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 $g(10) = 44$, find the minimum value of $b$.

Answer and solution

Correct answer: A

Solution

$g(x) = f(x+1) - f(x-1) = 4x + 2a$.
$g(10) = 40 + 2a = 44 \Rightarrow a = 2$.
For $f(x) \ge 0$ for all real $x$, the discriminant must be $\le 0$:
$a^2 - 4b \le 0 \Rightarrow 4 - 4b \le 0 \Rightarrow b \ge 1$.
Minimum value of $b = 1$.

Question 11Indices

MCQ · +3 / −1

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

Answer and solution

Correct answer: D

Solution

$343^{n-1}=7^{3(n-1)}=7^{3n-3}$.
$7^{3n-1}-7^{3n-3}=7^{3n-3}(7^2-1)=7^{3n-3}\times48=2352$.
$7^{3n-3}=49=7^2 \Rightarrow 3n-3=2 \Rightarrow n=\dfrac{5}{3}$.

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 $c=4$: $4(4+k+1)=40\Rightarrow k=5$. Check 2nd: $3(25)+4(5)+5=100$ ✓. $c+k=9$

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 x=₹100 cheques, y=₹250, z=₹1000. $x+y+z=90$ and $100x+250y+1000z=\text{total}$. Solve system for valid integer solutions.

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