The sum of all those terms which are rational numbers in the expansion of $\left( 2 ^ { \frac { 1 } { 3 } } + 3 ^ { \frac { 1 } { 4 } } \right) ^ { 12 }$ is: (1) 89 (2) 27 (3) 35 (4) 43
If the greatest value of the term independent of $x$ in the expansion of $\left( x \sin \alpha + a \frac { \cos \alpha } { x } \right) ^ { 10 }$ is $\frac { 10 ! } { ( 5 ! ) ^ { 2 } }$, then the value of $a$ is equal to: (1) - 1 (2) 1 (3) - 2 (4) 2
If ${ } ^ { n } P _ { r } = { } ^ { n } P _ { r + 1 }$ and ${ } ^ { n } C _ { r } = { } ^ { n } C _ { r - 1 }$, then the value of $r$ is equal to: (1) 1 (2) 4 (3) 2 (4) 3
The number of distinct real roots of $\left| \begin{array} { c c c } \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{array} \right| = 0$ in the interval $- \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 4 }$ is: (1) 4 (2) 1 (3) 2 (4) 3
Let the equation of the pair of lines, $y = p x$ and $y = q x$, can be written as $( y - p x ) ( y - q x ) = 0$. Then the equation of the pair of the angle bisectors of the lines $x ^ { 2 } - 4 x y - 5 y ^ { 2 } = 0$ is: (1) $x ^ { 2 } - 3 x y + y ^ { 2 } = 0$ (2) $x ^ { 2 } + 4 x y - y ^ { 2 } = 0$ (3) $x ^ { 2 } + 3 x y - y ^ { 2 } = 0$ (4) $x ^ { 2 } - 3 x y - y ^ { 2 } = 0$
If a tangent to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $B C$ as diameter passes through the point. (1) $( \sqrt { 3 } , 0 )$ (2) $( \sqrt { 2 } , 0 )$ (3) $( 1,1 )$ (4) $( - 1,1 )$
The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt { 13.44 }$, then the standard deviation of the second sample is: (1) 8 (2) 6 (3) 4 (4) 5
If $[ x ]$ be the greatest integer less than or equal to $x$, then $\sum _ { n = 8 } ^ { 100 } \left[ \frac { ( - 1 ) ^ { n } n } { 2 } \right]$ is equal to: (1) 0 (2) 4 (3) - 2 (4) 2
Consider function $f : A \rightarrow B$ and $g : B \rightarrow C ( A , B , C \subseteq R )$ such that $( g o f ) ^ { - 1 }$ exists, then: (1) $f$ and $g$ both are one-one (2) $f$ and $g$ both are onto (3) $f$ is one-one and $g$ is onto (4) $f$ is onto and $g$ is one-one
If $f ( x ) = \left\{ \begin{array} { l l } \int _ { 0 } ^ { x } ( 5 + | 1 - t | ) d t , & x > 2 \\ 5 x + 1 , & x \leq 2 \end{array} \right.$, then (1) $f ( x )$ is not continuous at $x = 2$ (2) $f ( x )$ is everywhere differentiable (3) $f ( x )$ is continuous but not differentiable at $x = 2$ (4) $f ( x )$ is not differentiable at $x = 1$