Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ be three points in $xy$-plane, whose position vector are given by $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\mathrm { a } \hat { i } + ( 1 - \mathrm { a } ) \hat { j }$ respectively with respect to the origin O. If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OB } }$ is $\frac { 9 } { \sqrt { 2 } }$, then the sum of all the possible values of $a$ is : (1) 2 (2) $9/2$ (3) 1 (4) 0
Let the coefficients of three consecutive terms $T _ { r } , T _ { r + 1 }$ and $T _ { r + 2 }$ in the binomial expansion of $( a + b ) ^ { 12 }$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $( \sqrt [ 4 ] { 3 } + \sqrt [ 3 ] { 4 } ) ^ { 12 }$. Then $\mathrm { p } + \mathrm { q }$ is equal to : (1) 283 (2) 287 (3) 295 (4) 299
Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is : (1) $\frac { 1 } { 2 }$ (2) $\frac { 1 } { 4 }$ (3) $\frac { 2 } { 3 }$ (4) $\frac { 1 } { 3 }$
Let $f$ be a real valued continuous function defined on the positive real axis such that $g ( x ) = \int _ { 0 } ^ { x } \mathrm { t } f ( \mathrm { t } ) \mathrm { dt }$. If $\mathrm { g } \left( x ^ { 3 } \right) = x ^ { 6 } + x ^ { 7 }$, then value of $\sum _ { r = 1 } ^ { 15 } f \left( \mathrm { r } ^ { 3 } \right)$ is : (1) 270 (2) 340 (3) 320 (4) 310
Let $f : [ 0,3 ] \rightarrow \mathrm { A }$ be defined by $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 7$ and $g : [ 0 , \infty ) \rightarrow B$ be defined by $\mathrm { g } ( x ) = \frac { x ^ { 2025 } } { x ^ { 2025 } + 1 }$. If both the functions are onto and $\mathrm { S } = \{ x \in \mathbf { Z } : x \in \mathrm {~A}$ or $x \in \mathrm {~B} \}$, then $\mathrm { n } ( \mathrm { S } )$ is equal to : (1) 29 (2) 30 (3) 31 (4) 36
Bag $B _ { 1 }$ contains 6 white and 4 blue balls, Bag $B _ { 2 }$ contains 4 white and 6 blue balls, and Bag $B _ { 3 }$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B _ { 2 }$, is : (1) $\frac { 4 } { 15 }$ (2) $\frac { 1 } { 3 }$ (3) $\frac { 2 } { 5 }$ (4) $\frac { 2 } { 3 }$
Let $\mathrm { f } : \mathbf { R } \rightarrow \mathbf { R }$ be a twice differentiable function such that $f ( 2 ) = 1$. If $\mathrm { F } ( x ) = x f ( x )$ for all $x \in \mathbf { R }$, $\int _ { 0 } ^ { 2 } x \mathrm {~F} ^ { \prime } ( x ) \mathrm { d } x = 6$ and $\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm {~F} ^ { \prime \prime } ( x ) \mathrm { d } x = 40$, then $\mathrm { F } ^ { \prime } ( 2 ) + \int _ { 0 } ^ { 2 } \mathrm {~F} ( x ) \mathrm { d } x$ is equal to : (1) 11 (2) 13 (3) 15 (4) 9
For positive integers $n$, if $4 a _ { n } = \left( n ^ { 2 } + 5 n + 6 \right)$ and $S _ { n } = \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { a _ { k } } \right)$, then the value of $507 S _ { 2025 }$ is : (1) 540 (2) 675 (3) 1350 (4) 135
Let $f : \mathbf { R } - \{ 0 \} \rightarrow ( - \infty , 1 )$ be a polynomial of degree 2, satisfying $f ( x ) f \left( \frac { 1 } { x } \right) = f ( x ) + f \left( \frac { 1 } { x } \right)$. If $f ( K ) = - 2 K$, then the sum of squares of all possible values of $K$ is : (1) 7 (2) 6 (3) 1 (4) 9
If $A$ and $B$ are the points of intersection of the circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ and a point P moves on the line $2 x - 3 y + 4 = 0$, then the centroid of $\triangle \mathrm { PAB }$ lies on the line : (1) $x + 9 y = 36$ (2) $4 x - 9 y = 12$ (3) $6 x - 9 y = 20$ (4) $9 x - 9 y = 32$
The square of the distance of the point $\left( \frac { 15 } { 7 } , \frac { 32 } { 7 } , 7 \right)$ from the line $\frac { x + 1 } { 3 } = \frac { y + 3 } { 5 } = \frac { z + 5 } { 7 }$ in the direction of the vector $\hat { i } + 4 \hat { j } + 7 \hat { k }$ is : (1) 54 (2) 44 (3) 41 (4) 66
If the midpoint of a chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ is $( \sqrt { 2 } , 4 / 3 )$, and the length of the chord is $\frac { 2 \sqrt { \alpha } } { 3 }$, then $\alpha$ is : (1) 20 (2) 22 (3) 18 (4) 26
If $\alpha + i \beta$ and $\gamma + i \delta$ are the roots of $x ^ { 2 } - ( 3 - 2 i ) x - ( 2 i - 2 ) = 0 , i = \sqrt { - 1 }$, then $\alpha \gamma + \beta \delta$ is equal to : (1) $-2$ (2) 6 (3) $-6$ (4) 2
Two equal sides of an isosceles triangle are along $- x + 2 y = 4$ and $x + y = 4$. If m is the slope of its third side, then the sum, of all possible distinct values of $m$, is : (1) $- 2 \sqrt { 10 }$ (2) 12 (3) 6 (4) $-6$
Let A and B be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If d denotes the distance between A and B, and a denotes the area of $\triangle S A B$, where $S$ is the focus of the parabola $y ^ { 2 } = 4 x$, then the value of $( a + d )$ is
The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6 ^ { \circ }$. If the largest interior angle of the polygon is $219 ^ { \circ }$, then n is equal to