A rod of mass $M$ and length $L$ is lying on a horizontal frictionless surface. A particle of mass $m$ travelling along the surface hits at one end of the rod with a velocity $u$ in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses ( $\frac { m } { M }$ ) is $\frac { 1 } { x }$. The value of $x$ will be
Let the tangent to the parabola $S : y ^ { 2 } = 2 x$ at the point $P ( 2,2 )$ meet the $x$-axis at $Q$ and normal at it meet the parabola $S$ at the point $R$. Then the area (in sq. units) of the triangle $P Q R$ is equal to: (1) $\frac { 25 } { 2 }$ (2) $\frac { 35 } { 2 }$ (3) $\frac { 15 } { 2 }$ (4) 25
The mean of 6 distinct observations is 6.5 and their variance is 10.25 . If 4 out of 6 observations are $2,4,5$ and 7, then the remaining two observations are: (1) 10,11 (2) 3,18 (3) 8,13 (4) 1,20
If in a triangle $A B C , A B = 5$ units, $\angle B = \cos ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ and radius of circumcircle of $\triangle A B C$ is 5 units, then the area (in sq. units) of $\triangle A B C$ is: (1) $10 + 6 \sqrt { 2 }$ (2) $8 + 2 \sqrt { 2 }$ (3) $6 + 8 \sqrt { 3 }$ (4) $4 + 2 \sqrt { 3 }$
Let $A = \left[ \begin{array} { l l } 2 & 3 \\ a & 0 \end{array} \right] , a \in R$ be written as $P + Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\operatorname { det } ( Q ) = 9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to: (1) 36 (2) 24 (3) 45 (4) 18
Let $[ x ]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $f ( x ) = \sqrt { \frac { | [ x ] | - 2 } { | [ x ] | - 3 } }$ is $( - \infty , a ) \cup [ b , c ) \cup [ 4 , \infty ) , a < b < c$, then the value of $a + b + c$ is: (1) 8 (2) 1 (3) $- 2$ (4) $- 3$
Let a function $f : R \rightarrow R$ be defined as, $f ( x ) = \begin{cases} \sin x - e ^ { x } & \text { if } x \leq 0 \\ a + [ - x ] & \text { if } 0 < x < 1 \\ 2 x - b & \text { if } x \geq 1 \end{cases}$ Where $[ x ]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then ( $a + b$ ) is equal to: (1) 4 (2) 3 (3) 2 (4) 5
Let $A = \left[ a _ { i j } \right]$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { c c } 1 , & \text { if } i = j \\ - x , & \text { if } | i - j | = 1 \\ 2 x + 1 , & \text { otherwise } \end{array} \right.$ Let a function $f : R \rightarrow R$ be defined as $f ( x ) = \operatorname { det } ( A )$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to: (1) $- \frac { 20 } { 27 }$ (2) $\frac { 88 } { 27 }$ (3) $\frac { 20 } { 27 }$ (4) $- \frac { 88 } { 27 }$
Let $a$ be a real number such that the function $f ( x ) = a x ^ { 2 } + 6 x - 15 , x \in R$ is increasing in $( - \infty , \frac { 3 } { 4 } )$ and decreasing in $\left( \frac { 3 } { 4 } , \infty \right)$. Then the function $g ( x ) = a x ^ { 2 } - 6 x + 15 , x \in R$ has a (1) local maximum at $x = - \frac { 3 } { 4 }$ (2) local minimum at $x = - \frac { 3 } { 4 }$ (3) local maximum at $x = \frac { 3 } { 4 }$ (4) local minimum at $x = \frac { 3 } { 4 }$
Let $a$ be a positive real number such that $\int _ { 0 } ^ { a } e ^ { x - [ x ] } d x = 10 e - 9$ where, $[ x ]$ is the greatest integer less than or equal to $x$. Then, $a$ is equal to: (1) $10 - \log _ { e } ( 1 + e )$ (2) $10 + \log _ { e } 2$ (3) $10 + \log _ { e } ( 1 + e )$ (4) $10 - \log _ { e } 2$
Let $f : R \rightarrow R$ be defined as $f ( x ) = e ^ { - x } \sin x$. If $F : [ 0,1 ] \rightarrow R$ is a differentiable function such that $F ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, then the value of $\int _ { 0 } ^ { 1 } \left( F ^ { \prime } ( x ) + f ( x ) \right) e ^ { x } d x$ lies in the interval
Let $\vec { a } = \hat { i } + \hat { j } + \hat { k }$ and $\vec { b } = \hat { j } - \hat { k }$. If $\vec { c }$ is a vector such that $\vec { a } \times \vec { c } = \vec { b }$ and $\vec { a } \cdot \vec { c } = 3$, then $\vec { a } \cdot ( \vec { b } \times \vec { c } )$ is equal to
The locus of the midpoints of the chord of the circle, $x ^ { 2 } + y ^ { 2 } = 25$ which is tangent to the hyperbola, $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ is:
The number of solutions of the equation $\log _ { 4 } ( x - 1 ) = \log _ { 2 } ( x - 3 )$ is
Q82
First order differential equations (integrating factor)View
If $y = y ( x )$ is the solution of the differential equation $\frac { d y } { d x } + ( \tan x ) y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$, with $y ( 0 ) = 0$, then $y \left( \frac { \pi } { 4 } \right)$ equal to
Let $P$ be a plane $l x + m y + n z = 0$ containing the line, $\frac { 1 - x } { 1 } = \frac { y + 4 } { 2 } = \frac { z + 2 } { 3 }$. If plane $P$ divides the line segment $A B$ joining points $A ( - 3 , - 6,1 )$ and $B ( 2,4 , - 3 )$ in ratio $k : 1$ then the value of $k$ is
Let $A = \{ n \in N : n$ is a 3-digit number $\}$, $B = \{ 9 k + 2 : k \in N \}$ and $C = \{ 9 k + l : k \in N \}$ for some $l ( 0 < l < 9 )$. If the sum of all the elements of the set $A \cap ( B \cup C )$ is $274 \times 400$, then $l$ is equal to
Let $f : [ - 1,1 ] \rightarrow R$ be defined as $f ( x ) = a x ^ { 2 } + b x + c$ for all $x \in [ - 1,1 ]$, where $a , b , c \in R$ such that $f ( - 1 ) = 2 , f ^ { \prime } ( - 1 ) = 1$ and for $x \in ( - 1,1 )$ the maximum value of $f ^ { \prime \prime } ( x )$ is $\frac { 1 } { 2 }$. If $f ( x ) \leq \alpha , x \in [ - 1,1 ]$, then the least value of $\alpha$ is equal to
Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots$ be squares such that for each $n \geq 1$, 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