CAT Inequalities Questions & Solutions

A sample of real CAT Inequalities past-year questions, each with a worked solution and the correct answer highlighted. AthenaPrep has 59 Inequalities questions in all — sign up free to practise them timed.

1. Q1.CAT 2025

   Let $3\leq x\leq6$ and $\left[x^{2}\right] =\left[x\right]^{2}$ , where $[x]$ is the greatest integer not exceeding $x$ . If set $S$ represents all feasible values of $x$, then a possible subset of $S$ is

   • $\left(3,\sqrt{10}\right)\cup \left[5,\sqrt{26}\right)\cup \left\{6\right\}$

   • $\left[3,\sqrt{10}\right] \cup \left[5,\sqrt{26}\right]$

   • $\left[3,\sqrt{10}\right]\cup \left[4,\sqrt{17}\right]\cup \left\{6\right\}$

   • $\left(4,\sqrt{18}\right)\cup\left[5,\sqrt{27}\right)\cup\left\{6\right\}$

2. Q2.CAT 2025

   If a,b,c and d are integers such that their sum is 46, then the minimum possible value of $(a-b)^{2}+(a-c)^{2}+(a-d)^{2}$ is

   Answer: 2

3. Q3.CAT 2023

   If $x$ and $y$ are real numbers such that $x^{2} + (x - 2y - 1)^{2} = -4y(x + y)$, then the value $x - 2y$ is

   • 0

   • 1

   • -1

   • 2

4. Q4.CAT 2023

   The population of a town in 2020 was 100000. The population decreased by y% from the year 2020 to 2021, and increased by x% from the year 2021 to 2022, where x and y are two natural numbers. If population in 2022 was greater than the population in 2020 and the difference between x and y is 10, then the lowest possible population of the town in 2021 was

   • 72000

   • 74000

   • 73000

   • 75000

5. Q5.CAT 2022

   The number of distinct integer values of n satisfying $\frac{4-\log_{2}n}{3-\log_{4}n} < 0$, is

   Answer: 47

6. Q6.CAT 2021

   If n is a positive integer such that $(\sqrt[7]{10})(\sqrt[7]{10})^{2}...(\sqrt[7]{10})^{n}>999$, then the smallest value of n is

   Answer: 6

7. Q7.CAT 2019

   If x is a real number, then $\sqrt{\log_{e}{\dfrac{4x - x^2}{3}}}$ is a real number if and only if

   • $1 \leq x \leq 3$

   • $1 \leq x \leq 2$

   • $-1 \leq x \leq 3$

   • $-3 \leq x \leq 3$

8. Q8.CAT 2006

   The number of solutions of the equation 2x + y = 40 where both x and y are positive integers and x <= y is:

   • 7

   • 13

   • 14

   • 18

   • 20

9. Q9.CAT 2003

   If x, y, z are distinct positive real numbers the $(x^2(y+z) + y^2(x+z) + z^2(x+y))/xyz$ would always be

   • Less than 6

   • greater than 8

   • greater than 6

   • Less than 8

10. Q10.CAT 2001

    If x > 5 and y < -1, then which of the following statements is true?

    • $(x + 4y) > 1$

    • $x > -4y$

    • $-4x < 5y$

    • None of these

11. Q11.CAT 2000

    If x>2 and y>-1,then which of the following statements is necessarily true?

    • $xy>-2$

    • $-x<2y$

    • $xy<-2$

    • $-x>2y$

12. Passage

    For these questions the following functions have been defined.
    
    $la(x, y, z) = min (x+y, y+z)$
    $le(x, y, z) = max(x -y, y-z)$
    $ma (x, y, z) = \frac{1}{2} (le (x, y, z) + la (x, y, z))$
    Q12.CAT 1997

    For x=15, y=10 and z=9 , find the value of le(x, min(y, x-z), le(9, 8, ma(x, y, z)).

    • 5

    • 12

    • 9

    • 4

CAT Inequalities previous year questions with solutions — AthenaPrep