If $m$ is chosen in the quadratic equation $\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 3 x + \left( m ^ { 2 } + 1 \right) ^ { 2 } = 0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is: (1) $4 \sqrt { 3 }$ (2) $10 \sqrt { 5 }$ (3) $8 \sqrt { 3 }$ (4) $8 \sqrt { 5 }$
Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is (1) 262 (2) 190 (3) 225 (4) 157
If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a value of its $11^{\text{th}}$ term is: (1) $- 25$ (2) $- 35$ (3) $25$ (4) $- 36$
If some three consecutive coefficients in the binomial expansion of $( x + 1 ) ^ { n }$ in powers of $x$ are in the ratio $2 : 15 : 70$, then the average of these three coefficients is: (1) 227 (2) 964 (3) 625 (4) 232
If the two lines $x + ( a - 1 ) y = 1$ and $2 x + a ^ { 2 } y = 1 , ( a \in R - \{ 0,1 \} )$ are perpendicular, then the distance of their point of intersection from the origin is (1) $\frac { 2 } { \sqrt { 5 } }$ (2) $\frac { \sqrt { 2 } } { 5 }$ (3) $\frac { 2 } { 5 }$ (4) $\sqrt { \frac { 2 } { 5 } }$
A rectangle is inscribed in a circle with a diameter lying along the line $3 y = x + 7$. If the two adjacent vertices of the rectangle are $( - 8,5 )$ and $( 6,5 )$, then the area of the rectangle (in sq. units) is: (1) 72 (2) 98 (3) 56 (4) 84
The area (in sq. units) of the smaller of the two circles that touch the parabola, $y ^ { 2 } = 4 x$ at the point $( 1,2 )$ and the $x$-axis is (1) $8 \pi ( 3 - 2 \sqrt { 2 } )$ (2) $8 \pi ( 2 - \sqrt { 2 } )$ (3) $4 \pi ( 3 + \sqrt { 2 } )$ (4) $4 \pi ( 2 - \sqrt { 2 } )$
If $f ( x ) = [ x ] - \left[ \frac { x } { 4 } \right] , x \in R$, where $[ x ]$ denotes the greatest integer function, then: (1) $\lim _ { x \rightarrow 4 + } f ( x )$ exists but $\lim _ { x \rightarrow 4 - } f ( x )$ does not exist (2) $f$ is continuous at $x = 4$ (3) $\lim _ { x \rightarrow 4 - } f ( x )$ exists but $\lim _ { x \rightarrow 4 + } f ( x )$ does not exist (4) Both $\lim _ { x \rightarrow 4 - } f ( x )$ and $\lim _ { x \rightarrow 4 + } f ( x )$ exist but are not equal
The mean and the median of the following ten numbers in increasing order $10,22,26,29,34 , x , 42,67,70 , y$ are 42 and 35 respectively, then $\frac { y } { x }$ is equal to: (1) $\frac { 9 } { 4 }$ (2) $\frac { 7 } { 3 }$ (3) $\frac { 7 } { 2 }$ (4) $\frac { 8 } { 3 }$
Two poles standing on a horizontal ground are of heights $5 m$ and $10 m$ respectively. The line joining their tops makes an angle of $15 ^ { \circ }$ with the ground. Then the distance (in $m$) between the poles, is (1) $10 ( \sqrt { 3 } - 1 )$ (2) $\frac { 5 } { 2 } ( 2 + \sqrt { 3 } )$ (3) $5 ( 2 + \sqrt { 3 } )$ (4) $5 ( \sqrt { 3 } + 1 )$
The total number of matrices $A = \left( \begin{array} { c c c } 0 & 2 y & 1 \\ 2 x & y & - 1 \\ 2 x & - y & 1 \end{array} \right) , ( x , y \in R , x \neq y )$ for which $A ^ { T } A = 3 I _ { 3 }$ is: (1) 6 (2) 3 (3) 4 (4) 2
If the system of equations $2 x + 3 y - z = 0 , x + k y - 2 z = 0$ and $2 x - y + z = 0$ has a non-trivial solution $( x , y , z )$, then $\frac { x } { y } + \frac { y } { z } + \frac { z } { x } + k$ is equal to (1) $- \frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) $- 4$ (4) $\frac { 3 } { 4 }$
A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$. Water is poured into it at a constant rate of $5$ cubic $\mathrm { m } / \mathrm { min }$. Then the rate (in $\mathrm { m } / \mathrm { min }$), at which the level of water is rising at the instant when the depth of water in the tank is $10 m$; is: (1) $\frac { 1 } { 10 \pi }$ (2) $\frac { 1 } { 15 \pi }$ (3) $\frac { 1 } { 5 \pi }$ (4) $\frac { 2 } { \pi }$
If $\int e ^ { \sec x } \left( \sec x \tan x f ( x ) + \left( \sec x \tan x + \sec ^ { 2 } x \right) \right) d x = e ^ { \sec x } f ( x ) + C$, then a possible choice of $f ( x )$ is: (1) $\sec x - \tan x - \frac { 1 } { 2 }$ (2) $\sec x + \tan x + \frac { 1 } { 2 }$ (3) $x \sec x + \tan x + \frac { 1 } { 2 }$ (4) $\sec x + x \tan x - \frac { 1 } { 2 }$
If $f : R \rightarrow R$ is a differentiable function and $f ( 2 ) = 6$, then $\lim _ { x \rightarrow 2 } \int _ { 6 } ^ { f ( x ) } \frac { 2 t d t } { ( x - 2 ) }$ is: (1) 0 (2) $2 f ^ { \prime } ( 2 )$ (3) $24 f ^ { \prime } ( 2 )$ (4) $12 f ^ { \prime } ( 2 )$
The area (in sq. units) of the region $A = \left\{ ( x , y ) : \frac { y ^ { 2 } } { 2 } \leq x \leq y + 4 \right\}$ is: (1) 30 (2) 18 (3) $\frac { 53 } { 3 }$ (4) 16
The vertices $B$ and $C$ of a $\triangle A B C$ lie on the line, $\frac { x + 2 } { 3 } = \frac { y - 1 } { 0 } = \frac { z } { 4 }$ such that $B C = 5$ units. Then the area (in sq. units) of this triangle, given the point $A ( 1 , - 1,2 )$, is (1) 6 (2) $2 \sqrt { 34 }$ (3) $\sqrt { 34 }$ (4) $5 \sqrt { 17 }$
Let $P$ be the plane, which contains the line of intersection of the planes, $x + y + z - 6 = 0$ and $2 x + 3 y + z + 5 = 0$ and it is perpendicular to the $x y$-plane. Then the distance of the point $( 0,0,256 )$ from $P$ is equal to: (1) $205 \sqrt { 5 }$ units (2) $\frac { 17 } { \sqrt { 5 } }$ units (3) $\frac { 11 } { \sqrt { 5 } }$ units (4) $63 \sqrt { 5 }$ units
Two newspapers $A$ and $B$ are published in a city. It is known that $25 \%$ of the city population reads $A$ and $20 \%$ reads $B$ while $8 \%$ reads both $A$ and $B$. Further, 30\% of those who read $A$ but not $B$ look into advertisements and $40 \%$ of those who read $B$ but not $A$ also look into advertisements, while $50 \%$ of those who read both $A$ and $B$ look into advertisements. Then the percentage of the population who look into advertisements is: (1) 13.5 (2) 12.8 (3) 13.9 (4) 13