The integer $k$, for which the inequality $x ^ { 2 } - 2 ( 3 k - 1 ) x + 8 k ^ { 2 } - 7 > 0$ is valid for every $x$ in $R$ is: (1) 4 (2) 2 (3) 3 (4) 0
Let the lines $( 2 - i ) z = ( 2 + i ) \bar { z }$ and $( 2 + i ) z + ( i - 2 ) \bar { z } - 4 i = 0$, (here $i ^ { 2 } = - 1$ ) be normal to a circle $C$. If the line $i z + \bar { z } + 1 + i = 0$ is tangent to this circle $C$, then its radius is : (1) $\frac { 3 } { \sqrt { 2 } }$ (2) $3 \sqrt { 2 }$ (3) $\frac { 3 } { 2 \sqrt { 2 } }$ (4) $\frac { 1 } { 2 \sqrt { 2 } }$
If $0 < \theta , \phi < \frac { \pi } { 2 } , x = \sum _ { n = 0 } ^ { \infty } \cos ^ { 2 n } \theta , y = \sum _ { n = 0 } ^ { \infty } \sin ^ { 2 n } \phi$ and $z = \sum _ { n = 0 } ^ { \infty } \cos ^ { 2 n } \theta \cdot \sin ^ { 2 n } \phi$ then : (1) $x y - z = ( x + y ) z$ (2) $x y + y z + z x = z$ (3) $x y + z = ( x + y ) z$ (4) $x y z = 4$
A tangent is drawn to the parabola $y ^ { 2 } = 6 x$ which is perpendicular to the line $2 x + y = 1$. Which of the following points does NOT lie on it? (1) $( 0,3 )$ (2) $( 4,5 )$ (3) $( 5,4 )$ (4) $( - 6,0 )$
If the curves, $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$ and $\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1$ intersect each other at an angle of $90 ^ { \circ }$, then which of the following relations is TRUE? (1) $a - c = b + d$ (2) $a - b = c - d$ (3) $a + b = c + d$ (4) $a b = \frac { c + d } { a + b }$
A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point $A$, with uniform speed. At that point, angle of depression of the boat with the man's eye is $30 ^ { \circ }$ (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point $B$, where the angle of depression is $45 ^ { \circ }$. Then the time taken (in seconds) by the boat from $B$ to reach the base of the tower is : (1) 10 (2) $10 ( \sqrt { 3 } - 1 )$ (3) $10 \sqrt { 3 }$ (4) $10 ( \sqrt { 3 } + 1 )$
Let $f , g : N \rightarrow N$ such that $f ( n + 1 ) = f ( n ) + f ( 1 ) \forall n \in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true? (1) If $f$ is onto, then $f ( n ) = n \forall n \in N$ (2) If $g$ is onto, then $f o g$ is one-one (3) $f$ is one-one (4) If $f \circ g$ is one-one, then $g$ is one-one
The value of $\int _ { - 1 } ^ { 1 } x ^ { 2 } e ^ { \left[ x ^ { 3 } \right] } d x$, where $[ t ]$ denotes the greatest integer $\leq t$, is : (1) $\frac { e + 1 } { 3 }$ (2) $\frac { e - 1 } { 3 e }$ (3) $\frac { 1 } { 3 e }$ (4) $\frac { e + 1 } { 3 e }$
If a curve passes through the origin and the slope of the tangent to it at any point $( x , y )$ is $\frac { x ^ { 2 } - 4 x + y + 8 } { x - 2 }$, then this curve also passes through the point: (1) $( 5,4 )$ (2) $( 4,4 )$ (3) $( 4,5 )$ (4) $( 5,5 )$
The coefficients $a , b$ and $c$ of the quadratic equation, $a x ^ { 2 } + b x + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is: (1) $\frac { 1 } { 72 }$ (2) $\frac { 1 } { 36 }$ (3) $\frac { 1 } { 54 }$ (4) $\frac { 5 } { 216 }$
When a missile is fired from a ship, the probability that it is intercepted is $\frac { 1 } { 3 }$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac { 3 } { 4 }$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is: (1) $\frac { 3 } { 8 }$ (2) $\frac { 1 } { 27 }$ (3) $\frac { 1 } { 8 }$ (4) $\frac { 3 } { 4 }$
The total number of numbers, lying between 100 and 1000 that can be formed with the digits $1,2,3,4,5$, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is
Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots$ be squares such that for each $n \geqslant 1$, the length of the side of $A _ { n }$ equals the length of diagonal of $A _ { n + 1 }$. If the length of $A _ { 1 }$ is 12 cm, then the smallest value of $n$ for which area of $A _ { n }$ is less than one, is
The locus of the point of intersection of the lines $( \sqrt { 3 } ) k x + k y - 4 \sqrt { 3 } = 0$ and $\sqrt { 3 } x - y - 4 ( \sqrt { 3 } ) k = 0$ is a conic, whose eccentricity is
Let $A = \left[ \begin{array} { l l l } x & y & z \\ y & z & x \\ z & x & y \end{array} \right]$, where $x , y$ and $z$ are real numbers such that $x + y + z > 0$ and $x y z = 2$. If $A ^ { 2 } = I _ { 3 }$, then the value of $x ^ { 3 } + y ^ { 3 } + z ^ { 3 }$ is
If the system of equations $$\begin{aligned}
& k x + y + 2 z = 1 \\
& 3 x - y - 2 z = 2 \\
& - 2 x - 2 y - 4 z = 3
\end{aligned}$$ has infinitely many solutions, then $k$ is equal to
The number of points, at which the function $f ( x ) = | 2 x + 1 | - 3 | x + 2 | + \left| x ^ { 2 } + x - 2 \right| , x \in R$ is not differentiable, is
Let $f ( x )$ be a polynomial of degree 6 in $x$, in which the coefficient of $x ^ { 6 }$ is unity and it has extrema at $x = - 1$ and $x = 1$. If $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } } = 1$, then $5 \cdot f ( 2 )$ is equal to
The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $A ^ { 4 }$ is equal to
Let $\vec { a } = \hat { i } + 2 \hat { j } - \widehat { k } , \vec { b } = \hat { i } - \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } - \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = \vec { c } \times \vec { a }$ and $\vec { r } \cdot \vec { b } = 0$, then $\vec { r } \cdot \vec { a }$ is equal to