In a car race, car A beats car B by 45 km, car B beats car C by 50 km, and car A beats car C by 90 km. The distance (in km) over which the race has been conducted is

From the interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the lengths of the perpendiculars is $s$. Then the area of the triangle is

In a group of 10 students, the mean of the lowest 9 scores is $42$ while the mean of the highest 9 scores is $47$. For the entire group of 10 students, the maximum possible mean exceeds the minimum possible mean by

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

The value of $\log_a(ab) + \log_b(ba)$, for $1 < a \leq b$ cannot be equal to

Let the $m$-th and $n$-th terms of a Geometric progression be $\dfrac{3}{4}$ and $12$, respectively, when $m < n$. If the common ratio of the progression is an integer $r$, then the smallest possible value of $r + n - m$ is

If $x$ and $y$ are positive real numbers satisfying $x + y = 102$, then the minimum possible value of $2601\left(1 + \dfrac{1}{x}\right)\left(1 + \dfrac{1}{y}\right)$ is

For the same principal amount, the compound interest for two years at $5\%$ per annum exceeds the simple interest for three years at $3\%$ per annum by Rs $1125$. Then the principal amount in rupees is

Let $C$ be a circle of radius $5$ meters having center at $O$. Let $PQ$ be a chord of $C$ that passes through points $A$ and $B$ where $A$ is located $4$ meters north of $O$ and $B$ is located $3$ meters east of $O$. Then, the length of $PQ$, in meters, is nearest to

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