Let $A ( a , 0 ) , B ( b , 2 b + 1 )$ and $C ( 0 , b ) , b \neq 0 , | b | \neq 1$, be points such that the area of triangle $A B C$ is 1 sq. unit, then the sum of all possible values of $a$ is: (1) $\frac { - 2 b } { b + 1 }$ (2) $\frac { 2 b ^ { 2 } } { b + 1 }$ (3) $\frac { - 2 b ^ { 2 } } { b + 1 }$ (4) $\frac { 2 b } { b + 1 }$
If two tangents drawn from a point $P$ to the parabola $y ^ { 2 } = 16 ( x - 3 )$ are at right angles, then the locus of point $P$ is: (1) $x + 4 = 0$ (2) $x + 2 = 0$ (3) $x + 3 = 0$ (4) $x + 1 = 0$
Two poles $A B$ of length $a$ metres and $C D$ of length $a + b ( b \neq a )$ metres are erected at the same horizontal level with bases at $B$ and $D$. If $B D = x$ and $\tan \angle A C B = \frac { 1 } { 2 }$, then: (1) $x ^ { 2 } + 2 ( a + 2 b ) x - b ( a + b ) = 0$ (2) $x ^ { 2 } + 2 ( a + 2 b ) x + a ( a + b ) = 0$ (3) $x ^ { 2 } - 2 a x + b ( a + b ) = 0$ (4) $x ^ { 2 } - 2 a x + a ( a + b ) = 0$
Let $Z$ be the set of all integers, $A = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ $B = \left\{ ( x , y ) \in Z \times Z : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ and $C = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \leq 4 \right\}$ If the total number of relations from $A \cap B$ to $A \cap C$ is $2 ^ { p }$, then the value of $p$ is: (1) 25 (2) 9 (3) 16 (4) 49
Let $[ \lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x + y + z = 4,3 x + 2 y + 5 z = 3,9 x + 4 y + ( 28 + [ \lambda ] ) z = [ \lambda ]$ has a solution is: (1) $R$ (2) $( - \infty , - 9 ) \cup [ - 8 , \infty )$ (3) $( - \infty , - 9 ) \cup ( - 9 , \infty )$ (4) $[ - 9 , - 8 )$
Let $A = \left[ \begin{array} { c c c } { [ x + 1 ] } & { [ x + 2 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 3 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 2 ] } & { [ x + 4 ] } \end{array} \right]$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. If $\operatorname { det } ( A ) = 192$, then the set of values of $x$ is in the interval: (1) $[ 62,63 )$ (2) $[ 65,66 )$ (3) $[ 60,61 )$ (4) $[ 68,69 )$
A box open from top is made from a rectangular sheet of dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to: (1) $\frac { a + b + \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (2) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 12 }$ (3) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (4) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } + a b } } { 6 }$
Let $M$ and $m$ respectively be the maximum and minimum values of the function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ in $\left[ 0 , \frac { \pi } { 2 } \right]$. Then the value of $\tan ( M - m )$ is equal to: (1) $2 - \sqrt { 3 }$ (2) $3 - 2 \sqrt { 2 }$ (3) $3 + 2 \sqrt { 2 }$ (4) $2 + \sqrt { 3 }$
The area of the region bounded by the parabola $( y - 2 ) ^ { 2 } = ( x - 1 )$, the tangent to it at the point whose ordinate is 3 and the $x$-axis, is: (1) 4 (2) 6 (3) 9 (4) 10
If the solution curve of the differential equation $\left( 2 x - 10 y ^ { 3 } \right) d y + y d x = 0$, passes through the points $( 0,1 )$ and $( 2 , \beta )$, then $\beta$ is a root of the equation? (1) $y ^ { 5 } - 2 y - 2 = 0$ (2) $y ^ { 5 } - y ^ { 2 } - 1 = 0$ (3) $2 y ^ { 5 } - y ^ { 2 } - 2 = 0$ (4) $2 y ^ { 5 } - 2 y - 1 = 0$
A differential equation representing the family of parabolas with axis parallel to $y$-axis and whose length of latus rectum is the distance of the point $( 2 , - 3 )$ from the line $3 x + 4 y = 5$, is given by: (1) $11 \frac { d ^ { 2 } x } { d y ^ { 2 } } = 10$ (2) $11 \frac { d ^ { 2 } y } { d x ^ { 2 } } = 10$ (3) $10 \frac { d ^ { 2 } y } { d x ^ { 2 } } = 11$ (4) $10 \frac { d ^ { 2 } x } { d y ^ { 2 } } = 11$
Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is: (1) $\frac { 5 } { 8 }$ (2) $\frac { 1 } { 8 }$ (3) $\frac { 5 } { 16 }$ (4) 1
Let $z _ { 1 }$ and $z _ { 2 }$ be two complex numbers such that $\arg \left( z _ { 1 } - z _ { 2 } \right) = \frac { \pi } { 4 }$ and $z _ { 1 } , z _ { 2 }$ satisfy the equation $| z - 3 | = \operatorname { Re } ( z )$. Then the imaginary part $z _ { 1 } + z _ { 2 }$ is equal to
Let $S = \{ 1,2,3,4,5,6,9 \}$. Then the number of elements in the set $T = \{ A \subseteq S : A \neq \phi$ and the sum of all the elements of $A$ is not a multiple of $3 \}$ is
Let $S$ be the sum of all solutions (in radians) of the equation $\sin ^ { 4 } \theta + \cos ^ { 4 } \theta - \sin \theta \cos \theta = 0$ in $[ 0,4 \pi ]$ then $\frac { 8 S } { \pi }$ is equal to
Two circles each of radius 5 units touch each other at the point $( 1,2 )$. If the equation of their common tangent is $4 x + 3 y = 10$, and $C _ { 1 } ( \alpha , \beta )$ and $C _ { 2 } ( \gamma , \delta ) , C _ { 1 } \neq C _ { 2 }$ are their centres, then $| ( \alpha + \beta ) ( \gamma + \delta ) |$ is equal to
Let $P ( a \sec \theta , b \tan \theta )$ and $Q ( a \sec \phi , b \tan \phi )$ where $\theta + \phi = \frac { \pi } { 2 }$, be two points on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the ordinate of the point of intersection of normals at $P$ and $Q$ is $- k \left( \frac { a ^ { 2 } + b ^ { 2 } } { 2 b } \right)$, then $k$ is equal to
An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2 . The variance of marks obtained by 30 girls is also 2 . The average marks of all 50 candidates is 15 . If $\mu$ is the average marks of girls and $\sigma ^ { 2 }$ is the variance of marks of 50 candidates, then $\mu + \sigma ^ { 2 }$ is equal to
$\int \frac { 2 e ^ { x } + 3 e ^ { - x } } { 4 e ^ { x } + 7 e ^ { - x } } d x = \frac { 1 } { 14 } \left( u x + v \log _ { e } \left( 4 e ^ { x } + 7 e ^ { - x } \right) \right) + C$, where $C$ is a constant of integration, then $u + v$ is equal to
Let $S$ be the mirror image of the point $Q ( 1,3,4 )$ with respect to the plane $2 x - y + z + 3 = 0$ and let $R ( 3,5 , \gamma )$ be a point of this plane. Then the square of the length of the line segment $S R$ is