# 50+ CAT Inequalities Questions With Video Solutions

Inequalities is one of the few topics in the quantitative part, which can throw up tricky questions. The questions are often asked in conjunction with other sections like ratio and proportion, progressions etc. The questions from this topic appearing in the CAT exam can be time-consuming if a candidate does not have a good understanding of the concepts. It requires a good understanding of algebraic expressions and equations. Solving previous years' question papers is a great way to get familiar with the exam pattern and also check out the free CAT mocks and understand the types of questions that are likely to appear on the exam.

We have compiled all the questions from this topic that appeared in the past CAT question papers, shown below. You can download all these questions in a PDF format along with the video solutions explained by the CAT experts. Click the link below to download the CAT linear equation questions PDF with detailed video solutions.

## CAT Inequalities Questions Weightage

 Year Weightage 2023 4 2022 2 2021 5 2020 4 2019 1 2018 1

## CAT Inequalities Formulas PDF

To help aspirants, we have made available the CAT Inequalities formulas PDF, which provides a comprehensive list of formulas and tips for solving CAT Inequalities questions. As mentioned before, solving inequalities questions may consume time. Being well-versed in the formulas helps you solve these quickly in the actual CAT examination. Click on the link below to download the CAT Inequalities formulas PDF.

1. Inequalities Formulae :

If a$$x^{2}$$+bx+c < 0 then (x-m)(x-n) < 0, and if n > m, then m < x < n

If a$$x^{2}$$+bx+c > 0 then (x-m)(x-n) > 0 and if m < n, then x < m and x > n

If a$$x^{2}$$+bx+c > 0 but m = n, then the value of x exists for all values, except x is equal to m, i.e., x < m and x > m but x ≠ m

If a, x, b are positive, ax > b => x > $$\dfrac{b}{a}$$ and ax < b => x < $$\dfrac{b}{a}$$

2.  Properties of inequalities

For any three real numbers X, Y and Z; if X > Y then X+Z > Y+Z

If X > Y and

Z is positive, then XZ > YZ

Z is negative, then XZ < YZ

If X and Y are of the same sign, $$\dfrac{1}{X}$$ < $$\dfrac{1}{Y}$$

If X and Y are of different signs, $$\dfrac{1}{X}$$ > $$\dfrac{1}{Y}$$

## CAT 2023 Inequalities questions

#### Question 1

Any non-zero real numbers x,y such that $$y\neq3$$ and $$\frac{x}{y}<\frac{x+3}{y-3}$$, Will satisfy the condition.

#### Question 2

Let n and m be two positive integers such that there are exactly 41 integers greater than $$8^m$$ and less than $$8^n$$, which can be expressed as powers of 2. Then, the smallest possible value of n + m is

#### Question 3

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

#### Question 4

Let n be any natural number such that $$5^{n-1} < 3^{n + 1}$$. Then, the least integer value of m that satisfies $$3^{n+1} < 2^{n+m}$$ for each such n, is

#### Question 5

The number of integer solutions of equation $$2|x|(x^{2}+1) = 5x^{2}$$ is

#### Question 6

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

#### Question 7

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is

#### Question 8

If a certain amount of money is divided equally among n persons, each one receives Rs 352. However, if two persons receive Rs 506 each and the remaining amount is divided equally among the other persons, each of them receive less than or equal to Rs 330. Then, the maximum possible value of n is

#### Question 9

If $$p^{2}+q^{2}-29=2pq-20=52-2pq$$, then the difference between the maximum and minimum possible value of $$(p^{3}-q^{3})$$

## CAT 2022 Inequalities questions

#### Question 1

The number of integer solutions of the equation $$\left(x^{2} - 10\right)^{\left(x^{2}- 3x- 10\right)} = 1$$ is

#### Question 2

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

## CAT 2021 Inequalities questions

#### Question 1

The number of distinct pairs of integers (m,n), satisfying $$\mid1+mn\mid<\mid m+n\mid<5$$ is:

#### Question 2

The number of integers n that satisfy the inequalities $$\mid n - 60 \mid < \mid n - 100 \mid < \mid n - 20 \mid$$ is

#### Question 3

For a real number x the condition $$\mid3x-20\mid+\mid3x-40\mid=20$$ necessarily holds if

#### Question 4

$$f(x) = \frac{x^2 + 2x - 15}{x^2 - 7x - 18}$$ is negative if and only if

#### Question 5

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

## CAT 2020 Inequalities questions

#### Question 1

Among 100 students, $$x_1$$ have birthdays in January, $$X_2$$ have birthdays in February, and so on. If $$x_0=max(x_1,x_2,....,x_{12})$$, then the smallest possible value of $$x_0$$ is

#### Question 2

The number of pairs of integers $$(x,y)$$ satisfying $$x\geq y\geq-20$$ and $$2x+5y=99$$

#### Question 3

For real x, the maximum possible value of $$\frac{x}{\sqrt{1+x^{4}}}$$ is

#### Question 4

if x and y are positive real numbers satisfying $$x+y=102$$, then the minimum possible valus of $$2601(1+\frac{1}{x})(1+\frac{1}{y})$$ is