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Let p, q and r be three natural numbers such that their sum is 900, and r is a perfect square whose value lies between 150 and 500. If p is not less than 0.3q and not more than 0.7q, then the sum of the maximum and minimum possible values of p is

All the values of x satisfying the inequality $\frac{1}{x+5} \leq \frac{1}{2 x-3}$ are

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

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

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

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

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

Let $m$ and $n$ be natural numbers such that $n$ is even and $0.2<\frac{\mathrm{m}}{20}, \frac{\mathrm{n}}{\mathrm{m}}, \frac{\mathrm{n}}{11}<0.5$. Then $m-2n$ equals

If $x$ and $y$ are non-negative integers such that $x+9=z, y+1=z$ and $x+y<z+5$, then the maximum possible value of $2 x+y$ ?

Let $a, b, x, y$ be real number such $a² + b² = 25, x² + y² = 169$, and $ax + by = 65.$ If $k = ay - bx,$ then

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

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

If $a$ and $b$ are integers such that $2x^2 - ax + 2 > 0$ and $x^2 −bx + 8 \geq 0$ for all real numbers $x,$ then the largest possible value of $2a-6b$ is

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

For how many integers $n$, will the inequality $(n - 5) (n - 10) - 3(n − 2) \leq 0$ be satisfied?