The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R _ { 0 } = 40 \mathrm {~m}$. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $\mathrm { v } = 20 \mathrm {~m} / \mathrm { s }$ on a horizontal surface? $\left( \mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$ (1) $30 ^ { \circ }$ (2) $60 ^ { \circ }$ (3) $75 ^ { \circ }$ (4) $45 ^ { \circ }$
Two blocks of masses m and M are connected by means of a metal wire of cross-sectional area A passing over a frictionless fixed pulley as shown in the figure. The system is then released. If $\mathrm { M } = 2 \mathrm {~m}$, then the stress produced in the wire is: (1) $\frac { 2 \mathrm { mg } } { 3 \mathrm {~A} }$ (2) $\frac { 4 \mathrm { mg } } { 3 \mathrm {~A} }$ (3) $\frac { \mathrm { mg } } { \mathrm { A } }$ (4) $\frac { 3 \mathrm { mg } } { 4 \mathrm {~A} }$
A uniform cylinder of length L and mass $M$ having cross-sectional area A is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. When the cylinder is given a downward push and released, it starts oscillating vertically with a small amplitude. The time period T of the oscillations of the cylinder will be: (1) Smaller than $2\pi \left[ \frac { M } { ( k + A\sigma g ) } \right] ^ { 1/2 }$ (2) $2\pi \sqrt { \frac { M } { k } }$ (3) Larger than $2\pi \left[ \frac { M } { ( k + A\sigma g ) } \right] ^ { 1/2 }$ (4) $2\pi \left[ \frac { M } { ( k + A\sigma g ) } \right] ^ { 1/2 }$
Let $z$ satisfy $| z | = 1$ and $z = 1 - \bar { z }$. Statement $1 : z$ is a real number. Statement 2 : Principal argument of z is $\frac { \pi } { 3 }$ (1) Statement 1 is true Statement 2 is true; Statement 2 is a correct explanation for Statement 1. (2) Statement 1 is false; Statement 2 is true. (3) Statement 1 is true, Statement 2 is false. (4) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.
5-digit numbers are to be formed using $2,3,5,7,9$ without repeating the digits. If $p$ be the number of such numbers that exceed 20000 and $q$ be the number of those that lie between 30000 and 90000, then $p : q$ is: (1) $6 : 5$ (2) $3 : 2$ (3) $4 : 3$ (4) $5 : 3$
Given a sequence of 4 numbers, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is: (1) 16 (2) 8 (3) 4 (4) 2
If for positive integers $r > 1 , n > 2$, the coefficients of the $( 3r ) ^ { \text {th} }$ and $( r + 2 ) ^ { \text {th} }$ powers of $x$ in the expansion of $( 1 + x ) ^ { 2n }$ are equal, then $n$ is equal to: (1) $2r + 1$ (2) $2r - 1$ (3) $3r$ (4) $r + 1$
If the image of point $\mathrm { P } ( 2,3 )$ in a line L is $\mathrm { Q } ( 4,5 )$, then the image of point $\mathrm { R } ( 0,0 )$ in the same line is: (1) $( 2,2 )$ (2) $( 4,5 )$ (3) $( 3,4 )$ (4) $( 7,7 )$
Statement 1: The only circle having radius $\sqrt { 10 }$ and a diameter along line $2x + y = 5$ is $x ^ { 2 } + y ^ { 2 } - 6x + 2y = 0$. Statement 2: $2x + y = 5$ is a normal to the circle $x ^ { 2 } + y ^ { 2 } - 6x + 2y = 0$. (1) Statement 1 is false; Statement 2 is true. (2) Statement 1 is true; Statement 2 is true, Statement 2 is a correct explanation for Statement 1. (3) Statement 1 is true; Statement 2 is false. (4) Statement 1 is true; Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.