A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height $h$. Find the ratio of the times in which it is at height $\frac { h } { 3 }$ while going up and coming down respectively. (1) $\frac { \sqrt { 2 } - 1 } { \sqrt { 2 } + 1 }$ (2) $\frac { \sqrt { 3 } - \sqrt { 2 } } { \sqrt { 3 } + \sqrt { 2 } }$ (3) $\frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 }$ (4) $\frac { 1 } { 3 }$
A ball is projected with kinetic energy $E$, at an angle of $60 ^ { \circ }$ to the horizontal. The kinetic energy of this ball at the highest point of its flight will become : (1) Zero (2) $\frac { E } { 2 }$ (3) $\frac { E } { 4 }$ (4) $E$
Two bodies of mass 1 kg and 3 kg have position vectors $\hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ and $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector : (1) $\hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ (2) $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ (3) $- 2 \hat { \mathrm { i } } + 2 \widehat { \mathrm { k } }$ (4) $- 2 \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$
An object is projected in the air with initial velocity $u$ at an angle $\theta$. The projectile motion is such that the horizontal range $R$, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be $\_\_\_\_$ degree.
Let the circumcentre of a triangle with vertices $A ( a , 3 ) , B ( b , 5 )$ and $C ( a , b ) , a b > 0$ be $P ( 1,1 )$. If the line $A P$ intersects the line $B C$ at the point $Q \left( k _ { 1 } , k _ { 2 } \right)$, then $k _ { 1 } + k _ { 2 }$ is equal to (1) 2 (2) $\frac { 4 } { 7 }$ (3) $\frac { 2 } { 7 }$ (4) 4
Let a line $L$ pass through the point of intersection of the lines $b x + 10 y - 8 = 0$ and $2 x - 3 y = 0$, $b \in R - \left\{ \frac { 4 } { 3 } \right\}$. If the line $L$ also passes through the point $( 1,1 )$ and touches the circle $17 \left( x ^ { 2 } + y ^ { 2 } \right) = 16$, then the eccentricity of the ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is (1) $\frac { 2 } { \sqrt { 5 } }$ (2) $\sqrt { \frac { 3 } { 5 } }$ (3) $\frac { 1 } { \sqrt { 5 } }$ (4) $\sqrt { \frac { 2 } { 5 } }$
Let the focal chord of the parabola $P : y ^ { 2 } = 4 x$ along the line $L : y = m x + c , m > 0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $O M F N$ is (1) $2 \sqrt { 6 }$ (2) $2 \sqrt { 14 }$ (3) $4 \sqrt { 6 }$ (4) $4 \sqrt { 14 }$
The angle of elevation of the top of a tower from a point $A$ due north of it is $\alpha$ and from a point $B$ at a distance of 9 units due west of $A$ is $\cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 13 } } \right)$. If the distance of the point $B$ from the tower is 15 units, then $\cot \alpha$ is equal to (1) $\frac { 6 } { 5 }$ (2) $\frac { 9 } { 5 }$ (3) $\frac { 4 } { 3 }$ (4) $\frac { 7 } { 3 }$
Let $R$ be a relation from the set $\{ 1,2,3 \ldots\ldots . , 60 \}$ to itself such that $R = \{ ( a , b ) : b = p q$, where $p , q \geq 3$ are prime numbers\}. Then, the number of elements in $R$ is (1) 600 (2) 660 (3) 540 (4) 720
Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $A B$ is a zero matrix. Then (1) The system of linear equations $A X = 0$ has a unique solution (2) The system of linear equations $A X = 0$ has infinitely many solutions (3) $B$ is an invertible matrix (4) $\operatorname { adj } ( A )$ is an invertible matrix
The number of points, where the function $f : R \rightarrow R , f ( x ) = | x - 1 | \cos | x - 2 | \sin | x - 1 | + ( x - 3 ) \left| x ^ { 2 } - 5 x + 4 \right|$, is NOT differentiable, is (1) 1 (2) 2 (3) 3 (4) 4
Let $f ( x ) = 3 ^ { \left( x ^ { 2 } - 2 \right) ^ { 3 } + 4 } , \mathrm { x } \in R$. Then which of the following statements are true? $P : x = 0$ is a point of local minima of $f$ $Q : x = \sqrt { 2 }$ is a point of inflection of $f$ $R : f ^ { \prime }$ is increasing for $x > \sqrt { 2 }$ (1) Only $P$ and $Q$ (2) Only $P$ and $R$ (3) Only $Q$ and $R$ (4) All $P , Q$ and $R$