Let $S = \{ x \in R : x \geq 0 \& 2 | \sqrt { x } - 3 | + \sqrt { x } ( \sqrt { x } - 6 ) + 6 = 0 \}$. Then $S$ : (1) Contains exactly four elements (2) Is an empty set (3) Contains exactly one element (4) Contains exactly two elements
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is: (1) At least 750 but less than 1000 (2) At least 1000 (3) Less than 500 (4) At least 500 but less than 750
Let $A$ be the sum of the first 20 terms and $B$ be the sum of the first 40 terms of the series $1 ^ { 2 } + 2 \cdot 2 ^ { 2 } + 3 ^ { 2 } + 2 \cdot 4 ^ { 2 } + 5 ^ { 2 } + 2 \cdot 6 ^ { 2 } + \ldots$ If $B - 2 A = 100 \lambda$, then $\lambda$ is equal to : (1) 496 (2) 232 (3) 248 (4) 464
The sum of the co-efficient of all odd degree terms in the expansion of $\left( x + \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } + \left( x - \sqrt { x ^ { 3 } - 1 } \right) ^ { 5 } , ( x > 1 )$ is (1) 2 (2) - 1 (3) 0 (4) 1
A straight line through a fixed point $( 2,3 )$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $O P R Q$ is completed, then the locus of $R$ is: (1) $3 x + 2 y = 6 x y$ (2) $3 x + 2 y = 6$ (3) $2 x + 3 y = x y$ (4) $3 x + 2 y = x y$
If the tangent at $( 1,7 )$ to the curve $x ^ { 2 } = y - 6$ touch the circle $x ^ { 2 } + y ^ { 2 } + 16 x + 12 y + c = 0$ then the value of $c$ is: (1) 95 (2) 195 (3) 185 (4) 85
Tangent and normal are drawn at $P ( 16,16 )$ on the parabola $y ^ { 2 } = 16 x$, which intersect the axis of the parabola at $A \& B$, respectively. If $C$ is the center of the circle through the points $P , A \& B$ and $\angle C P B = \theta$, then a value of $\tan \theta$ is: (1) $\frac { 4 } { 3 }$ (2) $\frac { 1 } { 2 }$ (3) 2 (4) 3
Tangents are drawn to the hyperbola $4 x ^ { 2 } - y ^ { 2 } = 36$ at the points $P$ and $Q$. If these tangents intersect at the point $T ( 0,3 )$ then the area (in sq. units) of $\triangle P T Q$ is: (1) $36 \sqrt { 5 }$ (2) $45 \sqrt { 5 }$ (3) $54 \sqrt { 3 }$ (4) $60 \sqrt { 3 }$
For each $t \in R$, let $[ t ]$ be the greatest integer less than or equal to $t$. Then $\lim _ { x \rightarrow 0 ^ { + } } x \left( \left[ \frac { 1 } { x } \right] + \left[ \frac { 2 } { x } \right] + \ldots + \left[ \frac { 15 } { x } \right] \right)$ (1) does not exist (in $R$ ) (2) is equal to 0 (3) is equal to 15 (4) is equal to 120
$P Q R$ is a triangular park with $P Q = P R = 200 \mathrm {~m}$. A T.V. tower stands at the mid-point of $Q R$. If the angles of elevation of the top of the tower at $P , Q$ and $R$ are respectively, $45 ^ { \circ } , 30 ^ { \circ }$ and $30 ^ { \circ }$, then the height of the tower (in m) is: (1) $50 \sqrt { 2 }$ (2) 100 (3) 50 (4) $100 \sqrt { 3 }$
Let the orthocentre and centroid of a triangle be $A ( - 3,5 )$ and $B ( 3,3 )$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $A C$ as diameter, is: (1) $\frac { 3 \sqrt { } 5 } { 2 }$ (2) $\sqrt { 10 }$ (3) $2 \sqrt { 10 }$ (4) $3 \sqrt { \frac { 5 } { 2 } }$
If the system of linear equations $x + k y + 3 z = 0$ $3 x + k y - 2 z = 0$ $2 x + 4 y - 3 z = 0$ has a non-zero solution $( x , y , z )$, then $\frac { x z } { y ^ { 2 } }$ is equal to: (1) 30 (2) - 10 (3) 10 (4) - 30
$\left| \begin{array} { c c c } x - 4 & 2 x & 2 x \\ 2 x & x - 4 & 2 x \\ 2 x & 2 x & x - 4 \end{array} \right| = ( A + B x ) ( x - A ) ^ { 2 }$, then the ordered pair $( A , B )$ is equal to (1) $( 4,5 )$ (2) $( - 4 , - 5 )$ (3) $( - 4,3 )$ (4) $( - 4,5 )$
Let $S = \left\{ t \in R : f ( x ) = | x - \pi | \cdot \left( e ^ { | x | } - 1 \right) \sin | x |\right.$ is not differentiable at $\left. t \right\}$. Then, the set $S$ is equal to: (1) $\{ 0 , \pi \}$ (2) $\phi$ (an empty set) (3) $\{ 0 \}$ (4) $\{ \pi \}$
If the curves $y ^ { 2 } = 6 x , 9 x ^ { 2 } + b y ^ { 2 } = 16$ intersect each other at right angles, then the value of $b$ is: (1) $\frac { 9 } { 2 }$ (2) 6 (3) $\frac { 7 } { 2 }$ (4) 4
Let $f ( x ) = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }$ and $g ( x ) = x - \frac { 1 } { x } , x \in R - \{ - 1,0,1 \}$. If $h ( x ) = \frac { f ( x ) } { g ( x ) }$, then the local minimum value of $h ( x )$ is: (1) $2 \sqrt { 2 }$ (2) 3 (3) - 3 (4) $- 2 \sqrt { 2 }$
Let $\vec { u }$ be a vector coplanar with the vectors $\vec { a } = 2 \hat { i } + 3 \hat { j } - \widehat { k }$ and $\vec { b } = \hat { j } + \widehat { k }$. If $\vec { u }$ is perpendicular to $\vec { a }$ and $\vec { u } \cdot \vec { b } = 24$, then $| \vec { u } | ^ { 2 }$ is equal to: (1) 84 (2) 336 (3) 315 (4) 256
If $L _ { 1 }$ is the line of intersection of the planes $2 x - 2 y + 3 z - 2 = 0 , x - y + z + 1 = 0$ and $L _ { 2 }$ is the line of intersection of the planes $x + 2 y - z - 3 = 0,3 x - y + 2 z - 1 = 0$, then the distance of the origin from the plane, containing the lines $L _ { 1 }$ and $L _ { 2 }$ is (1) $\frac { 1 } { \sqrt { 2 } }$ (2) $\frac { 1 } { 4 \sqrt { 2 } }$ (3) $\frac { 1 } { 3 \sqrt { 2 } }$ (4) $\frac { 1 } { 2 \sqrt { 2 } }$
A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is: (1) $\frac { 3 } { 4 }$ (2) $\frac { 3 } { 10 }$ (3) $\frac { 2 } { 5 }$ (4) $\frac { 1 } { 5 }$