Moment of inertia of a cylinder of mass m, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is $I = M \left( \frac { R ^ { 2 } } { 4 } + \frac { L ^ { 2 } } { 12 } \right)$. If such a cylinder is to be made for a given mass of a material, the ratio $\frac { L } { R }$ for it to have minimum possible $I$ is: (1) $\frac { 2 } { 3 }$ (2) $\frac { 3 } { 2 }$ (3) $\sqrt { \frac { 3 } { 2 } }$ (4) $\sqrt { \frac { 2 } { 3 } }$
A block of mass $\mathrm { m } = 1 \mathrm {~kg}$ slides with velocity $\mathrm { v } = 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle $\theta$ before momentarily coming to rest. if the rod has mass $M = 2 \mathrm {~kg}$, and length $\ell = 1 \mathrm {~m}$, the value of $\theta$ is approximately (take $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$) (1) $63 ^ { \circ }$ (2) $55 ^ { \circ }$ (3) $69 ^ { \circ }$ (4) $49 ^ { \circ }$
A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius $\mathrm { R } _ { \mathrm { e } }$. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become $\sqrt { \frac { 3 } { 2 } }$ times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is $R$. Value of $R$ is: (1) $4 \mathrm { R } _ { \mathrm { e } }$ (2) $2.5 \mathrm { R } _ { \mathrm { e } }$ (3) $3 R _ { e }$ (4) $2 \mathrm { R } _ { \mathrm { e } }$
A cricket ball of mass 0.15 kg is thrown vertically up by a bowling machine so that it rises to a maximum height of 20 m after leaving the machine. If the part pushing the ball applies a constant force $F$ on the ball and moves horizontally a distance of 0.2 m while launching the ball, the value of $F$ (in N) is $\left( g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right)$
A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre?
Consider the two sets: $A = \left\{ m \in R : \right.$ both the roots of $x ^ { 2 } - ( m + 1 ) x + m + 4 = 0$ are real $\}$ and $B = [ - 3,5 )$ Which of the following is not true? (1) $A - B = ( - \infty , - 3 ) \cup ( 5 , \infty )$ (2) $A \cap B = \{ - 3 \}$ (3) $B - A = ( - 3,5 )$ (4) $A \cup B = R$
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is (1) $\frac { 1 } { 6 }$ (2) $\frac { 1 } { 5 }$ (3) $\frac { 1 } { 4 }$ (4) $\frac { 1 } { 7 }$
If the number of integral terms in the expansion of $\left( 3 ^ { \frac { 1 } { 2 } } + 5 ^ { \frac { 1 } { 8 } } \right) ^ { n }$ is exactly 33, then the least value of $n$ is (1) 264 (2) 128 (3) 256 (4) 248
Let P be a point on the parabola, $y ^ { 2 } = 12 x$ and N be the foot of the perpendicular drawn from $P$, on the axis of the parabola. A line is now drawn through the mid-point $M$ of $P N$, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line NQ is $\frac { 4 } { 3 }$, then: (1) $P N = 4$ (2) $M Q = \frac { 1 } { 3 }$ (3) $M Q = \frac { 1 } { 4 }$ (4) $P N = 3$
For the frequency distribution: Variate $( x ) : x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots , x _ { 15 }$ Frequency $( f ) : f _ { 1 } , f _ { 2 } , f _ { 3 } , \ldots , f _ { 15 }$ where $0 < x _ { 1 } < x _ { 2 } < x _ { 3 } < \ldots < x _ { 15 } = 10$ and $\sum _ { i = 1 } ^ { 15 } f _ { i } > 0$, the standard deviation cannot be (1) 4 (2) 1 (3) 6 (4) 2
If $\Delta = \left| \begin{array} { c c c } x - 2 & 2 x - 3 & 3 x - 4 \\ 2 x - 3 & 3 x - 4 & 4 x - 5 \\ 3 x - 5 & 5 x - 8 & 10 x - 17 \end{array} \right| = A x ^ { 3 } + B x ^ { 2 } + C x + D$, then $B + C$ is equal to: (1) $- 1$ (2) $1$ (3) $- 3$ (4) $9$
The foot of the perpendicular drawn from the point $( 4,2,3 )$ to the line joining the points $( 1 , - 2,3 )$ and $( 1,1,0 )$ lies on the plane (1) $2 x + y - z = 1$ (2) $x - y - 2 z = 1$ (3) $x - 2 y + z = 1$ (4) $x + 2 y - z = 1$
The lines $\vec { r } = ( \hat { i } - \hat { j } ) + l ( 2 \hat { i } + \widehat { k } )$ and $\vec { r } = ( 2 \hat { i } - \hat { j } ) + m ( \hat { i } + \hat { j } - \widehat { k } )$ (1) Do not intersect for any values of $l$ and $m$ (2) Intersect for all values of $l$ and $m$ (3) Intersect when $l = 2$ and $m = \frac { 1 } { 2 }$ (4) Intersect when $l = 1$ and $m = 2$
A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared at least once is (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 3 }$ (3) $\frac { 1 } { 8 }$ (4) $\frac { 1 } { 9 }$