If $\log x-5\log 3=-2$, then $x$ equals
Find the value of $\log_{20} 100 + \log_{20} 1000 + \log_{20} 10000 \quad \bigl[\textit{Assume that } \log 2 = 0.3\bigr].$
Find the remainder when the $41$-digit number $1234\ldots$ is divided by $8$.
In a survey of $500$ TV viewers: $285$ watch football (F), $195$ hockey (H), $115$ basketball (B); $45$ watch F&B, $70$ watch F&H, $50$ watch H&B, and $50$ watch none. How many watch exactly one of the three games?
Nine squares are chosen at random on a chessboard. What is the probability that they form a square of size $3\times 3$?
$A,B,C,D$ are four towns, any three of which are non-colinear. In how many ways can we construct three roads (each road joins a pair of towns) so that the roads do not form a triangle?
If $x^{2}-ax-21=0$ and $x^{2}-3ax+35=0$ with $a>0$ have a common root, then $a$ equals:
If $a,b,c$ are distinct positive real numbers and $a^2+b^2+c^2=1$, then $ab+bc+ca$ is
In the figure, $\triangle APB$ is formed by three tangents to a circle with centre $O$. If $\angle APB=40^\circ$, then the measure of $\angle BOA$ is
If $1+\sin^2(2A)=3\sin A\cos A$, then what are the possible values of $\tan A$?
If \[ 2\sin\alpha + 15\cos^{2}\alpha = 7, \quad 0^\circ < \alpha < 90^\circ, \] find \(\cot\alpha\).
In the given figure, $ABCD$ is a rectangle. $P$ and $Q$ are the midpoints of sides $CD$ and $BC$ respectively. Then the ratio of area of shaded portion to the area of unshaded portion is:
In the adjoining figure, points $A,B,C,D$ lie on a circle. $AD=24$ and $BC=12$. What is the ratio of the area of $\triangle CBE$ to that of $\triangle ADE$?
In $\triangle ABC$, $D$ is a point on $BC$ such that $3BD=BC$. If each side of the triangle is $12\,$cm, then $AD$ equals
On dividing $x^{3} - 3x^{2} + x + 2$ by a polynomial $g(x)$, the quotient and remainder were $x - 2$ and $-2x + 4$ respectively. Find $g(x)$.
At the end of a business conference, the ten people present all shake hands with each other once. How many handshakes will there be altogether?
In the Adjoining figure, if O is the centre of the circle of radius 5cm, OP=13 cm, where PQ and PR are tangents to the circle from point P. Length PR is :
