Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11
Let the triangle PQR be the image of the triangle with vertices $( 1,3 ) , ( 3,1 )$ and $( 2,4 )$ in the line $x + 2 y = 2$. If the centroid of $\triangle \mathrm { PQR }$ is the point $( \alpha , \beta )$, then $15 ( \alpha - \beta )$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22
Let the parabola $y = x ^ { 2 } + \mathrm { p } x - 3$, meet the coordinate axes at the points $\mathrm { P } , \mathrm { Q }$ and R. If the circle C with centre at $( - 1 , - 1 )$ passes through the points $P , Q$ and $R$, then the area of $\triangle P Q R$ is: (1) 7 (2) 4 (3) 6 (4) 5
Let $f ( x )$ be a real differentiable function such that $f ( 0 ) = 1$ and $f ( x + y ) = f ( x ) f ^ { \prime } ( y ) + f ^ { \prime } ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. Then $\sum _ { \mathrm { n } = 1 } ^ { 100 } \log _ { \mathrm { e } } f ( \mathrm { n } )$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406
From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is: (1) 5148 (2) 6084 (3) 4356 (4) 14950
Let $f : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. If $f ^ { \prime } ( 0 ) = 4 \mathrm { a }$ and $f$ satisfies $f ^ { \prime \prime } ( x ) - 3 \mathrm { a } f ^ { \prime } ( x ) - f ( x ) = 0 , \mathrm { a } > 0$, then the area of the region $\mathrm { R } = \{ ( x , y ) \mid 0 \leq y \leq f ( \mathrm { a } x ) , 0 \leq x \leq 2 \}$ is: (1) $e ^ { 2 } - 1$ (2) $\mathrm { e } ^ { 2 } + 1$ (3) $e ^ { 4 } + 1$ (4) $e ^ { 4 } - 1$
Let the foci of a hyperbola be $( 1,14 )$ and $( 1 , - 12 )$. If it passes through the point $( 1,6 )$, then the length of its latus-rectum is: (1) $\frac { 24 } { 5 }$ (2) $\frac { 25 } { 6 }$ (3) $\frac { 144 } { 5 }$ (4) $\frac { 288 } { 5 }$
A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma ^ { 2 }$ denote the mean and variance of $X$, then the value of $64 \left( \mu + \sigma ^ { 2 } \right)$ is: (1) 51 (2) 64 (3) 32 (4) 48
A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $( 2,5 )$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $( \alpha , \beta )$, then $3 \beta - 2 \alpha$ is equal to: (1) 10 (2) 15 (3) 12 (4) 14
Let $A = \{ 1,2,3 , \ldots , 10 \}$ and $B = \left\{ \frac { m } { n } : m , n \in A , m < n \right.$ and $\left. \operatorname { gcd } ( m , n ) = 1 \right\}$. Then $n ( B )$ is equal to: (1) 36 (2) 31 (3) 37 (4) 29
Let $A$ be a square matrix of order 3 such that $\operatorname { det } ( A ) = - 2$ and $\operatorname { det } ( 3 \operatorname { adj } ( - 6 \operatorname { adj } ( 3 A ) ) ) = 2 ^ { \mathrm { m } + \mathrm { n } } \cdot 3 ^ { \mathrm { mn } } , \mathrm { m } > \mathrm { n }$. Then $4 \mathrm {~m} + 2 \mathrm { n }$ is equal to $\_\_\_\_$
Let $\vec { c }$ be the projection vector of $\vec { b } = \lambda \hat { i } + 4 \hat { k } , \lambda > 0$, on the vector $\vec { a } = \hat { i } + 2 \hat { j } + 2 \hat { k }$. If $| \vec { a } + \vec { c } | = 7$, then the area of the parallelogram formed by the vectors $\vec { b }$ and $\vec { c }$ is $\_\_\_\_$
Let the function, $f ( x ) = \left\{ \begin{array} { l l } - 3 a x ^ { 2 } - 2 , & x < 1 \\ a ^ { 2 } + b x , & x \geqslant 1 \end{array} \right.$ be differentiable for all $x \in \mathbf { R }$, where $\mathbf { a } > 1 , \mathbf { b } \in \mathbf { R }$. If the area of the region enclosed by $y = f ( x )$ and the line $y = - 20$ is $\alpha + \beta \sqrt { 3 } , \alpha , \beta \in Z$, then the value of $\alpha + \beta$ is $\_\_\_\_$
Let $\mathrm { L } _ { 1 } : \frac { x - 1 } { 3 } = \frac { y - 1 } { - 1 } = \frac { z + 1 } { 0 }$ and $\mathrm { L } _ { 2 } : \frac { x - 2 } { 2 } = \frac { y } { 0 } = \frac { z + 4 } { \alpha } , \alpha \in \mathbf { R }$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A ( 1,1 , - 1 )$ on $L _ { 2 }$, then the value of $26 \alpha ( \mathrm {~PB} ) ^ { 2 }$ is $\_\_\_\_$