Let $A = \left\{ z \in C : \left| \frac { z + 1 } { z - 1 } \right| < 1 \right\}$ and $B = \left\{ z \in C : \arg \left( \frac { z - 1 } { z + 1 } \right) = \frac { 2 \pi } { 3 } \right\}$. Then $A \cap B$ is (1) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second and third quadrants only (2) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second quadrant only (3) an empty set (4) a portion of a circle of radius $\frac { 2 } { \sqrt { 3 } }$ that lies in the third quadrant only
Let $R$ be the point $( 3,7 )$ and let $P$ and $Q$ be two points on the line $x + y = 5$ such that $PQR$ is an equilateral triangle. Then the area of $\triangle PQR$ is (1) $\frac { 25 } { 4 \sqrt { 3 } }$ (2) $\frac { 25 \sqrt { 3 } } { 2 }$ (3) $\frac { 25 } { \sqrt { 3 } }$ (4) $\frac { 25 } { 2 \sqrt { 3 } }$
Let $C$ be a circle passing through the points $A ( 2 , - 1 )$ and $B ( 3,4 )$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $( x - 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 13 } { 2 }$, then $r ^ { 2 }$ is equal to (1) 32 (2) $\frac { 65 } { 2 }$ (3) $\frac { 61 } { 2 }$ (4) 30
Let the normal at the point $P$ on the parabola $y ^ { 2 } = 6 x$ pass through the point $( 5 , - 8 )$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is (1) $\frac { - 9 } { 4 }$ (2) $\frac { 9 } { 4 }$ (3) $\frac { - 5 } { 2 }$ (4) $- 3$
The mean of the numbers $a , b , 8 , 5 , 10$ is 6 and their variance is 6.8. If $M$ is the mean deviation of the numbers about the mean, then $25M$ is equal to (1) 60 (2) 55 (3) 50 (4) 75
The ordered pair $( a , b )$, for which the system of linear equations $3 x - 2 y + z = b$ $5 x - 8 y + 9 z = 3$ $2 x + y + a z = - 1$ has no solution, is (1) $\left( 3 , \frac { 1 } { 3 } \right)$ (2) $\left( - 3 , \frac { 1 } { 3 } \right)$ (3) $\left( - 3 , - \frac { 1 } { 3 } \right)$ (4) $\left( 3 , - \frac { 1 } { 3 } \right)$
$f , g : R \rightarrow R$ be two real valued function defined as $f ( x ) = \left\{ \begin{array} { c l } - | x + 3 | & , x < 0 \\ e ^ { x } & , x \geq 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } + k _ { 1 } x , & x < 0 \\ 4 x + k _ { 2 } , & x \geq 0 \end{array} \right.$, where $k _ { 1 }$ and $k _ { 2 }$ are real constants. If $gof$ is differentiable at $x = 0$, then $gof ( - 4 ) + gof ( 4 )$ is equal to (1) $4 \left( e ^ { 4 } + 1 \right)$ (2) $2 \left( 2 e ^ { 4 } + 1 \right)$ (3) $4 e ^ { 4 }$ (4) $2 \left( 2 e ^ { 4 } - 1 \right)$
Let $S$ be the set of all the natural numbers, for which the line $\frac { x } { a } + \frac { y } { b } = 2$ is a tangent to the curve $\left( \frac { x } { a } \right) ^ { n } + \left( \frac { y } { b } \right) ^ { n } = 2$ at the point $( a , b ) , ab \neq 0$. Then (1) $S = \phi$ (2) $n ( S ) = 1$ (3) $S = \{ 2k : k \in N \}$ (4) $S = N$
Let $f ( x ) = 2 \cos ^ { - 1 } x + 4 \cot ^ { - 1 } x - 3 x ^ { 2 } - 2 x + 10 , x \in [ - 1 , 1 ]$. If $[ a , b ]$ is the range of the function, then $4a - b$ is equal to (1) 11 (2) $11 - \pi$ (3) $11 + \pi$ (4) $15 - \pi$
If $\vec { a } \cdot \vec { b } = 1 , \vec { b } \cdot \vec { c } = 2$ and $\vec { c } \cdot \vec { a } = 3$, then the value of $[ \vec { a } \times ( \vec { b } \times \vec { c } ) \quad \vec { b } \times ( \vec { c } \times \vec { a } ) \quad \vec { c } \times ( \vec { b } \times \vec { a } ) ]$ is (1) 0 (2) $- 6 \vec { a } \cdot ( \vec { b } \times \vec { c } )$ (3) $12 \vec { c } \cdot ( \vec { a } \times \vec { b } )$ (4) $- 12 \vec { b } \cdot ( \vec { c } \times \vec { a } )$
Let the plane $2 x + 3 y + z + 20 = 0$ be rotated through a right angle about its line of intersection with the plane $x - 3 y + 5 z = 8$. If the mirror image of the point $\left( 2 , - \frac { 1 } { 2 } , 2 \right)$ in the rotated plane is $B ( a , b , c )$, then (1) $\frac { a } { 8 } = \frac { b } { 5 } = \frac { c } { - 4 }$ (2) $\frac { a } { 4 } = \frac { b } { 5 } = \frac { c } { - 2 }$ (3) $\frac { a } { 8 } = \frac { b } { - 5 } = \frac { c } { 4 }$ (4) $\frac { a } { 4 } = \frac { b } { 5 } = \frac { c } { 2 }$
Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is (1) $\frac { 46 } { 6 ^ { 4 } }$ (2) $\frac { 275 } { 6 ^ { 5 } }$ (3) $\frac { 41 } { 5 ^ { 5 } }$ (4) $\frac { 36 } { 5 ^ { 4 } }$
There are ten boys $B _ { 1 } , B _ { 2 } , \ldots , B _ { 10 }$ and five girls $G _ { 1 } , G _ { 2 } , \ldots G _ { 5 }$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B _ { 1 }$ and $B _ { 2 }$ together should not be the members of a group, is $\_\_\_\_$.
Let the common tangents to the curves $4 \left( x ^ { 2 } + y ^ { 2 } \right) = 9$ and $y ^ { 2 } = 4 x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac { l } { e ^ { 2 } }$ is equal to $\_\_\_\_$.
Let $A = \{ n \in N :$ H.C.F.$( n , 45 ) = 1 \}$ and let $B = \{ 2k : k \in \{ 1 , 2 , \ldots , 100 \} \}$. Then the sum of all the elements of $A \cap B$ is $\_\_\_\_$.
Let the solution curve $y = y ( x )$ of the differential equation $\left( 4 + x ^ { 2 } \right) dy - 2 x \left( x ^ { 2 } + 3 y + 4 \right) dx = 0$ pass through the origin. Then $y ( 2 )$ is equal to $\_\_\_\_$.
Let $S = ( 0 , 2 \pi ) - \left\{ \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 } \right\}$. Let $y = y ( x ) , x \in S$, be the solution curve of the differential equation $\frac { dy } { dx } = \frac { 1 } { 1 + \sin 2 x } , y \left( \frac { \pi } { 4 } \right) = \frac { 1 } { 2 }$. If the sum of abscissas of all the points of intersection of the curve $y = y ( x )$ with the curve $y = \sqrt { 2 } \sin x$ is $\frac { k \pi } { 12 }$, then $k$ is equal to $\_\_\_\_$.