the spacing: Ctrl+1 is single, Ctrl+2 is double. Ctrl+5 is the odd one and means 1.5.

⚡ Case cycle. Press Shift+F3 again and again on selected text. Word cycles through the case styles. No menu is needed.

Mail merge, references, review and macros

⚡ No style, no TOC. A table of contents is built from heading styles. If headings are only bolded by hand, the TOC comes out empty.

⚡ Footnote is F, Endnote is D. Ctrl+Alt+F is Footnote, at the bottom of the page. Ctrl+Alt+D is enDnote, at the end of the document.

⚡ Merge field marks. Placeholders look like <<Name>>. The data comes from the data source. The layout lives in the main document.

File types and defaults

⚡ X means XML, 2007 and later. docx, xlsx and pptx are all 2007 and later. The three-letter doc, xls and ppt are 97-2003. The X stands for XML.

⚡ Font eras. Word 2003 used Times New Roman 12. Word 2007 to 2021 use Calibri 11. Microsoft 365 from 2023 uses Aptos 12.

MS Excel

Workbook, cells and references

⚡ Dollar locks what follows. A dollar sign freezes the letter or number right after it. $A$1 is frozen both ways. $A1 freezes the column only. A$1 freezes the row only.…

⚡ Count family. COUNT wants numbers only. COUNTA counts anything filled in. COUNTBLANK counts the empty ones. COUNTIF needs a condition.

Functions you must know

⚡ MOD is the leftover. MOD(a, b) is what is left after a is divided by b. MOD(17,5) is 2, and MOD(10,5) is 0.

⚡ MID has three inputs. MID(text, start, count). MID("COMPUTER",3,3) starts at the third letter and takes three letters. That gives MPU.

⚡ FALSE means exact. The last input of VLOOKUP decides how it matches. FALSE means exact match. TRUE means nearest smaller match.

Formula errors, charts and data tools

⚡ Read the error's name. DIV is division. NAME is a spelling problem. VALUE is a wrong type. REF is a reference that is gone. N/A is not available. NUM is a number problem.

⚡ Pie needs one series. A pie chart shows parts of one whole. It uses a single data series. Use line for trends and scatter for two-number links.

Excel shortcuts and file facts

⚡ Semicolon for date and time. Ctrl+; enters the date. Add Shift and it enters the time.

⚡ Space picks a line. Ctrl+Space picks the column. Shift+Space picks the row. Remember: Ctrl is the taller shape, a column.

⚡ D goes down, R goes right. Ctrl+D fills down. Ctrl+R fills right.

MS PowerPoint

Slides, placeholders and file types

⚡ Layout, theme, template. Layout arranges boxes. Theme sets colours and fonts. Template packages a theme and layouts in one file, .potx.

⚡ One change for all slides. To change all slides at once, edit the Slide Master on the View tab.

Views, slide master, transitions vs animations

⚡ Between or within. Transition happens between slides. Animation happens within a slide, on an object.

⚡ Four animation kinds. Entrance brings an object in, Emphasis makes it stand out, Exit takes it away, Motion Path moves it along a line.

Slideshow and editing shortcuts

⚡ Shift means here. F5 starts from the beginning. Add Shift and it starts from the current slide.

⚡ Colour keys. B makes the screen black and W makes it white. Press the same key again to bring the slide back.

⚡ New vs duplicate. Ctrl+M adds a fresh slide. Ctrl+D makes a copy of the selected one. Ctrl+N is a whole new presentation.

Internet, Browsers & E-mail

Internet, WWW, URL and DNS

⚡ Internet vs WWW. The Internet is the hardware network of cables, routers and servers. The WWW is the content service of linked pages on top of it. The Internet is older (1969). The Web is 1989.

⚡ DNS is the phonebook. DNS turns names into numbers. It never assigns IPs (that is DHCP) and never sends mail (that is SMTP). Any line that says 'converts domain names into IP addresses' means DNS.

Browsers, search engines, downloading and uploading

⚡ Browser is the car, engine is the GPS. Chrome, Edge, Firefox, Safari and Opera are browsers, the program you install. Google, Bing, Yahoo and DuckDuckGo are search engines, the websites you visit inside a browser.

⚡ Ctrl-letter map. T is a new Tab. W closes the tab. D is a Bookmark (Drop a pin). H is History. J is the downloads list. N is a new window, and Shift+N makes it incognito. F5 refreshes.

⚡ Up or down. Upload sends a file up to the server. Download brings it down to you. The direction is always relative to your machine.

E-mail: structure, To/CC/BCC and protocols

⚡ CC vs BCC. CC is a Carbon Copy that everybody sees. BCC is Blind, so To and CC receivers cannot see the blind list.

⚡ SMTP sends, POP picks, IMAP in-sync. SMTP = Send (outgoing). POP3 = Pick up on one device (server copy usually removed). IMAP = in all devices (server copy stays, everything syncs).

⚡ At sign. In [email protected] the @ separates who from where: user at domain. An option with no @, or with a space or comma, is wrong.

e-Banking and digital payments

⚡ NEFT vs RTGS. RTGS is real time, one by one, for large sums: minimum Rs. 2 lakh. NEFT settles in half-hourly batches, for any amount.

⚡ IFSC is 11. IFSC has 11 characters: 4 letters for the bank, a zero, then 6 characters for the branch. MICR is the 9-digit cheque strip.

⚡ 2FA needs two keys. Two-factor means a password (you know) plus an OTP or fingerprint (you have or are). One stolen password is not enough.

Networking: Devices, Topologies & Protocols

Network types and topologies

⚡ Mesh half-row sum. Full-mesh links for n nodes are n(n-1)/2. Compute it each time. Do not memorise per-n values.

⚡ PLMW span ladder. PAN < LAN < MAN < WAN: Personal, Local, Metropolitan, Wide. The Internet is the largest WAN.

⚡ Star means central device. If the question says the central device fails and the whole network stops, it describes star topology.

Network devices

⚡ Hub, switch, router ladder. Hub hears and shouts to all (layer 1). Switch is selective, by MAC (layer 2). Router routes between networks by IP (layer 3). 'Broadcasts to all' means hub. 'MAC address table' means switch. 'Connects two different netwo…

⚡ Modem is modulator plus demodulator. The full form is the answer. Digital data is modulated onto an analog carrier and demodulated back. 'Digital to analog conversion' in a question means modem.

OSI and TCP/IP reference models

⚡ OSI mnemonics. Bottom-up: Please Do Not Throw Sausage Pizza Away (Physical, Data link, Network, Transport, Session, Presentation, Application). Top-down: All People Seem To Need Data Processing.

⚡ Layer-device pairs. Router is layer 3. Switch and bridge are layer 2. Hub and repeater are layer 1. Use the address type: IP is 3, MAC is 2, no address is 1.

IP addressing: IPv4 classes and IPv6

⚡ Class ruler. Remember the cut-points 1, 128, 192, 224, 240. A starts at 1, B at 128, C at 192, D (multicast) at 224, E at 240. The number 127 is loopback, between A and B.

⚡ Loopback is 127. 127.0.0.1 is localhost. Pinging it tests your own TCP/IP stack, not the cable. Any question with 127.x wants loopback.

⚡ IPv4 vs IPv6 size. IPv4 is 32 bits in dotted decimal. IPv6 is 128 bits in colon-separated hexadecimal. If addresses are running out, IPv6 is the fix.

Protocols and port numbers; TCP vs UDP

⚡ Port anchors. Memorise six first: 80 HTTP, 443 HTTPS, 25 SMTP, 53 DNS, 110 POP3, 21 FTP. Then 22 SSH and 23 Telnet. Telnet is the insecure one.

⚡ TCP tracks, UDP does not. TCP tracks every packet, so it is reliable but slower. UDP is unreliable but quick. 'Guaranteed delivery' in a question means TCP.

Cyber Security

Malware types

⚡ Host test. A virus needs a host file. A worm needs none. A trojan copies nothing.

⚡ Money test. If files are locked and money is demanded, it is ransomware. WannaCry in 2017 is the standard example.

Cyber attacks and social engineering

⚡ V for voice, S for SMS. Vishing uses voice calls. Smishing uses SMS. Both are phishing with a different bait.

⚡ Pharm needs no click. In pharming you type the right address and still reach a fake site. The address system was tampered with.

Preventive measures and security tools

⚡ Door and room. A firewall guards the door, so it watches traffic. Antivirus cleans the room, so it removes malware inside.

⚡ Two factors. 2FA needs something you know, such as a password, plus something you have or are, such as an OTP or fingerprint.

IT Act 2000, offences and authorities

⚡ 66 family. 66 is hacking, 66C is identity theft, 66D is cheating by personation, 66E is privacy violation and 66F is cyber terrorism.

⚡ 66A is gone. If Section 66A is offered as a valid law, it is a trap. The Supreme Court struck it down in 2015.

Keyboard Shortcuts

Windows 10/11 and File Explorer shortcuts

⚡ First-letter rule. E is Explorer, L is Lock, R is Run, D is Desktop, S is Search, I is settIngs. The letter hints at the job.

⚡ Direct vs menu. Ctrl+Shift+Esc opens Task Manager straight away. Ctrl+Alt+Del first opens a menu, and Task Manager is one option there.

Universal Ctrl keys and MS Word shortcuts

⚡ N-O-S-P and X-C-V. New, Open, Save, Print are N, O, S, P. Cut, Copy, Paste are X, C, V. Undo and Redo are Z and Y.

⚡ App decides. Ctrl+H is Replace in Word and History in a browser. Look for the app name in the question before choosing.

MS Excel and PowerPoint shortcuts

⚡ Space pair. Ctrl+Space selects a column. Shift+Space selects a row.

⚡ Semicolon pair. Ctrl+; enters the date. Adding Shift enters the time.

⚡ Shift means here. F5 starts from the first slide. Shift+F5 starts from the slide you are on.

Web browser shortcuts (Chrome, Edge, Firefox)

⚡ Tab trio. T opens a tab, W closes it and Shift+T reopens the one you closed.

⚡ H, J, D. H is History, J is downloads (the next letter) and D is a bookmark (Dog-ear a page).

Function keys F1-F12

⚡ Alt closes programs. Alt+F4 closes the whole program. Ctrl+F4 closes only the current document or tab.

⚡ Twelve is Save As. F12 opens Save As in Office. In a browser the same key opens developer tools.