In the $XY$-plane, the area, in sq. units, of the region defined by the inequalities $y \geq x + 4$ and $-4 \leq x^2 + y^2 + 4(x - y) \leq 0$ is

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

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

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 $y \neq 3$ and $\frac{x}{y} < \frac{x+3}{y-3}$ will satisfy the condition.

Let $n$ be any natural number such that $5n−1<3n+1$ $5^{n-1} < 3^{n+1}$. Then, the least integer value of $m$ that satisfies $3n+1<2n+m$ $3^{n+1} < 2^{n+m}$ for each such $n$, is

Let $n$ be any natural number such that $5^{n-1} \lt 3^{n+1}$ . Then, the least integer value of $m$ that satisfies $3^{n+1} \lt 2^{n+m}$ for each such $n$ , is

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

In an examination, there were 75 questions. 3 marks were awarded for each correct answer, 1 mark was deducted for each wrong answer and 1 mark was awarded for each unattempted question. Rayan scored a total of 97 marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is

If $a$ and $b$ are non-negative real numbers such that $a+2 b=6$ , then the average of the maximum and minimum possible values of $(a+b)$ is

If $c=\frac{16 x}{y}+\frac{49 y}{x}$ 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 all real numbers x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

For all possible integers $n$ satisfying $2.25 \leq 2 + 2^{\,n+2} \leq 202$, the number of integer values of $3 + 3^{\,n+1}$ is

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

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

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0, and y ≤ 1 is

Given two positive numbers $x$ and $y$ such that their product $xy=2601$. Find the minimum possible value of $(x+1)(y+1)$.

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 2x + y equals

The number of integers that satisfy the equation

The number of pairs of integers(x,y) satisfying x ≥ y ≥ -20 and 2x + 5y = 99 is

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

The smallest integer n such that n³ - 11n² + 32n - 28 > 0 is [TITA]

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