Let a complex number be $w = 1 - \sqrt { 3 } i$. Let another complex number $z$ be such that $| z w | = 1$ and $\arg ( z ) - \arg ( w ) = \frac { \pi } { 2 }$. Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to (1) 4 (2) $\frac { 1 } { 2 }$ (3) $\frac { 1 } { 4 }$ (4) 2
Let $S _ { 1 }$ be the sum of first $2 n$ terms of an arithmetic progression. Let $S _ { 2 }$ be the sum of first $4 n$ terms of the same arithmetic progression. If ( $S _ { 2 } - S _ { 1 }$ ) is 1000 , then the sum of the first $6 n$ terms of the arithmetic progression is equal to: (1) 1000 (2) 7000 (3) 5000 (4) 3000
Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x + y = 3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\triangle ABC$, then $( R + r )$ is equal to: (1) $\frac { 9 } { \sqrt { 2 } }$ (2) $7 \sqrt { 2 }$ (3) $2 \sqrt { 2 }$ (4) $3 \sqrt { 2 }$
Let a tangent be drawn to the ellipse $\frac { x ^ { 2 } } { 27 } + y ^ { 2 } = 1$ at $( 3 \sqrt { 3 } \cos \theta , \sin \theta )$ where $\theta \in \left( 0 , \frac { \pi } { 2 } \right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to: (1) $\frac { \pi } { 8 }$ (2) $\frac { \pi } { 4 }$ (3) $\frac { \pi } { 6 }$ (4) $\frac { \pi } { 3 }$
Consider a hyperbola $H : x ^ { 2 } - 2 y ^ { 2 } = 4$. Let the tangent at a point $P ( 4 , \sqrt { 6 } )$ meet the $x$-axis at $Q$ and latus rectum at $R \left( x _ { 1 } , y _ { 1 } \right) , x _ { 1 } > 0$. If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\triangle QFR$ (in sq. units) is equal to (1) $4 \sqrt { 6 }$ (2) $\sqrt { 6 } - 1$ (3) $\frac { 7 } { \sqrt { 6 } } - 2$ (4) $4 \sqrt { 6 } - 1$
Let in a series of $2 n$ observations, half of them are equal to $a$ and remaining half are equal to $- a$. Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become 5 and 20 , respectively. Then the value of $a ^ { 2 } + b ^ { 2 }$ is equal to: (1) 425 (2) 650 (3) 250 (4) 925
A pole stands vertically inside a triangular park $ABC$. Let the angle of elevation of the top of the pole from each corner of the park be $\frac { \pi } { 3 }$. If the radius of the circumcircle of $\triangle ABC$ is 2 , then the height of the pole is equal to: (1) $\frac { 2 \sqrt { 3 } } { 3 }$ (2) $2 \sqrt { 3 }$ (3) $\sqrt { 3 }$ (4) $\frac { 1 } { \sqrt { 3 } }$