Three particles $P , Q$ and $R$ are moving along the vectors $\vec { A } = \hat { \mathrm { i } } + \hat { \mathrm { j } } , \overrightarrow { \mathrm { B } } = \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ and $\vec { C } = - \hat { \mathrm { i } } + \hat { \mathrm { j } }$, respectively. They strike on a point and start to move in different directions. Now particle $P$ is moving normal to the plane which contains vector $\vec { A }$ and $\vec { B }$. Similarly particle $Q$ is moving normal to the plane which contains vector $\vec { A }$ and $\vec { C }$. The angle between the direction of motion of $P$ and $Q$ is $\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { x } } \right)$. Then the value of $x$ is $\_\_\_\_$ .
The position of the centre of mass of a uniform semi-circular wire of radius $R$ placed in $x - y$ plane with its centre at the origin and the line joining its ends as $x$-axis is given by, $\left( 0 , \frac { x R } { \pi } \right)$. Then, the value of $| x |$ is $\_\_\_\_$ .
The centre of a wheel rolling on a plane surface moves with a speed $v _ { 0 }$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $\sqrt { x } v _ { 0 }$. Then the value of $x$ is $\_\_\_\_$ .
Let $n$ denote the number of solutions of the equation $z ^ { 2 } + 3 \bar { z } = 0$, where $z$ is a complex number. Then the value of $\sum _ { k = 0 } ^ { \infty } \frac { 1 } { n ^ { k } }$ is equal to (1) 1 (2) $\frac { 4 } { 3 }$ (3) $\frac { 3 } { 2 }$ (4) 2
Let $S _ { n }$ denote the sum of first $n$-terms of an arithmetic progression. If $S _ { 10 } = 530 , S _ { 5 } = 140$, then $S _ { 20 } - S _ { 6 }$ is equal to: (1) 1862 (2) 1842 (3) 1852 (4) 1872
Let the circle $S : 36 x ^ { 2 } + 36 y ^ { 2 } - 108 x + 120 y + C = 0$ be such that it neither intersects nor touches the coordinate axes. If the point of intersection of the lines, $x - 2 y = 4$ and $2 x - y = 5$ lies inside the circle $S$, then: (1) $\frac { 25 } { 9 } < C < \frac { 13 } { 3 }$ (2) $100 < C < 165$ (3) $81 < C < 156$ (4) $100 < C < 156$
Let $E _ { 1 } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a > b$. Let $E _ { 2 }$ be another ellipse such that it touches the end points of major axis of $E _ { 1 }$ and the foci of $E _ { 2 }$ are the end points of minor axis of $E _ { 1 }$. If $E _ { 1 }$ and $E _ { 2 }$ have same eccentricities, then its value is: (1) $\frac { - 1 + \sqrt { 5 } } { 2 }$ (2) $\frac { - 1 + \sqrt { 8 } } { 2 }$ (3) $\frac { - 1 + \sqrt { 3 } } { 2 }$ (4) $\frac { - 1 + \sqrt { 6 } } { 2 }$
Let a line $L : 2 x + y = k , k > 0$ be a tangent to the hyperbola $x ^ { 2 } - y ^ { 2 } = 3$. If $L$ is also a tangent to the parabola $y ^ { 2 } = \alpha x$, then $\alpha$ is equal to: (1) 12 (2) - 12 (3) 24 (4) - 24