Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of GP is $3 r ^ { 2 }$, then $r ^ { 2 } - d$ is equal to : (1) $7 - \sqrt { 3 }$ (2) $7 + 3 \sqrt { 3 }$ (3) $7 - 7 \sqrt { 3 }$ (4) $7 + \sqrt { 3 }$
The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S ( > \mathrm { R } )$ respectively from the origin, is : (1) $2 ( S - R )$ (2) $2 ( S + R )$ (3) $4 ( S - R )$ (4) $4 ( S + R )$
A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is : (1) $8 \sqrt { 10 }$ (2) $6 \sqrt { 10 }$ (3) $12 \sqrt { 10 }$ (4) $12 \sqrt { 15 }$
The function $f ( x ) = \left| x ^ { 2 } - 2 x - 3 \right| \cdot \mathrm { e } ^ { 9 x ^ { 2 } - 12 x + 4 }$ is not differentiable at exactly : (1) Four points (2) Two points (3) three points (4) one point
Let $\vec { a }$ and $\vec { b }$ be two vectors such that $| 2 \vec { a } + 3 \vec { b } | = | 3 \vec { a } + \vec { b } |$ and the angle between $\vec { a }$ and $\vec { b }$ is $60 ^ { \circ }$. If $\frac { 1 } { 8 } \vec { a }$ is a unit vector, then $| \vec { b } |$ is equal to : (1) 8 (2) 4 (3) 6 (4) 5
Let the equation of the plane, that passes through the point $( 1,4 , - 3 )$ and contains the line of intersection of the planes $3 x - 2 y + 4 z - 7 = 0$ and $x + 5 y - 2 z + 9 = 0$, be $\alpha x + \beta y + \gamma z + 3 = 0$, then $\alpha + \beta + \gamma$ is equal to : (1) $- 15$ (2) 15 (3) $- 23$ (4) 23
A point $z$ moves in the complex plane such that $\arg \left( \frac { z - 2 } { z + 2 } \right) = \frac { \pi } { 4 }$, then the minimum value of $| z - 9 \sqrt { 2 } - 2 i | ^ { 2 }$ is equal to
The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is
If $\left( \frac { 3 ^ { 6 } } { 4 ^ { 4 } } \right) k$ is the term, independent of $x$, in the binomial expansion of $\left( \frac { x } { 4 } - \frac { 12 } { x ^ { 2 } } \right) ^ { 12 }$, then $k$ is equal to
If the variable line $3 x + 4 y = \alpha$ lies between the two circles $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$ and $( x - 9 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 4$, without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is
If $R$ is the least value of $a$ such that the function $f ( x ) = x ^ { 2 } + \mathrm { a } x + 1$ is increasing on $[ 1,2 ]$ and $S$ is the greatest value of $a$ such that the function $f ( x ) = x ^ { 2 } + a x + 1$ is decreasing on $[ 1,2 ]$, then the value of $| R - S |$ is
Let $[ t ]$ denote the greatest integer $\leq \mathrm { t }$. Then the value of $8 \cdot \int _ { - \frac { 1 } { 2 } } ^ { 1 } ( [ 2 x ] + | x | ) \mathrm { d } x$ is
The square of the distance of the point of intersection of the line $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z + 1 } { 6 }$ and the plane $2 x - y + z = 6$ from the point $( - 1 , - 1,2 )$ is
An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8 . The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p$, then $98p$ is equal to