IPMAT Inequalities Questions 2026 With Solutions, Download PDF

The length of the line segment joining the two intersection points of the curves $$y = 4970 - |x|$$ and $$y = x^{2}$$ is_________.

Given that
$$f(x)=|x|+2|x−1|+|x−2|+|x−4|+|x−6|+2|x−10|$$, $$x \epsilon (-\infty, \infty)$$
the minimum value of f(x) is _________.

The set of all possible values of f(x) for which $$(81)^{x} + (81)^{(f(x)} = 3$$ is

The sum of the squares of all the roots of the equation $$x^{2} + |x + 4| + |x − 4| − 35 = 0$$ is

For a > b > c > 0, the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| is

The set of all real values of x satisfying the inequality $$\frac{x^{2}(x + 1)}{(x - 1)(2x + 1)^{3}}> 0$$ is

Statements: $$M = V; R \geq S; V < S; M > A; R \leq U$$
Conclusions:
I. $$U > S$$
II. $$R = S$$
III. $$R > S$$

If $$\mid x + 1 \mid + (y + 2)^2 = 0$$ and $$ax - 3ay = 1$$, Then the value of a is

The number of solutions of the equation $$x_1 + x_2 + x_3 + x_4 = 50$$, where $$x_1 , x_2 , x_3 , x_4$$ are integers with $$x_1 \geq1, x_2 \geq 2, x_3 \geq 0, x_4 \geq 0$$ is

The set of values of x which satisfy the inequality $$0.7^{2x^{2}-3x+4} < 0.343$$ is

The number of pairs (x, y) of integers satisfying the inequality $$\mid x - 5 \mid + \mid y - 5 \mid \leq 6$$ is:

If |x|<100 and |y|<100, then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is

If a, b, c are real numbers $$a^{2} + b^{2} + c^{2} = 1$$, then the set of values $$ab+bc+ca$$ can take is:

The minimum value of $$f(x)=|3-x|+|2+x|+|5-x|$$ is equal to  _____________.

Let [x] denote the greatest integer not exceeding x and {x} = x -[x].
If n is a natural number, then the sum of all values of x satisfying the equation 2[x] = x + n{x} is

For all real values of x, $$\dfrac{3x^{2} - 6x + 12}{x^{2} + 2x + 4}$$ lies between 1 and k, and does not take any value above k. Then k equals...........

If a, b, c are three distinct natural numbers, all less than 100, such that $$\mid a - b \mid + \mid b - c \mid = \mid c - a \mid$$, then the maximum possible value of b is ______

If minimum value of $$f(x) = x^{2} + 2bx + 2c^{2}$$ is greater than the maximum value of $$g(x) = - x^{2} - 2cx + b^{2}$$, then for real value of x.

The set of all real numbers x for which $$x^{2} - |x + 2 |+ x > 0$$, is

Given $$A =2^{65}$$ and $$B = (2^{64} + 2^{63} + 2^{62} + ... + 2^{0})$$, which of the following is true?

Consider the following statements:
(i) When 0 < x < 1, then $$\dfrac{1}{1+x} < 1 - x + x^{2}$$.
(ii) When 0 < x < 1, then $$\dfrac{1}{1+x} > 1 - x + x^{2}$$.
(iii) When -1 < x < 0, then $$\dfrac{1}{1+x} < 1 - x + x^{2}$$.
(iv) When -1 < x < 0, then $$\dfrac{1}{1+x} > 1 - x + x^{2}$$.

Statements: $$A < P = C \geq D; S > U \leq B < A; C < Q > S \geq V$$
Conclusions:
I. $$U > V$$
II. $$B < C$$
III. $$Q > D$$

The total number of positive integer solutions of $$21 \leq a + b + c \leq 25$$ is __________.

If x ∈ (a, b) satisfies the inequality, $$\dfrac{x-3}{x^2+3x+2}\ge1$$ then the largest possible value of b - a is

For some non-zero real values of $$a, b$$ and $$c$$, it is given that $$\mid \frac{c}{a} \mid = 4, \mid \frac{a}{b} \mid = \frac{1}{3}$$ and $$\frac{b}{c} = -\frac{3}{4}$$. If $$ac > 0$$, then $$\left(\frac{b + c}{a}\right)$$

The area enclosed by $$2|x| + 3|y| \leq 6$$ is____________ sq. units

The inequality $$\log_{2} \frac{3x - 1}{2 - x} < 1$$ holds true for

Statements:
Lady's Finger is tastier than cabbage
Cauliflower is tastier than Lady's Finger
Cabbage is not tastier than peas