Points $A$, $P$, $Q$ and $B$ lie on the same line such that $P$, $Q$ and $B$ are, respectively, $100\text{ km}$, $200\text{ km}$ and $300\text{ km}$ away from $A$. Cars 1 and 2 leave $A$ at the same time and move towards $B$. Simultaneously, car 3 leaves $B$ and moves towards $A$. Car 3 meets Car 1 at $Q$, and Car 2 at $P$. If each car is moving at uniform speed, then the ratio of the speed of Car 2 to that of Car 1 is

Let $a_1, a_2, \ldots, a_{52}$ be positive integers such that $a_1 < a_2 < \cdots < a_{52}$. Suppose their arithmetic mean is one less than the arithmetic mean of $a_2, a_3, \ldots, a_{52}$. If $a_{52} = 100$, then the largest possible value of $a_1$ is

There are two drums, each containing a mixture of paints $A$ and $B$. In drum 1, $A$ and $B$ are in the ratio $18:7$. The mixtures from drums 1 and 2 are mixed in the ratio $3:4$, and in this final mixture, $A$ and $B$ are in the ratio $13:7$. In drum 2, $A$ and $B$ were in the ratio

On a triangle $ABC$, a circle with diameter $BC$ is drawn, intersecting $AB$ and $AC$ at points $P$ and $Q$, respectively. If the lengths of $AB$, $AC$, and $CP$ are $30\text{ cm}$, $25\text{ cm}$, and $20\text{ cm}$ respectively, then the length of $BQ$, in cm, is

Let $t_1, t_2, \ldots$ be real numbers such that $t_1 + t_2 + \cdots + t_n = 2n^2 + 9n + 13$, for every positive integer $n \ge 2$. If $t_k = 103$, then $k$ equals

From a rectangle $ABCD$ of area $768$ sq cm, a semicircular part with diameter $AB$ and area $72\pi$ sq cm is removed. The perimeter of the leftover portion, in cm, is

If $N$ and $x$ are positive integers such that $N^N = 2^{160}$ and $N^2 + 2^N$ is an integral multiple of $2^x$, then the largest possible $x$ is

A chord of length $5\text{ cm}$ subtends an angle of $60^{\circ}$ at the centre of a circle. The length, in cm, of a chord that subtends an angle of $120^{\circ}$ at the centre of the same circle is

If $p^3 = q^4 = r^5 = s^6$, then the value of $\log_s(pqr)$ is equal to

In a tournament, there are $43$ junior level and $51$ senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is $153$, while the number of boy versus boy matches in senior level is $276$. The number of matches a boy plays against a girl is