🔢 Quant

Algebra

Equations, inequalities, functions, logarithms. Appears heavily in CAT Quant.

34%

of Quant

Why This Topic Matters

Total PYQs📊

111

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~34%

of section questions

Est. hours⏱️

~30h

to master

~7/22

2021

7/22

2022

~8/22

2023

~9/22

2024

~8/22

2025

The second pillar of CAT Quant

Algebra is about 7 of the 22 Quant questions in a slot (34%) — second only to arithmetic, and often the deciding ground for a 99-percentile Quant score. The sub-topics — linear equations, quadratics, inequalities, indices & logarithms, sequences & series, functions — each have a dedicated page (functions is covered right here). This overview arms you with the high-frequency tools and the graph-reading that turns a hard algebra question into a one-line answer.

What CAT 2021–2025 actually asked

Sub-skill	2021	2022	2023	2024	2025	Avg/slot
Equations (linear, quadratic & polynomial)	2.0	2.3	3.0	2.7	3.3	2.7
Progressions & series	1.0	1.3	2.0	1.0	1.3	1.3
Functions	0.7	2.0	0.3	1.3	1.0	1.1
Inequalities & modulus	0.7	1.3	0.7	1.7	0.7	1.0
Logarithms	1.0	–	1.0	0.7	1.0	0.7
Surds & indices	–	–	0.7	1.0	–	0.3

🎯PYQ Evidence

Four fixtures, every single year: equations (the biggest single block in all of Quant — and growing: from ~2 to ~3 per slot across 2021–2025), progressions (~1–2 per slot, never absent), functions (every year, ~1 per slot) and inequalities (~1–2 per slot). Logarithms skipped only 2022. And 2025 brought a first: a linear-programming/optimisation question. If your algebra prep is "quadratics plus formulas," you're missing the half of CAT algebra that lives in functions, progressions and inequalities.

Quadratics — the most-tested object

For $ax^2+bx+c=0$ :

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

You rarely need the formula, though, because Vieta's relations answer most CAT questions directly:

$\text{sum of roots}=-\frac{b}{a},\qquad \text{product of roots}=\frac{c}{a}$

The discriminant $D=b^2-4ac$ tells you the nature of the roots at a glance:

$D$	Roots	Parabola vs x-axis
$D>0$	two distinct real	cuts at two points
$D=0$	one repeated real	just touches
$D<0$	none real (complex)	never meets

Read the graph, skip the algebra

A parabola $y=ax^2+bx+c$ opens up if $a>0$ (has a minimum) and down if $a<0$ (a maximum), with its vertex at $x=-\dfrac{b}{2a}$ .

For modulus / absolute value, $y=|x-a|$ is a V with its corner at $(a,0)$ . A CAT favourite: the sum of distances $|x-a|+|x-b|$ has minimum value $\mathbf{|a-b|}$ , attained anywhere on the interval $[a,b]$ .

Functions — the every-year topic without a formula sheet

About one question per slot, and they all reduce to four patterns:

Evaluate a composition: $f(g(x))$ — work inside-out, and never assume $f(g(x)) = g(f(x))$ .
Functional equations: given a rule like $f(x+y)=f(x)f(y)$ , plug small smart values ( $x=0$ , $y=1$ , $x=y$ ) to force the form.
Iterated functions: $f(f(f(x)))$ — compute one layer at a time and look for a cycle (the values usually repeat with period 2 or 3).
Max/min of a defined function — usually a disguised parabola or modulus-sum; sketch it.

💡Exam Tip

In functional-equation questions, the substitution $x = y$ and the substitution $x = 0$ (or $1$ ) crack the vast majority. Write what each substitution gives you before hunting anything fancier.

Inequalities — the wavy-curve method

To solve a factored inequality like $(x-1)(x-3)(x+2)>0$ : mark the roots on a line, and starting from the far right (always +), alternate signs across each root. Pick the intervals matching the inequality. (A repeated factor of even multiplicity does not flip the sign.)

A worked example

✏️Worked Example

Find the minimum value of $f(x)=|x-2|+|x-7|$ .

This is the sum of distances from $x$ to $2$ and to $7$ . By the rule above, the minimum is the gap between them:

$\min f(x)=|7-2|=\mathbf{5},\quad\text{for every }x\in[2,7].$

No calculus, no casework — just the geometry of the modulus.

Logarithms — the three laws that cover CAT

$\log_b(xy)=\log_b x+\log_b y,\quad \log_b\frac{x}{y}=\log_b x-\log_b y,\quad \log_b(x^n)=n\log_b x$

…plus the change of base $\log_b x=\dfrac{\log x}{\log b}$ . Almost every CAT log question is one of these in disguise.

⚠️CAT Trap

Domain first, algebra second. Log and root questions hide their trap in the domain: $\log$ needs a positive argument and a positive base $\ne 1$ ; an even root needs a non-negative radicand. CAT's wrong options are usually the solutions you'd get by manipulating first and checking never. Before solving, write the domain; after solving, test each root against it.

Where to go deeper

Dedicated pages: Linear Equations, Quadratic Equations, Inequalities & Modulus, Indices & Logarithms, Sequences & Series.

🎯PYQ Evidence

Three algebra reflexes: a sum of squares pins values, a common base linearises, and floors group. : move everything to one side until it reads (x+2y)² + (x−2y−1)² = 0; since real squares can't be negative, each must vanish, forcing x−2y = 1. : rewrite every term in base 1/8 (note 32768 = 8⁵), equate exponents to get a quadratic in k, and read the sum of roots straight off as −b/a without solving. : ⌊√n⌋ stays constant between consecutive squares, taking value k on 2k+1 terms, so a₁+…+a₅₀ becomes a few block products instead of 50 additions. Recognise the structure first; the algebra is then almost mechanical.