If equations $a x ^ { 2 } + b x + c = 0 , ( a , b , c \in R , a \neq 0 )$ and $2 x ^ { 2 } + 3 x + 4 = 0$ have a common root, then $a : b : c$ equals : (1) $2 : 3 : 4$ (2) $4 : 3 : 2$ (3) $1 : 2 : 3$ (4) $3 : 2 : 1$
Let $w ( \operatorname { Im } w \neq 0 )$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w - \bar { w } z = k ( 1 - z )$, for some real number $k$, is (1) $\{ z : z \neq 1 \}$ (2) $\{ z : | z | = 1 , z \neq 1 \}$ (3) $\{ z : z = \bar { z } \}$ (4) $\{ z : | z | = 1 \}$
The sum of the digits in the unit's place of all the 4-digit numbers formed by using the numbers $3,4,5$ and $6$, without repetition is: (1) 18 (2) 36 (3) 108 (4) 432
Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its $4^{\text{th}}$ term is: (1) 8 (2) 24 (3) 20 (4) 16
The number of terms in the expansion of $( 1 + x ) ^ { 101 } \left( 1 - x + x ^ { 2 } \right) ^ { 100 }$ in powers of $x$ is (1) 301 (2) 302 (3) 101 (4) 202
Given three points $P , Q , R$ with $P ( 5,3 )$ and $R$ lies on the $x$-axis. If the equation of $RQ$ is $x - 2 y = 2$ and $PQ$ is parallel to the $x$-axis, then the centroid of $\triangle PQR$ lies on the line (1) $x - 2 y + 1 = 0$ (2) $2 x + y - 9 = 0$ (3) $2 x - 5 y = 0$ (4) $5 x - 2 y = 0$
Let $a$ and $b$ be any two numbers satisfying $\frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } } = \frac { 1 } { 4 }$. Then, the foot of perpendicular from the origin on the variable line $\frac { x } { a } + \frac { y } { b } = 1$ lies on: (1) A circle of radius $= 2$ (2) A hyperbola with each semi-axis $= \sqrt { 2 }$. (3) A hyperbola with each semi-axis $= 2$ (4) A circle of radius $= \sqrt { 2 }$
If the point $( 1,4 )$ lies inside the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + p = 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval (1) $( 25,39 )$ (2) $( 25,29 )$ (3) $( 0,25 )$ (4) $( 9,25 )$
If $OB$ is the semi-minor axis of an ellipse, $F _ { 1 }$ and $F _ { 2 }$ are its focii and the angle between $F _ { 1 } B$ and $F _ { 2 } B$ is a right angle, then the square of the eccentricity of the ellipse is (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { \sqrt { 2 } }$ (3) $\frac { 1 } { 2 }$ (4) $\frac { 1 } { 2 \sqrt { 2 } }$
In a set of $2n$ distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then, the mean of the new set of observations: (1) Increases by 2. (2) Increase by 1. (3) Decreases by 2. (4) Decreases by 1.
If $a , b , c$ are non-zero real numbers and if the system of equations $$( a - 1 ) x = y + z$$ $$( b - 1 ) y = x + z$$ $$( c - 1 ) z = x + y$$ has a non-trivial solution, then $ab + bc + ca$ equals: (1) $-1$ (2) $a + b + c$ (3) $abc$ (4) 1
If $y = e ^ { n x }$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } \cdot \frac { d ^ { 2 } x } { d y ^ { 2 } }$ is equal to: (1) $n e ^ { - n x }$ (2) $- n e ^ { - n x }$ (3) $n e ^ { n x }$ (4) 1
If $f ( x ) = \left( \frac { 3 } { 5 } \right) ^ { x } + \left( \frac { 4 } { 5 } \right) ^ { x } - 1 , x \in R$, then the equation $f ( x ) = 0$ has: (1) No solution (2) More than two solutions (3) One solution (4) Two solutions
If the Rolle's theorem holds for the function $f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x$ in the interval $[ - 1,1 ]$ for the point $c = \frac { 1 } { 2 }$, then the value of $2 a + b$ is: (1) $-1$ (2) 2 (3) 1 (4) $-2$
If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left( x ^ { 2 } - y ^ { 2 } \right) \frac { d y } { d x } = g ( x ) y$, then $g ( x )$ equals (1) $\frac { 1 } { 2 } x ^ { 2 }$ (2) $2 x$ (3) $\frac { 1 } { 2 } x$ (4) $2 x ^ { 2 }$
Equation of the plane which passes through the point of intersection of lines $\frac { x - 1 } { 3 } = \frac { y - 2 } { 1 } = \frac { z - 3 } { 2 }$ and $\frac { x - 3 } { 1 } = \frac { y - 1 } { 2 } = \frac { z - 2 } { 3 }$ and has the largest distance from the origin is: (1) $4 x + 3 y + 5 z = 50$ (2) $3 x + 4 y + 5 z = 49$ (3) $5 x + 4 y + 3 z = 57$ (4) $7 x + 2 y + 4 z = 54$
If $A$ and $B$ are two events such that $P ( A \cup B ) = P ( A \cap B )$, then the incorrect statement amongst the following statements is: (1) $P ( A ) + P ( B ) = 1$ (2) $P \left( A \cap B ^ { \prime } \right) = 0$ (3) $A \& B$ are equally likely (4) $P \left( A ^ { \prime } \cap B \right) = 0$