A body of mass 2 kg moves under a force of $( 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } + 5 \widehat { \mathrm { k } } ) \mathrm { N }$. It starts from rest and was at the origin initially. After 4 s, its new coordinates are $( 8 , b , 20 )$. The value of $b$ is $\_\_\_\_$. (Round off to the Nearest Integer)
A swimmer can swim with velocity of $12 \mathrm {~km} / \mathrm { h }$ in still water. Water flowing in a river has velocity $6 \mathrm {~km} / \mathrm { h }$. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is $\_\_\_\_$. (Round off to the Nearest Integer) (find the angle in degree)
A force $\vec { F } = 4 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } + 4 \widehat { \mathrm { k } }$ is applied on an intersection point of $x = 2$ plane and $x$-axis. The magnitude of torque of this force about a point $( 2,3,4 )$ is $\_\_\_\_$. (Round off to the Nearest Integer)
Consider a rectangle $ABCD$ having $5,6,7,9$ points in the interior of the line segments $AB , BC , CD , DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta - \alpha)$ is equal to (1) 795 (2) 1173 (3) 1890 (4) 717
Let $A ( - 1,1 ) , B ( 3,4 )$ and $C ( 2,0 )$ be given three points. A line $y = m x , m > 0$, intersects lines $AC$ and $BC$ at point $P$ and $Q$ respectively. Let $A _ { 1 }$ and $A _ { 2 }$ be the areas of $\triangle ABC$ and $\triangle PQC$ respectively, such that $A _ { 1 } = 3 A _ { 2 }$, then the value of $m$ is equal to : (1) $\frac { 4 } { 15 }$ (2) 1 (3) 2 (4) 3
Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x ^ { 2 } + y ^ { 2 } + ax + 2ay + c = 0 , ( a < 0 )$ be $2 \sqrt { 2 }$ and $2 \sqrt { 5 }$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x + 2y = 0$, is equal to : (1) $\sqrt { 11 }$ (2) $\sqrt { 7 }$ (3) $\sqrt { 6 }$ (4) $\sqrt { 10 }$
Let $C$ be the locus of the mirror image of a point on the parabola $y ^ { 2 } = 4x$ with respect to the line $y = x$. Then the equation of tangent to $C$ at $P ( 2,1 )$ is : (1) $x - y = 1$ (2) $2x + y = 5$ (3) $x + 3y = 5$ (4) $x + 2y = 4$
If the points of intersection of the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the circle $x ^ { 2 } + y ^ { 2 } = 4b , b > 4$ lie on the curve $y ^ { 2 } = 3x ^ { 2 }$, then $b$ is equal to : (1) 12 (2) 5 (3) 6 (4) 10
Let $A = \{ 2,3,4,5 , \ldots , 30 \}$ and $\sim$ be an equivalence relation on $A \times A$, defined by $( a , b ) \simeq ( c , d )$, if and only if $ad = bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $( 4,3 )$ is equal to : (1) 5 (2) 6 (3) 8 (4) 7
Let $\alpha \in R$ be such that the function $f ( x ) = \left\{ \begin{array} { l l } \frac { \cos ^ { - 1 } \left( 1 - \{ x \} ^ { 2 } \right) \sin ^ { - 1 } ( 1 - \{ x \} ) } { \{ x \} - \{ x \} ^ { 3 } } , & x \neq 0 \\ \alpha , & x = 0 \end{array} \right.$ is continuous at $x = 0$, where $\{ x \} = x - [ x ] , [ x ]$ is the greatest integer less than or equal to $x$. Then : (1) $\alpha = \frac { \pi } { \sqrt { 2 } }$ (2) $\alpha = 0$ (3) no such $\alpha$ exists (4) $\alpha = \frac { \pi } { 4 }$
Let $f : S \rightarrow S$ where $S = ( 0 , \infty )$ be a twice differentiable function such that $f ( x + 1 ) = x f ( x )$. If $g : S \rightarrow R$ be defined as $g ( x ) = \log _ { \mathrm { e } } f ( x )$, then the value of $\left| g ^ { \prime \prime } ( 5 ) - g ^ { \prime \prime } ( 1 ) \right|$ is equal to : (1) $\frac { 205 } { 144 }$ (2) $\frac { 197 } { 144 }$ (3) $\frac { 187 } { 144 }$ (4) 1
Consider the integral $I = \int _ { 0 } ^ { 10 } \frac { [ x ] e ^ { [ x ] } } { e ^ { x - 1 } } d x$ where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to : (1) $9 ( e - 1 )$ (2) $45 ( e + 1 )$ (3) $45 ( e - 1 )$ (4) $9 ( e + 1 )$
Let $P ( x ) = x ^ { 2 } + bx + c$ be a quadratic polynomial with real coefficients such that $\int _ { 0 } ^ { 1 } P ( x ) d x = 1$ and $P ( x )$ leaves remainder 5 when it is divided by $( x - 2 )$. Then the value of $9 ( b + c )$ is equal to: (1) 9 (2) 15 (3) 7 (4) 11
Q75
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 $y \left( \frac { \pi } { 4 } \right)$ is equal to (1) $\frac { 1 } { 4 } \log _ { e } 2$ (2) $\left( \frac { 1 } { 2 \sqrt { 2 } } \right) \log _ { e } 2$ (3) $\log _ { e } 2$ (4) $\frac { 1 } { 2 } \log _ { e } 2$