$\lim_{x \to \frac{\pi}{2}} \tan^2 x \left[(2\sin^2 x + 3\sin x + 4)^{\frac{1}{2}} - (\sin^2 x + 6\sin x + 2)^{\frac{1}{2}}\right]$ is equal to
(1) $\frac{1}{12}$
(2) $-\frac{1}{18}$
(3) $-\frac{1}{12}$
(4) $\frac{1}{6}$

Negation of the Boolean expression $p \leftrightarrow (q \rightarrow p)$ is
(1) $\sim p \wedge q$
(2) $p \wedge \sim q$
(3) $\sim p \vee \sim q$
(4) $\sim p \wedge \sim q$

For $\alpha \in \mathbb{N}$, consider a relation $R$ on $\mathbb{N}$ given by $R = \{(x, y) : 3x + \alpha y \text{ is a multiple of } 7\}$. The relation $R$ is an equivalence relation if and only if
(1) $\alpha = 14$
(2) $\alpha$ is a multiple of 4
(3) 4 is the remainder when $\alpha$ is divided by 10
(4) 4 is the remainder when $\alpha$ is divided by 7

The negation of the Boolean expression $(\sim q \wedge p) \Rightarrow (\sim p \vee q)$ is logically equivalent to
(1) $p \Rightarrow q$
(2) $q \Rightarrow p$
(3) $\sim p \Rightarrow q$
(4) $\sim q \Rightarrow p$

Consider the following statements: $P$: Ramu is intelligent. $Q$: Ramu is rich. $R$: Ramu is not honest. The negation of the statement ``Ramu is intelligent and honest if and only if Ramu is not rich'' can be expressed as:
(1) $((P \wedge (\sim R)) \wedge Q) \wedge ((\sim Q) \wedge ((\sim P) \vee R))$
(2) $((P \wedge R) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \vee (\sim R)))$
(3) $((P \wedge R) \wedge Q) \wedge ((\sim Q) \wedge ((\sim P) \vee (\sim R)))$
(4) $((P \wedge (\sim R)) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \wedge R))$

Let $R _ { 1 }$ and $R _ { 2 }$ be two relations defined on $\mathbb { R }$ by $a \mathrm { R } _ { 1 } b \Leftrightarrow a b \geq 0$ and $a R _ { 2 } b \Leftrightarrow a \geq b$, then
(1) $R _ { 1 }$ is an equivalence relation but not $R _ { 2 }$
(2) $R _ { 2 }$ is an equivalence relation but not $R _ { 1 }$
(3) both $R _ { 1 }$ and $R _ { 2 }$ are equivalence relations
(4) neither $R _ { 1 }$ nor $R _ { 2 }$ is an equivalence relation

$\lim_{n \rightarrow \infty} \frac{1}{2^n} \left( \frac{1}{\sqrt{1 - \frac{1}{2^n}}} + \frac{1}{\sqrt{1 - \frac{2}{2^n}}} + \frac{1}{\sqrt{1 - \frac{3}{2^n}}} + \ldots + \frac{1}{\sqrt{1 - \frac{2^n - 1}{2^n}}} \right)$ is equal to
(1) $\frac{1}{2}$
(2) 1
(3) 2
(4) $-2$

A person travels $x$ distance with velocity $v _ { 1 }$ and then $x$ distance with velocity $v _ { 2 }$ in the same direction. The average velocity of the person is $v$, then the relation between $v , \quad v _ { 1 }$ and $v _ { 2 }$ will be
(1) $v = \frac { v _ { 1 } + v _ { 2 } } { 2 }$
(2) $\frac { 1 } { v } = \frac { 1 } { v _ { 1 } } + \frac { 1 } { v _ { 2 } }$
(3) $v = v _ { 1 } + v _ { 2 }$
(4) $\frac { 2 } { v } = \frac { 1 } { v _ { 1 } } + \frac { 1 } { v _ { 2 } }$

Two projectiles are projected at $30 ^ { \circ }$ and $60 ^ { \circ }$ with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is:
(1) $\sqrt { 3 } : 1$
(2) $1 : \sqrt { 3 }$
(3) $2 : \sqrt { 3 }$
(4) $1 : 3$

A body of mass $(5 \pm 0.5)$ kg is moving with a velocity of $(20 \pm 0.4)$ m s$^{-1}$. Its kinetic energy will be
(1) $(1000 \pm 0.14)$ J
(2) $(500 \pm 0.14)$ J
(3) $(500 \pm 140)$ J
(4) $(1000 \pm 140)$ J

The time period of a satellite, revolving above earth's surface at a height equal to R will be (Given $g = \pi ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 } , R =$ radius of earth)
(1) $\sqrt { 2 \mathrm { R } }$
(2) $\sqrt { 8 \mathrm { R } }$
(3) $\sqrt { 32 \mathrm { R } }$
(4) $\sqrt { 4 \mathrm { R } }$

Two planets A and B of radii $R$ and $1.5 R$ have densities $\rho$ and $\frac { \rho } { 2 }$ respectively. The ratio of acceleration due to gravity at the surface of $B$ to $A$ is:
(1) $2 : 3$
(2) $2 : 1$
(3) $3 : 4$
(4) $4 : 3$

The radii of two planets $A$ and $B$ are $R$ and $4R$ and their densities are $\rho$ and $\frac{\rho}{3}$ respectively. The ratio of acceleration due to gravity at their surfaces $g_A : g_B$ will be
(1) $4:3$
(2) $1:16$
(3) $3:16$
(4) $3:4$

Young's moduli of the material of wires $A$ and $B$ are in the ratio of $1 : 4$, while its area of cross sections are in the ratio of $1 : 3$. If the same amount of load is applied to both the wires, the amount of elongation produced in the wires A and B will be in the ratio of [Assume length of wires $A$ and $B$ are same]
(1) $12 : 1$
(2) $1 : 36$
(3) $36 : 1$
(4) $1 : 12$

A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing $W$ on earth will weigh on that planet:
(1) $2^{\frac{1}{4}}\mathrm{~W}$
(2) $2^{\frac{1}{3}}\mathrm{~W}$
(3) $2W$
(4) $2^{\frac{2}{3}}\mathrm{~W}$

Moment of inertia of a disc of mass $M$ and radius ' $R$ ' about any of its diameter is $\frac { M R ^ { 2 } } { 4 }$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $\frac { x } { 2 } M R ^ { 2 }$. The value of $x$ is $\_\_\_\_$ .

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $\pi : 22$ then, the value of its angular speed will be $\_\_\_\_$ rad s$^{-1}$.

A gas mixture consists of 2 moles of oxygen and 4 moles of neon at temperature $T$. Neglecting all vibrational modes, the total internal energy of the system will be:
(1) $11 R T$
(2) $8 R T$
(3) $4 R T$
(4) $16 R T$

The escape velocities of two planets $A$ and $B$ are in the ratio $1 : 2$. If the ratio of their radii respectively is $1 : 3$, then the ratio of acceleration due to gravity of planet $A$ to the acceleration of gravity of planet $B$ will be:
(1) $\frac { 4 } { 3 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$

A small ball of mass $M$ and density $\rho$ is dropped in a viscous liquid of density $\rho_0$. After some time, the ball falls with a constant velocity. What is the viscous force on the ball?
(1) $F = Mg\left(1 + \frac{\rho_0}{\rho}\right)$
(2) $F = Mg\left(1 + \frac{\rho}{\rho_0}\right)$
(3) $F = Mg\left(1 - \frac{\rho_0}{\rho}\right)$
(4) $F = Mg\left(1 \pm \rho\rho_0\right)$

A planet having mass $9M_e$ and radius $4R_e$, where $M_e$ and $R_e$ are mass and radius of earth respectively, has escape velocity in km s$^{-1}$ given by: (Given escape velocity on earth $V_e = 11.2 \times 10^3$ m s$^{-1}$)
(1) 67.2
(2) 16.8
(3) 11.2
(4) 33.6

For a periodic motion represented by the equation $y = \sin \omega t + \cos \omega t$ the amplitude of the motion is
(1) 1
(2) 0.5
(3) 2
(4) $\sqrt { 2 }$

For a body projected at an angle with the horizontal from the ground, choose the correct statement
(1) Gravitational potential energy is maximum at the highest point.
(2) The horizontal component of velocity is zero at highest point.
(3) The vertical component of momentum is maximum at the highest point.
(4) The kinetic energy (K.E.) is zero at the highest point of projectile motion.

The elastic potential energy stored in a steel wire of length 20 m stretched through 2 cm is 80 J. The cross sectional area of the wire is $\_\_\_\_$ mm$^2$. (Given, $Y = 2.0 \times 10^{11}$ N m$^{-2}$)

A Carnot engine operating between two reservoirs has efficiency $\frac { 1 } { 3 }$. When the temperature of cold reservoir raised by $x$, its efficiency decreases to $\frac { 1 } { 6 }$. The value of $x$, if the temperature of hot reservoir is $99 ^ { \circ } C$, will be
(1) 16.5 K
(2) 33 K
(3) 66 K
(4) 62 K