Algebra
🔒 Log in to trackIdentities, expressions, , surds, linear equations and simple max–min. Algebra gives 2–4 marks-rich questions in every CGL shift, and almost all of them fall to a handful of identities plus value-putting — the highest return per hour of study in advanced maths.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
4 exam-level questions worked step by step.
78 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (78 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Given x + 1/x, find x³ + 1/x³
very common"If x + 1/x = 6, find x³ + 1/x³."
For x² + 1/x²: square the given value, then subtract 2.
For x³ + 1/x³: cube the given value, then subtract 3 times the value.
If x + 1/x = 6, then x³ + 1/x³ is:
Cube the value: 6 × 6 × 6 = 216
Subtract 3 × 6 = 18
216 − 18 = 198
Given a + b and ab, find a³ + b³
very common"If a + b = 7 and ab = 12, find a³ + b³."
Do not solve for a and b.
Cube (a + b), then subtract 3 × ab × (a + b).
If a + b = 7 and ab = 12, then a³ + b³ is:
Cube of the sum: 7 × 7 × 7 = 343
3 × ab × (a + b) = 3 × 12 × 7 = 252
343 − 252 = 91
Value with a root, like 2 + √3
very commonx is given as a number with a root, e.g. x = 2 + √3, and you must find x² + 1/x².
Find 1/x. It is usually the same number with the sign flipped (2 − √3).
Add them to get x + 1/x, then square it and subtract 2.
If x = 2 + √3, then x² + 1/x² is:
1/x = 2 − √3 (because (2 + √3)(2 − √3) = 4 − 3 = 1)
x + 1/x = 2 + √3 + 2 − √3 = 4
Square: 4 × 4 = 16
16 − 2 = 14
a³ + b³ + c³ − 3abc
common"If a + b + c = 6 and ab + bc + ca = 11, find a³ + b³ + c³ − 3abc."
If a + b + c = 0, then a³ + b³ + c³ is simply 3abc.
Otherwise: (a + b + c) × (a² + b² + c² − ab − bc − ca).
If a + b + c = 6 and ab + bc + ca = 11, then a³ + b³ + c³ − 3abc is:
a² + b² + c² = 6 × 6 − 2 × 11 = 14
Bracket: 14 − 11 = 3
6 × 3 = 18
Squares adding up to zero
commonOne equation with x², y² and plain x, y terms, all equal to 0.
Turn it into perfect squares like (x − 4)².
A sum of squares is 0 only when each square is 0.
If x² + y² − 8x + 6y + 25 = 0, then x − y is:
x² − 8x + 16 = (x − 4)² and y² + 6y + 9 = (y + 3)² (16 + 9 = 25)
(x − 4)² + (y + 3)² = 0, so x = 4 and y = −3
x − y = 4 − (−3) = 7
Big cubes over a matching square
commonA fraction with cubes on top and a matching three-term expression below.
Match it to the pattern (p³ + q³) ÷ (p² − pq + q²) = p + q.
For (p³ − q³) ÷ (p² + pq + q²), the answer is p − q.
The value of (4.7³ + 2.3³) ÷ (4.7² − 4.7 × 2.3 + 2.3²) is:
Top has p³ + q³, with p = 4.7 and q = 2.3
Bottom is p² − pq + q², the matching pattern
So the fraction = p + q = 4.7 + 2.3 = 7
Remainder when dividing by (x − a)
occasional"Find the remainder when x³ − 3x² + 4x − 5 is divided by (x − 2)."
Find the x that makes the divisor zero (x − 2 = 0 gives x = 2).
Put that value in the expression. The result is the remainder.
The remainder when x³ − 3x² + 4x − 5 is divided by (x − 2) is:
x − 2 = 0, so x = 2
Put x = 2: 8 − 12 + 8 − 5
= −1
Greatest or least value
occasional"Greatest value of 7 + 4x − x²" or "least value of 4x + 9/x".
Write the expression as a square plus a number. A square is never negative.
For 4x + 9/x, the least value is 2 × √(4 × 9) = 12.
The greatest value of 7 + 4x − x² is:
7 + 4x − x² = 11 − (x − 2)²
(x − 2)² is never below 0
Best case is (x − 2)² = 0, at x = 2
Greatest value = 11 − 0 = 11