A ball is projected vertically upward with an initial velocity of $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $t = 0 \mathrm {~s}$. At $t = 2 \mathrm {~s}$, another ball is projected vertically upward with same velocity. At $t =$ $\_\_\_\_$ s, second ball will meet the first ball $\left( \mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right)$.
A system of 10 balls each of mass 2 kg are connected via massless and unstretchable string. The system is allowed to slip over the edge of a smooth table as shown in figure. Tension on the string between the $7 ^ { \text {th} }$ and $8 ^ { \text {th} }$ ball is $\_\_\_\_$ N when $6 ^ { \text {th} }$ ball just leaves the table.
A batsman hits back a ball of mass 0.4 kg straight in the direction of the bowler without changing its initial speed of $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The impulse imparted to the ball is $\_\_\_\_$ Ns.
A set of 20 tuning forks is arranged in a series of increasing frequencies. If each fork gives 4 beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is $\_\_\_\_$ Hz.
If $m$ is the slope of a common tangent to the curves $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 12$, then $12 \mathrm {~m} ^ { 2 }$ is equal to (1) 6 (2) 9 (3) 10 (4) 12
The locus of the mid-point of the line segment joining the point $( 4,3 )$ and the points on the ellipse $x ^ { 2 } + 2 y ^ { 2 } = 4$ is an ellipse with eccentricity (1) $\frac { \sqrt { 3 } } { 2 }$ (2) $\frac { 1 } { 2 \sqrt { 2 } }$ (3) $\frac { 1 } { \sqrt { 2 } }$ (4) $\frac { 1 } { 2 }$
Let the mean of 50 observations is 15 and the standard deviation is 2. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is 70. If the mean of the correct set of observations is 16, then the variance of the correct set is equal to (1) 10 (2) 36 (3) 43 (4) 60
If the system of equations $\alpha x + y + z = 5 , x + 2 y + 3 z = 4 , x + 3 y + 5 z = \beta$ has infinitely many solutions, then the ordered pair $( \alpha , \beta )$ is equal to (1) $( 1 , - 3 )$ (2) $( - 1,3 )$ (3) $( 1,3 )$ (4) $( - 1 , - 3 )$
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as $f ( x ) = x - 1$ and $g : R \rightarrow \{ 1 , - 1 \} \rightarrow \mathbb { R }$ be defined as $g ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }$. Then the function $f o g$ is: (1) One-one but not onto (2) onto but not one-one (3) Both one-one and onto (4) Neither one-one nor onto
Let $f ( x ) = \min \{ 1,1 + x \sin x \} , 0 \leq x \leq 2 \pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not continuous, then the ordered pair $( m , n )$ is equal to (1) $( 2,0 )$ (2) $( 1,0 )$ (3) $( 1,1 )$ (4) $( 2,1 )$
Consider a cuboid of sides $2 x , 4 x$ and $5 x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is constant $k$, then the ratio $x : r$, for which the sum of their volumes is maximum, is (1) $2 : 5$ (2) $19 : 45$ (3) $3 : 8$ (4) $19 : 15$
If $y = y ( x )$ is the solution of the differential equation $x \frac { d y } { d x } + 2 y = x e ^ { x } , y ( 1 ) = 0$ then the local maximum value of the function $z ( x ) = x ^ { 2 } y ( x ) - e ^ { x } , x \in R$ is (1) $1 - e$ (2) 0 (3) $\frac { 1 } { 2 }$ (4) $\frac { 4 } { e } - e$
If $\frac { d y } { d x } + e ^ { x } \left( x ^ { 2 } - 2 \right) y = \left( x ^ { 2 } - 2 x \right) \left( x ^ { 2 } - 2 \right) e ^ { 2 x }$ and $y ( 0 ) = 0$, then the value of $y ( 2 )$ is (1) $-1$ (2) 1 (3) 0 (4) $e$
Let $\vec { a } = \hat { i } + \hat { j } + 2 \widehat { k } , \vec { b } = 2 \hat { i } - 3 \hat { j } + \widehat { k }$ and $\vec { c } = \hat { i } - \hat { j } + \widehat { k }$ be the three given vectors. Let $\vec { v }$ be a vector in the plane of $\vec { a }$ and $\vec { b }$ whose projection on $\vec { c }$ is $\frac { 2 } { \sqrt { 3 } }$. If $\vec { v } \cdot \hat { j } = 7$, then $\vec { v } \cdot ( \hat { i } + \hat { k } )$ is equal to (1) 6 (2) 7 (3) 8 (4) 9
If the plane $2 x + y - 5 z = 0$ is rotated about its line of intersection with the plane $3 x - y + 4 z - 7 = 0$ by an angle of $\frac { \pi } { 2 }$, then the plane after the rotation passes through the point (1) $( 2 , - 2,0 )$ (2) $( - 2,2,0 )$ (3) $( 1,0,2 )$ (4) $( - 1,0 , - 2 )$
If $p$ and $q$ are real numbers such that $p + q = 3 , p ^ { 4 } + q ^ { 4 } = 369$, then the value of $\left( \frac { 1 } { p } + \frac { 1 } { q } \right) ^ { - 2 }$ is equal to (if the full expression were available).