Top 40 CAT Inequalities Questions With Solutions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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?

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

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?

$$m$$ 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 $$m$$?

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 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

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

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

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

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

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

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

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

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

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

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