# Top 40 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 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

## CAT 2019 Inequalities questions

#### Question 1

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

## CAT 2018 Inequalities questions

#### Question 1

The smallest integer $$n$$ such that $$n^3-11n^2+32n-28>0$$ is

## CAT 2017 Inequalities questions

#### Question 1

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

#### Question 2

The minimum possible value of the sum of the squares of the roots of the equation $$x^2+(a+3)x-(a+5)=0$$ is

#### Question 3

For how many integers n, will the inequality $$(n - 5) (n - 10) - 3(n - 2)\leq0$$ be satisfied?

## CAT 2006 Inequalities questions

#### Question 1

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

#### Question 2

What values of x satisfy $$x^{2/3} + x^{1/3} - 2 <= 0$$?

## CAT 2003 Inequalities questions

#### Question 1

Let a, b, c, d be four integers such that a+b+c+d = 4m+1 where m is a positive integer. Given m, which one of the following is necessarily true?

#### Question 2

Given that $$-1 \leq v \leq 1, -2 \leq u \leq -0.5$$ and $$-2 \leq z \leq -0.5$$ and $$w = vz /u$$ , then which of the following is necessarily true?

#### Question 3

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

## CAT 2002 Inequalities questions

#### Question 1

If x, y and z are real numbers such that x + y + z = 5 and xy + yz + zx = 3, what is the largest value that x can have?

#### Question 2

If $$x^2 + 5y^2 + z^2 = 2y(2x+z)$$, then which of the following statements is(are) necessarily true?

A. x = 2y B. x = 2z C. 2x = z

#### Question 3

If u, v, w and m are natural numbers such that $$u^m + v^m = w^m$$, then which one of the following is true?

#### Question 4

If pqr = 1, the value of the expression $$1/(1+p+q^{-1}) + 1/(1+q+r^{-1}) + 1/(1+r+p^{-1})$$

## CAT 2001 Inequalities questions

#### Question 1

x and y are real numbers satisfying the conditions 2 < x < 3 and - 8 < y < -7. Which of the following expressions will have the least value?

#### Question 2

$$m$$ is the smallest positive integer such that for any integer $$n \geq m$$, the quantity $$n^3 - 7n^2 + 11n - 5$$ is positive. What is the value of $$m$$?

#### Question 3

If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1+b)(1+c)(1+d)?

#### Question 4

Let x and y be two positive numbers such that $$x + y = 1.$$

Then the minimum value of $$(x+\frac{1}{x})^2+(y+\frac{1}{y})^2$$ is

#### Question 5

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

## CAT 2000 Inequalities questions

#### Question 1

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

## CAT 1999 Inequalities questions

#### Question 1

The number of positive integer valued pairs (x, y), satisfying 4x - 17 y = 1 and x < 1000 is:

#### Question 2

If | r - 6 | = 11 and | 2q - 12 | = 8, what is the minimum possible value of q / r?

## CAT 1997 Inequalities questions

Instruction for set 1: For these questions the following functions have been defined.

$$la(x, y, z) = min (x+y, y+z)$$
$$le(x, y, z) = max(x -y, y-z)$$
$$ma (x, y, z) = \frac{1}{2} (le (x, y, z) + la (x, y, z))$$

#### Question 1

Given that $$x >y> z> 0$$. Which of the following is necessarily true?

Instruction for set 1: For these questions the following functions have been defined.

$$la(x, y, z) = min (x+y, y+z)$$
$$le(x, y, z) = max(x -y, y-z)$$
$$ma (x, y, z) = \frac{1}{2} (le (x, y, z) + la (x, y, z))$$

#### Question 2

What is the value of ma(10, 4, le((la10, 5, 3), 5, 3))?

Instruction for set 1: For these questions the following functions have been defined.

$$la(x, y, z) = min (x+y, y+z)$$
$$le(x, y, z) = max(x -y, y-z)$$
$$ma (x, y, z) = \frac{1}{2} (le (x, y, z) + la (x, y, z))$$

#### Question 3

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)).

## CAT 1996 Inequalities questions

#### Question 1

Which of the following values of x do not satisfy the inequality $$(x^2 - 3x + 2 > 0)$$ at all?

## CAT 1991 Inequalities questions

#### Question 1

The number of integers n satisfying -n+2 ≥ 0 and 2n ≥ 4 is

#### Question 2

x, y and z are three positive integers such that x > y > z. Which of the following is closest to the product xyz?

## CAT 1990 Inequalities questions

#### Question 1

From any two numbers $$x$$ and $$y$$, we define $$x* y = x + 0.5y - xy$$ . Suppose that both $$x$$ and $$y$$ are greater than 0.5. Then
$$x* x < y* y$$ if