CAT Algebra Questions & Solutions

A sample of real CAT Algebra past-year questions, each with a worked solution and the correct answer highlighted. AthenaPrep has 396 Algebra questions in all — sign up free to practise them timed.

1. Q1.CAT 2025

   A value of $c$ for which the minimum value of $f(x)=x^{2}-4cx+8c$ is greater than the maximum value of $g(x)=-x^{2}+3cx-2c$, is

   • $2$

   • $\dfrac{1}{2}$

   • $-\dfrac{1}{2}$

   • $-2$

2. Q2.CAT 2024

   The sum of the infinite series $\cfrac{1}{5}\left(\cfrac{1}{5} - \cfrac{1}{7}\right) + \left(\cfrac{1}{5}\right)^2 \left(\left(\cfrac{1}{5}\right)^2 - \left(\cfrac{1}{7}\right)^2\right) + \left(\cfrac{1}{5}\right)^3 \left(\left(\cfrac{1}{5}\right)^3 - \left(\cfrac{1}{7}\right)^3\right) + ......$ is equal to

   • $\cfrac{7}{816}$

   • $\cfrac{5}{408}$

   • $\cfrac{7}{408}$

   • $\cfrac{5}{816}$

3. Q3.CAT 2023

   A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is

   Answer: 340

4. Q4.CAT 2022

   The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is

   Answer: 548

5. Q5.CAT 2020

   The number of distinct real roots of the equation $(x+\frac{1}{x})^{2}-3(x+\frac{1}{x})+2=0$ equals

   Answer: 1

6. Q6.CAT 2019

   The number of common terms in the two sequences: 15, 19, 23, 27, . . . . , 415 and 14, 19, 24, 29, . . . , 464 is

   • 21

   • 20

   • 18

   • 19

7. Q7.CAT 2017

   If a and b are integers of opposite signs such that $(a + 3)^{2} : b^{2} = 9 : 1$ and $(a -1)^{2}:(b - 1)^{2} = 4:1$, then the ratio $a^{2} : b^{2}$ is

   • 9:4

   • 81:4

   • 1:4

   • 25:4

8. Q8.CAT 2007

   A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value of f (x) at x = 10?

   • -119

   • -159

   • -110

   • -180

   • -105

9. Q9.CAT 2004

   Let $u = ({\log_2 x})^2 - 6 {\log_2 x} + 12$ where x is a real number. Then the equation $x^u = 256$, has

   • no solution for x

   • exactly one solution for x

   • exactly two distinct solutions for x

   • exactly three distinct solutions for x

10. Q10.CAT 2002

    Let S denotes the infinite sum $2 + 5x + 9x^2 + 14x^3 + 20x^4 + ...$ , where |x| < 1 and the coefficient of $x^{n - 1}$ is n( n + 3 )/2 , ( n = 1, 2 , . . . ) . Then S equals:

    • $(2-x)/(1-x)^3$

    • $(2-x)/(1+x)^3$

    • $(2+x)/(1-x)^3$

    • $(2+x)/(1+x)^3$

11. Passage

    Directions for the next 2 questions:

    For real numbers x, y, let

    f(x, y) = Positive square-root of (x + y), if $(x + y)^{0.5}$ is real
    f(x, y) = $(x + y)^2$; otherwise

    g(x, y) = $(x + y)^2$, if $\sqrt{(x + y)}$ is real
    g(x, y) = $- (x + y)$ otherwise

    Q11.CAT 2000

    Under which of the following conditions is f(x, y) necessarily greater than g(x, y)?

    • Both x and y are less than -1

    • Both x and y are positive

    • Both x and y are negative

    • $y>x$

12. Q12.CAT 1996

    Given the quadratic equation $x^2 - (A - 3)x - (A - 2)$, for what value of $A$ will the sum of the squares of the roots be zero?

    • -2

    • 3

    • 6

    • None of these

CAT Algebra previous year questions with solutions — AthenaPrep