CAT 2026 Slot 1 Quant: Algebra Questions with Solutions

4 questions: Q5, Q11, Q16, Q19. Solve each one here: tap the answer, press Check, then open the solution.

Question 5Algebra – Quadratics & Functions

MCQ · +3 / −1

Let $f(x) = x^2 + ax + b$ and $g(x) = f(x + 1) - f(x - 1)$. If $f(x) \ge 0$ for all real $x$ and $g(15) = 48$, then the smallest possible value of $b$ is:

Answer and solution

Correct answer: C

Solution

$f(x+1) = x^2 + 2x + 1 + ax + a + b$
$f(x-1) = x^2 - 2x + 1 + ax - a + b$
$g(x) = f(x+1) - f(x-1) = 4x + 2a$
$g(15) = 60 + 2a = 48 \Rightarrow a = -6$
$f(x) = x^2 - 6x + b = (x - 3)^2 + b - 9$
Its least value is $b - 9$ (at $x = 3$). For $f(x) \ge 0$ always: $b - 9 \ge 0 \Rightarrow b \ge 9$
Smallest $b = 9$

Question 11Indices

MCQ · +3 / −1

Find $n$ if $7^{3n-1} - 343^{\,n-1} = 2352$.

Answer and solution

Correct answer: D

Solution

$343 = 7^3$, so $343^{\,n-1} = 7^{3n-3}$
$7^{3n-1} - 7^{3n-3} = 7^{3n-3}(7^2 - 1) = 48 \cdot 7^{3n-3} = 2352$
$7^{3n-3} = 49 = 7^2 \Rightarrow 3n - 3 = 2 \Rightarrow n = \dfrac{5}{3}$

Question 16Algebra – Equations

MCQ · +3 / −1

Let $c$ and $k$ be positive integers such that $c^2 + ck + c = 40$ and $3k^2 + ck + k = 100$. What is the value of $\dfrac{c + 2k}{2}$?

Answer and solution

Correct answer: D

Solution

First equation: $c(c + k + 1) = 40$. Try values of $c$:
$c = 1 \Rightarrow k = 38$ ✗; $c = 2 \Rightarrow k = 17$ ✗; $c = 3 \Rightarrow k = \tfrac{28}{3}$ ✗
$c = 4 \Rightarrow 4k = 20 \Rightarrow k = 5$. Check: $3(25) + 20 + 5 = 100$ ✓
$\dfrac{c + 2k}{2} = \dfrac{4 + 10}{2} = 7$

Question 19Algebra – Linear Equations

TITA (type the answer) · +3 / 0

A donation box can receive only cheques of ₹100, ₹250 and ₹1000. On the last day of the month, the box contained exactly 90 cheques totalling ₹42600. If the number of ₹1000 cheques was the maximum possible, what is the sum of the numbers of ₹100 and ₹250 cheques?

Your answer

Answer and solution

Correct answer: 53

Solution

Let the counts be $x$ (₹100), $y$ (₹250), $z$ (₹1000).
$x + y + z = 90$ and $100x + 250y + 1000z = 42600 \Rightarrow 2x + 5y + 20z = 852$
Subtract $2(x + y + z) = 180$: $3y + 18z = 672 \Rightarrow y = 224 - 6z$
$x = 90 - y - z = 5z - 134$
$x \ge 0 \Rightarrow z \ge 27$; $y \ge 0 \Rightarrow z \le 37$. Maximum $z = 37$
$x = 51,\; y = 2 \Rightarrow x + y = 53$

An expected paper by Cetking, not the actual CAT paper. CAT is conducted by the IIMs.