Inequalities & Modulus — CAT Previous-Year Questions

The value of x satisfying the inequality $\frac{1}{x+5}$ ≤ $\frac{1}{2x-3}$ are

If x and y satisfy the equations |x| + x + y = 15 and x + |y| - y = 20, then (x - y) equals

The number of distinct real values of x, satisfying the equation 

max{x, 2} - min{x, 2} = |x + 2| - |x - 2|, is

The number of distinct integer solutions (x, y) of the equation |x + y| + |x - y| = 2, is

Any non-zero real numbers x, y such that y ≠ 3 and $\frac{\mathrm{x}}{\mathrm{y}}$ < $\frac{\mathrm{x}+3}{\mathrm{y}-3}$, will satisfy the condition

The largest real value of a for which the equation |x + a| + |x - 1| = 2 has an infinite number of solutions for x is

Let 0 ≤ a ≤ x ≤ 100 and f(x) = |x - a| + |x - 100| + |x - a - 50|. Then the maximum value of f(x) becomes 100 when a is equal to

The minimum possibe value of $\frac{x^2-6x+10}{3-x}$, for x < 3, is

If c = $\frac{16x}{y}$ + $\frac{49y}{x}$ for some non-zero real numbers x and y, then c cannot take the value

The number of integers n that satisfy the inequalities |n - 60| < |n - 100|  < |n - 20| is

For a real number x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

If 3x + 2|y| + y = 7 and x + |x| + 3y = 1, then x + 2y is

The number of distinct pairs of integers (m, n) satisfying |1 + mn| < |m + n| < 5 is

If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied?

If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)² + (a - c)² + (a - d)² is

If f(x) = x³ – 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is necessarily true?

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?

Given that −1 ≤ v ≤ 1, −2 ≤ u ≤ −0.5 and −2 ≤ z ≤ −0.5 and w = vz/u, then which of the following is necessarily true?

If x, y, z are distinct positive real numbers, then $\frac{x^2(y+z)+y^2(x+z)+z^2(x+y)}{xyz}$ would be

If |b| ≥ 1 and x = –|a|b, then which one of the following is necessarily true?

A real number x satisfying $1-\frac{1}{n}<x\le 3+\frac{1}{n},$ for every positive integer n, is best described by:

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

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?

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)?

Let x, y be two positive numbers such that x + y = 1. Then, the minimum value of ${\left(x+\frac{1}{x}\right)}^2+{\left(y+\frac{1}{y}\right)}^2$ is ______.

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

If x² + y² = 0.1 and |x – y| = 0.2, then |x| + |y| is equal to

If |r − 6| = 11 and |2q − 12| = 8,what is the minimum possible value of $\frac{q}{r}$?

Which of the following values of x do not satisfy the inequality (x² – 3x + 2 > 0) at all?

What is the value of m which satisfies 3m² – 21m + 30 < 0?