Consider a force $\vec { F } = - x \hat { i } + y \hat { j }$. The work done by this force in moving a particle from point $A ( 1,0 )$ to $B ( 0,1 )$ along the line segment is : (all quantities are in SI units) (1) 2 (2) $\frac { 1 } { 2 }$ (3) 1 (4) $\frac { 3 } { 2 }$
Two particles of equal mass $m$ have respective initial velocities $u \hat { i }$ and $u \left( \frac { \hat { i } + \hat { j } } { 2 } \right)$. They collide completely inelastically. The energy lost in the process is: (1) $\frac { 1 } { 3 } m u ^ { 2 }$ (2) $\frac { 1 } { 8 } m u ^ { 2 }$ (3) $\frac { 3 } { 4 } m u ^ { 2 }$ (4) $\sqrt { \frac { 2 } { 3 } } m u ^ { 2 }$
Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$. The ratio $\frac { I _ { 0 } } { I _ { A } }$ of moment of inertia $I _ { 0 }$ of the system about an axis passing the centroid and about center of any of the spheres $I _ { A }$ and perpendicular to the plane of the triangle is: (1) $\frac { 13 } { 23 }$ (2) $\frac { 15 } { 13 }$ (3) $\frac { 23 } { 13 }$ (4) $\frac { 13 } { 15 }$
A body A of mass $m$ is moving in a circular orbit of radius $R$ about a planet. Another body B of mass $\frac { m } { 2 }$ collides with A with a velocity which is half $\left( \frac { \vec { v } } { 2 } \right)$ the instantaneous velocity $\vec { v }$ of A. The collision is completely inelastic. Then, the combined body: (1) continues to move in a circular orbit (2) Escapes from the Planet's Gravitational field (3) Falls vertically downwards towards the planet (4) starts moving in an elliptical orbit around the planet
The distance $x$ covered by a particle in one dimensional motion varies with time $t$ as $x ^ { 2 } = a t ^ { 2 } + 2 b t + c$. If the acceleration of the particle depends on $x$ as $x ^ { - n }$, where $n$ is an integer, the value of $n$ is $\_\_\_\_$
One end of a straight uniform $1 m$ long bar is pivoted on horizontal table. It is released from rest when it makes an angle $30 ^ { \circ }$ from the horizontal (see figure). Its angular speed when it hits the table is given as $\sqrt { n } \mathrm { rad } s ^ { - 1 }$, where $n$ is an integer. The value of $n$ is $\_\_\_\_$
A body of mass $\mathrm { m } = 10 \mathrm {~kg}$ is attached to one end of a wire of length 0.3 m . What is the maximum angular speed (in $\mathrm { rad } \mathrm { s } ^ { - 1 }$ ) with which it can be rotated about its other end in a space station without breaking the wire? [Breaking stress of wire $( \sigma ) = 4.8 \times 10 ^ { 7 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$ and area of cross-section of the wire $= 10 ^ { - 2 } \mathrm {~cm} ^ { 2 }$ ]
Let $z$ be a complex number such that $\left| \frac { z - i } { z + 2 i } \right| = 1$ and $| z | = \frac { 5 } { 2 }$. Then, the value of $| z + 3 i |$ is (1) $\sqrt { 10 }$ (2) $\frac { 7 } { 2 }$ (3) $\frac { 15 } { 4 }$ (4) $2 \sqrt { 3 }$
If the number of five digit numbers with distinct digits and 2 at the $10 ^ { \text {th} }$ place is $336 k$, then $k$ is equal to: (1) 4 (2) 6 (3) 7 (4) 8
A circle touches the $y$-axis at the point $( 0,4 )$ and passes through the point $( 2,0 )$. Which of the following lines is not a tangent to this circle? (1) $4 x - 3 y + 17 = 0$ (2) $3 x - 4 y - 24 = 0$ (3) $3 x + 4 y - 6 = 0$ (4) $4 x + 3 y - 8 = 0$
If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of the ellipse $\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 4 } = 1$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ respectively and $\left( e _ { 1 } , e _ { 2 } \right)$ is a point on the ellipse $15 x ^ { 2 } + 3 y ^ { 2 } = k$, then the value of $k$ is equal to (1) 16 (2) 17 (3) 15 (4) 14
If for some $\alpha$ and $\beta$ in $R$, the intersection of the following three planes $x + 4 y - 2 z = 1$ $x + 7 y - 5 z = \beta$ $x + 5 y + \alpha z = 5$ is a line in $R ^ { 3 }$, then $\alpha + \beta$ is equal to: (1) 0 (2) 10 (3) 2 (4) - 10
Let $f$ be any function continuous on $[ a , b ]$ and twice differentiable on $( a , b )$. If all $x \in ( a , b ) , f ^ { \prime } ( x ) > 0$ and $f ^ { \prime \prime } ( x ) < 0$, then for any $c \in ( a , b ) , \frac { f ( c ) - f ( a ) } { f ( b ) - f ( c ) }$ (1) $\frac { b + a } { b - a }$ (2) 1 (3) $\frac { b - c } { c - a }$ (4) $\frac { c - a } { b - c }$
A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $50 \mathrm {~cm} ^ { 3 } / \mathrm { min }$. When the thickness of ice is 5 cm , then the rate (in $\mathrm { cm } / \mathrm { min }$.) at which of the thickness of ice decreases, is: (1) $\frac { 5 } { 6 \pi }$ (2) $\frac { 1 } { 54 \pi }$ (3) $\frac { 1 } { 36 \pi }$ (4) $\frac { 1 } { 18 \pi }$
Let $D$ be the centroid of the triangle with vertices $( 3 , - 1 ) , ( 1,3 )$ and $( 2,4 )$. Let P be the point of intersection of the lines $x + 3 y - 1 = 10$ and $3 x - y + 1 = 0$. Then, the line passing through the points $D$ and P also passes through the point: (1) $( - 9 , - 6 )$ (2) $( 9,7 )$ (3) $( 7,6 )$ (4) $( - 9 , - 7 )$
In a box, there are 20 cards, out of which 10 are labelled as $A$ and the remaining 10 are labelled as $B$. Cards are drawn at random, one after the other and with replacement, till a second $A$ card is obtained. The probability that the second $A$ card appears before the third $B$ card is: (1) $\frac { 9 } { 16 }$ (2) $\frac { 11 } { 16 }$ (3) $\frac { 13 } { 16 }$ (4) $\frac { 15 } { 16 }$