60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50 ^ { \text {th} }$ word is : (1) JBBOH (2) OBBJH (3) OBBHJ (4) HBBJO
For $x \geqslant 0$, the least value of K , for which $4 ^ { 1 + x } + 4 ^ { 1 - x } , \frac { \mathrm {~K} } { 2 } , 16 ^ { x } + 16 ^ { - x }$ are three consecutive terms of an A.P., is equal to : (1) 8 (2) 4 (3) 10 (4) 16
Let $A ( - 1,1 )$ and $B ( 2,3 )$ be two points and $P$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm { PAB }$ is 10 . If the locus of P is $\mathrm { a } x + \mathrm { b } y = 15$, then $5 \mathrm { a } + 2 \mathrm {~b}$ is : (1) 6 (2) $- \frac { 6 } { 5 }$ (3) 4 (4) $- \frac { 12 } { 5 }$
Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment AB and the point F is on the diagonal AC . Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies: (1) $r = 0$ (2) $2 r ^ { 2 } - 4 r + 1 = 0$ (3) $2 r ^ { 2 } - 8 r + 7 = 0$ (4) $r ^ { 2 } - 8 r + 8 = 0$
Let the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 2 ( x + y ) + 1 = 0$ and $C _ { 2 }$ be a circle having centre at $( - 1,0 )$ and radius 2 . If the line of the common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ intersects the $y$-axis at the point P , then the square of the distance of P from the centre of $\mathrm { C } _ { 1 }$ is : (1) 2 (2) 1 (3) 4 (4) 6
Let the set $S = \{ 2,4,8,16 , \ldots , 512 \}$ be partitioned into 3 sets $A , B , C$ with equal number of elements such that $\mathrm { A } \cup \mathrm { B } \cup \mathrm { C } = \mathrm { S }$ and $\mathrm { A } \cap \mathrm { B } = \mathrm { B } \cap \mathrm { C } = \mathrm { A } \cap \mathrm { C } = \phi$. The maximum number of such possible partitions of $S$ is equal to: (1) 1680 (2) 1640 (3) 1520 (4) 1710
Let $\alpha \beta \neq 0$ and $A = \left[ \begin{array} { r r r } \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2 \alpha \end{array} \right]$. If $B = \left[ \begin{array} { r r r } 3 \alpha & - 9 & 3 \alpha \\ - \alpha & 7 & - 2 \alpha \\ - 2 \alpha & 5 & - 2 \beta \end{array} \right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname { det } ( A B )$ is equal to : (1) 64 (2) 216 (3) 343 (4) 125
The values of $m , n$, for which the system of equations \begin{align*} x + y + z &= 4, 2x + 5y + 5z &= 17, x + 2y + \mathrm{m}z &= \mathrm{n} \end{align*} has infinitely many solutions, satisfy the equation: (1) $m ^ { 2 } + n ^ { 2 } - m n = 39$ (2) $m ^ { 2 } + n ^ { 2 } - m - n = 46$ (3) $m ^ { 2 } + n ^ { 2 } + m + n = 64$ (4) $m ^ { 2 } + n ^ { 2 } + m n = 68$
Let $f , g : \mathbf { R } \rightarrow \mathbf { R }$ be defined as : $f ( x ) = | x - 1 |$ and $g ( x ) = \begin{cases} \mathrm { e } ^ { x } , & x \geq 0 \\ x + 1 , & x \leq 0 \end{cases}$ Then the function $f ( g ( x ) )$ is (1) neither one-one nor onto. (2) one-one but not onto. (3) onto but not one-one. (4) both one-one and onto.
Let $f : [ - 1,2 ] \rightarrow \mathbf { R }$ be given by $f ( x ) = 2 x ^ { 2 } + x + \left[ x ^ { 2 } \right] - [ x ]$, where $[ t ]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is : (1) 5 (2) 6 (3) 3 (4) 4
The area enclosed between the curves $y = x | x |$ and $y = x - | x |$ is : (1) $\frac { 4 } { 3 }$ (2) 1 (3) $\frac { 2 } { 3 }$ (4) $\frac { 8 } { 3 }$
The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is : (1) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } - 2 x y \right) \mathrm { d } y$ (2) $\left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } y$ (3) $\left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } y$ (4) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } y$
Consider three vectors $\vec { a } , \vec { b } , \vec { c }$. Let $| \vec { a } | = 2 , | \vec { b } | = 3$ and $\vec { a } = \vec { b } \times \vec { c }$. If $\alpha \in \left[ 0 , \frac { \pi } { 3 } \right]$ is the angle between the vectors $\vec { b }$ and $\vec { c }$, then the minimum value of $27 | \vec { c } - \vec { a } | ^ { 2 }$ is equal to: (1) 110 (2) 124 (3) 121 (4) 105
Let $( \alpha , \beta , \gamma )$ be the image of the point $( 8,5,7 )$ in the line $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 2 } { 5 }$. Then $\alpha + \beta + \gamma$ is equal to : (1) 16 (2) 20 (3) 14 (4) 18
The coefficients $a , b , c$ in the quadratic equation $a x ^ { 2 } + b x + c = 0$ are from the set $\{ 1,2,3,4,5,6 \}$. If the probability of this equation having one real root bigger than the other is $p$, then 216 p equals : (1) 57 (2) 76 (3) 38 (4) 19
The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$
Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y ^ { 2 } = 4 ( x - 9 )$ at the point $P$. The distance of the point $P$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 14 x - 8 y + 56 = 0$ is $\_\_\_\_$
Let the maximum and minimum values of $\left( \sqrt { 8 x - x ^ { 2 } - 12 } - 4 \right) ^ { 2 } + ( x - 7 ) ^ { 2 } , x \in \mathbf { R }$ be M and m , respectively. Then $\mathrm { M } ^ { 2 } - \mathrm { m } ^ { 2 }$ is equal to $\_\_\_\_$
If $f ( t ) = \int _ { 0 } ^ { \pi } \frac { 2 x \mathrm {~d} x } { 1 - \cos ^ { 2 } \mathrm { t } \sin ^ { 2 } x } , 0 < \mathrm { t } < \pi$, then the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \pi ^ { 2 } \mathrm { dt } } { f ( \mathrm { t } ) }$ equals $\_\_\_\_$
Q89
First order differential equations (integrating factor)View
Let $y = y ( x )$ be the solution of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } y = x \mathrm { e } ^ { \frac { 1 } { \left( 1 + x ^ { 2 } \right) } } ; y ( 0 ) = 0$. Then the area enclosed by the curve $f ( x ) = y ( x ) \mathrm { e } ^ { - \frac { 1 } { \left( 1 + x ^ { 2 } \right) } }$ and the line $y - x = 4$ is $\_\_\_\_$
Let the point $( - 1 , \alpha , \beta )$ lie on the line of the shortest distance between the lines $\frac { x + 2 } { - 3 } = \frac { y - 2 } { 4 } = \frac { z - 5 } { 2 }$ and $\frac { x + 2 } { - 1 } = \frac { y + 6 } { 2 } = \frac { z - 1 } { 0 }$. Then $( \alpha - \beta ) ^ { 2 }$ is equal to $\_\_\_\_$