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
All the values of x satisfying the inequality $\cfrac{1}{x + 5} \leq \cfrac{1}{2x - 3}$ are
If x and y satisfy the equations $\mid x \mid + x + y = 15$ and $x + \mid y \mid - y = 20$, then $(x - y)$ equals
The number of distinct integer solutions (x, y) of the equation $\mid x + y \mid + \mid x - y \mid = 2$, is
If x and y are real numbers such that $x^2+(x-2 y-1)^2=-4 y(x+y)$, then the value x−2y is
The number of integer solutions of equation $2|x|\left(x^2+1\right)=5 x^2$ is
In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is
Any non-zero real numbers x,y such that $y ≠ 3$ and $\frac{x}{y} \lt \frac{x+3}{y-3}$, will satisfy the condition
If a certain amount of money is divided equally among n persons, each one receives Rs 352 . However, if two persons receive Rs 506 each and the remaining amount is divided equally among the other persons, each of them receive less than or equal to Rs 330 . Then, the maximum possible value of n is
If $p^2+q^2-29=2 p q-20=52-2 p q$, then the difference between the maximum and minimum possible value of $\left(p^3-q^3\right)$ is
Let n and m be two positive integers such that there are exactly 41 integers greater than $8^m$ and less then $8^n$, which can be expressed as powers of 2. Then, the smallest possible value of n + m 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 population of a town in 2020 was 100000 . The population decreased by y% from the year 2020 to 2021 , and increased by x% from the year 2021 to 2022, where x and y are two natural numbers. If population in 2022 was greater than the population in 2020 and the difference between x and y is 10 , then the lowest possible population of the town in 2021 was
The number of distinct integer values of n satisfying $\frac{4-\log _2 n}{3-\log _4 n}\lt0$, is
The number of integer solutions of the equation $\left(x^2-10\right)^{\left(x^2-3 x-10\right)}=1$ is
If a, b, c and d are integers such that their sum is 46, then the minimum possible value of $(a-b)^{2}+(a-c)^{2}+(a-d)^{2}$ is
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
