Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be 5 . Let the sum of its first five terms be $\frac { 98 } { 25 }$. Then the sum of the first 21 terms of an AP, whose first term is $10 a r , n ^ { \text {th } }$ term is $a _ { n }$ and the common difference is $10 a r ^ { 2 }$, is equal to (1) $21 a _ { 11 }$ (2) $22 a _ { 11 }$ (3) $15 a _ { 16 }$ (4) $14 a _ { 16 }$
The equations of the sides $A B , B C$ and $C A$ of a triangle $A B C$ are $2 x + y = 0 , x + p y = 39$ and $x - y = 3$ respectively and $P ( 2,3 )$ is its circumcentre. Then which of the following is NOT true (1) $( A C ) ^ { 2 } = 9 p$ (2) $( A C ) ^ { 2 } + p ^ { 2 } = 136$ (3) $32 <$ area $( \triangle A B C ) < 36$ (4) $34 <$ area $( \triangle A B C ) < 38$
A circle $C _ { 1 }$ passes through the origin $O$ and has diameter 4 on the positive $x$-axis. The line $y = 2 x$ gives a chord $O A$ of a circle $C _ { 1 }$. Let $C _ { 2 }$ be the circle with $O A$ as a diameter. If the tangent to $C _ { 2 }$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $Q A : A P$ is equal to (1) $1 : 4$ (2) $1 : 5$ (3) $2 : 5$ (4) $1 : 3$
If the length of the latus rectum of a parabola, whose focus is $( a , a )$ and the tangent at its vertex is $x + y = a$, is 16 , then $| a |$ is equal to (1) $2 \sqrt { 2 }$ (2) $2 \sqrt { 3 }$ (3) $4 \sqrt { 2 }$ (4) 4
The angle of elevation of the top $P$ of a vertical tower $P Q$ of height 10 from a point $A$ on the horizontal ground is $45 ^ { \circ }$. Let $R$ be a point on $A Q$ and from a point $B$, vertically above $R$, the angle of elevation of $P$ is $60 ^ { \circ }$. If $\angle B A Q = 30 ^ { \circ } , A B = d$ and the area of the trapezium $P Q R B$ is $\alpha$, then the ordered pair ( $d , \alpha$ ) is (1) $( 10 ( \sqrt { 3 } - 1 ) , 25 )$ (2) $\left( 10 ( \sqrt { 3 } - 1 ) , \frac { 25 } { 2 } \right)$ (3) $( 10 ( \sqrt { 3 } + 1 ) , 25 )$ (4) $\left( 10 ( \sqrt { 3 } + 1 ) , \frac { 25 } { 2 } \right)$
Let $A = \left( \begin{array} { c c } 4 & - 2 \\ \alpha & \beta \end{array} \right)$. If $A ^ { 2 } + \gamma A + 18 I = O$, then $\operatorname { det } ( A )$ is equal to (1) - 18 (2) 18 (3) - 50 (4) 50
Consider a curve $y = y ( x )$ in the first quadrant as shown in the figure. Let the area $A _ { 1 }$ is twice the area $A _ { 2 }$. Then the normal to the curve perpendicular to the line $2 x - 12 y = 15$ does NOT pass through the point (1) $( 6,21 )$ (2) $( 8,9 )$ (3) $( 10 , - 4 )$ (4) $( 12 , - 15 )$
If the length of the perpendicular drawn from the point $P ( a , 4,2 ) , a > 0$ on the line $\frac { x + 1 } { 2 } = \frac { y - 3 } { 3 } = \frac { z - 1 } { - 1 }$ is $2 \sqrt { 6 }$ units and $Q \left( \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } \right)$ is the image of the point $P$ in this line, then $a + \sum _ { i = 1 } ^ { 3 } \alpha _ { i }$ is equal to (1) 7 (2) 8 (3) 12 (4) 14
Let $X$ have a binomial distribution $B ( n , p )$ such that the sum and the product of the mean and variance of $X$ are 24 and 128 respectively. If $P ( X > n - 3 ) = \frac { k } { 2 ^ { n } }$, then $k$ is equal to (1) 528 (2) 529 (3) 629 (4) 630
A six faced die is biased such that $3 \times P ($ a prime number $) = 6 \times P ($ a composite number $) = 2 \times P ( 1 )$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is (1) $\frac { 3 } { 11 }$ (2) $\frac { 5 } { 11 }$ (3) $\frac { 7 } { 11 }$ (4) $\frac { 8 } { 11 }$
Let for the $9 ^ { \text {th } }$ term in the binomial expansion of $( 3 + 6 x ) ^ { n }$, in the increasing powers of $6 x$, to be the greatest for $x = \frac { 3 } { 2 }$, the least value of $n$ is $n _ { 0 }$. If $k$ is the ratio of the coefficient of $x ^ { 6 }$ to the coefficient of $x ^ { 3 }$, then $k + n _ { 0 }$ is equal to $\_\_\_\_$ .
The number of functions $f$, from the set $A = \left\{ x \in N : x ^ { 2 } - 10 x + 9 \leq 0 \right\}$ to the set $B = \left\{ n ^ { 2 } : n \in N \right\}$ such that $f ( x ) \leq ( x - 3 ) ^ { 2 } + 1$, for every $x \in A$, is $\_\_\_\_$ .
For the curve $C : \left( x ^ { 2 } + y ^ { 2 } - 3 \right) + \left( x ^ { 2 } - y ^ { 2 } - 1 \right) ^ { 5 } = 0$, the value of $3 y ^ { \prime } - y ^ { 3 } y ^ { \prime \prime }$, at the point $( \alpha , \alpha ) , \alpha > 0$, on $C$, is equal to $\_\_\_\_$ .
A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is $\tan ^ { - 1 } \frac { 3 } { 4 }$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is $\_\_\_\_$ .
Let $f ( x ) = \min \{ [ x - 1 ] , [ x - 2 ] , \ldots , [ x - 10 ] \}$ where $[ t ]$ denotes the greatest integer $\leq t$. Then $\int _ { 0 } ^ { 10 } f ( x ) d x + \int _ { 0 } ^ { 10 } ( f ( x ) ) ^ { 2 } d x + \int _ { 0 } ^ { 10 } | f ( x ) | d x$ is equal to $\_\_\_\_$ .
Let $f$ be a differentiable function satisfying $f ( x ) = \frac { 2 } { \sqrt { 3 } } \int _ { 0 } ^ { \sqrt { 3 } } f \left( \frac { \lambda ^ { 2 } x } { 3 } \right) d \lambda , x > 0$ and $f ( 1 ) = \sqrt { 3 }$. If $y = f ( x )$ passes through the point $( \alpha , 6 )$, then $\alpha$ is equal to $\_\_\_\_$ .
Let $\vec { a } , \vec { b } , \vec { c }$ be three non-coplanar vectors such that $\vec { a } \times \vec { b } = \overrightarrow { 4 c } , \vec { b } \times \vec { c } = 9 \vec { a }$ and $\vec { c } \times \vec { a } = \alpha \vec { b } , \alpha > 0$. If $| \vec { a } | + | \vec { b } | + | \vec { c } | = 36$, then $\alpha$ is equal to $\_\_\_\_$ .