If $x=\sin \theta, y=\sin ^3 \theta$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$ at $\theta=\frac{\pi}{2}$ is

Let $P, Q, R$ and $S$ be the points on the plane with position vectors $-2 \hat{i}-\hat{j}, 4 \hat{i}, 3 \hat{i}+3 \hat{

If $y=\frac{x^{\frac{2}{3}}-x^{\frac{-1}{3}}}{x^{\frac{2}{3}}+x^{\frac{-1}{3}}}, x \neq 0$, then $(x+1)^2 y_1=$

The Number of values of C that satisfy the conclusion of Rolle's theorem in case of following function $\mathrm{f}(x)=\s

An open tank with a square bottom, to contain 4000 cubic cm . of liquid, is to be constructed. The dimensions of the tan

$$\int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x d x=$$

The contrapositive of the inverse of $\mathrm{p} \rightarrow(\mathrm{p} \rightarrow \mathrm{q})$ is

The area enclosed between the parabola $y^2=4 x$ and the line $y=2 x-4$ is

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 q

The order of the differential equation, whose solution is $y=\left(C_1+C_2\right) \mathrm{e}^x+C_3 \mathrm{e}^{x+C_4}$,

If the sides of a rectangle are given by the equations $x=-2, x=6, y=-2, y=5$, then the equation of the circle, drawn on

The solution set of the equation $\tan x+\sec x=2 \cos x$, in the interval $[0,2 \pi]$ is

If one of the lines represented by $a x^2+2 h x y+b y^2=0$ is perpendicular to $\mathrm{m} x+\mathrm{n} y=18$, then

If $\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2}$, then the val

The graphical solution set of the system of inequations $2 x+3 y \leq 6, x+4 y \geq 4, x \geq 0, y \geq 0$ is given by

The derivative of $\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)$ w.r.t. $\sin ^{-1}\left(3 x-4 x^3\right)$ is

If $A=\left[\begin{array}{cc}5 a & -b \\ 3 & 2\end{array}\right]$ and $A \cdot \operatorname{adj} A=A A^T$, then $5 a+b$

The function $\mathrm{f}(x)=2 x^3-9 x^2+12 x+2$ is decreasing in

$\int\left(1+x-\frac{1}{x}\right) \mathrm{e}^{x+\frac{1}{x}} \mathrm{~d} x$ is equal to

If the function $f(x)= \begin{cases}-2 \sin x & \text {, if } x \leq \frac{-\pi}{2} \\ A \sin x+B & , \text { if } \frac

$$\int\limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} d x=$$

If the complex number $z=x+i y$, where $i=\sqrt{-1}$, satisfies the condition $|z+1|=1$, then $z$ lies on

The general solution of the differential equation $\frac{d y}{d x}=\frac{3 e^{2 x}+3 e^{4 x}}{e^x+e^{-x}}$ is

If $\cos ^{-1} x=\alpha(0

If the points $\mathrm{P}, \mathrm{Q}$ and R are with the position vectors $\hat{i}-2 \hat{j}+3 \hat{k},-2 \hat{i}+3 \ha

A line makes $45^{\circ}$ angle with positive X -axis and makes equal angles with positive Y -axis ad Z-axis respectivel

If the lines $\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}$

The vector equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$, which is

The equation of motion of a particle is $s=a t^2+b t+c$. If the displacement after 1 second is 20 m , velocity after 2 s

One side and one diagonal of a parallelogram are represented by $3 \hat{i}+\hat{j}-\hat{k}$ and $2 \hat{i}+\hat{j}-2 \ha

If $\int \mathrm{e}^{x^2} \cdot x^3 \mathrm{dx}=\mathrm{e}^{x^2} \mathrm{f}(x)+\mathrm{c}$ and $\mathrm{f}(1)=0$ (where

If $\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x$, then the value of $\sin x$ is

If $[x]^2-5[x]+6=0$, where $[\cdot]$ denotes the greatest integer function, then

The value of the integral $\int_0^{\frac{\pi}{2}} \frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} \mathrm{dx}$ is

Variance of first n natural numbers is ________.

 Suppose three coins are tossed simultaneously. If $X$ denotes the number of heads, then probability distribution of x i

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three o

If $0

If the vector $\overline{\mathrm{c}}$ lies in the plane of $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$, where $\

The equation of a line passing through the point $(2,-1,1)$ and parallel to the line joining the points $\hat{i}+2 \hat{

The foot of the perpendicular drawn from origin to a plane is $\mathrm{M}(2,1,-2)$, then vector equation of the plane is

If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$

In a triangle ABC , with usual notations, $\frac{\cos \mathrm{B}+\cos \mathrm{C}}{\mathrm{b}+\mathrm{c}}+\frac{\cos \mat

If $\mathrm{f}(x)=\frac{x}{x+1}, x \neq-1$ and (fof) $(x)=\mathrm{F}(x)$, then $\int \mathrm{F}(x) \mathrm{d} x$ is

If p : The total prime numbers between 2 to 100 are 26. q : Zero is a complex number. $r$ : Least common multiple (L.C.M

$$\lim _\limits{x \rightarrow 2} \frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}}=$$

$$\int_0^{\frac{\pi}{4}} \frac{\sec ^2 x}{(1+\tan x)(2+\tan x)} d x=$$

If two fair dice are rolled, then the probability that the sum of the numbers on the upper faces is at least 9, is

After $t$ seconds, the acceleration of a particle, which starts from rest and moves in a straight line is $\left(8-\frac

If $\mathrm{A}+\mathrm{B}=225^{\circ}$, then $\frac{\cot \mathrm{A}}{1+\cot \mathrm{A}} \cdot \frac{\cot \mathrm{B}}{1+\

The figure shows currents in a part of electric circuit. Then current I is

A metal disc of radius R rotates with an angular velocity $\omega$ about an axis perpendicular to its plane passing thro

A body of mass $m$ slides down an incline and reaches the bottom with a velocity V . If the same mass were in the form o

For a particle executing S.H.M. having amplitude A, the speed of the article is $\left(\frac{1}{3}\right)^{\text {rd }}$

The minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in

The core used in transformers are laminated to

The end correction for the vibrations of air column in a tube of circular cross-section will be more if the tube is

If the work done in blowing a soap bubble of volume ' V ' is ' W ', then the work done in blowing a soap bubble of volum

A coil has inductance H . The ratio of its reactance when it is connected first to an a.c. source and then to d.c. sourc

A particle of mass m collides with another stationary particle of mass $M$. The particle $m$ stops just after collision.

In the circuit shown in the following figure, the potential difference cross $3 \mu \mathrm{~F}$ capacitor is

Sodium light $\left(\lambda=6 \times 10^{-7} \mathrm{~m}\right)$ is used to produce interference pattern. The observed f

The motion of a particle is described by the equation $a=-b x$ where ' $a$ ' is the acceleration, x is the displacement

Acceleration of an electron in the first Bohr's orbit is proportional to $\mathrm{m}=$ mass of electron, $\mathrm{r}=$ r

A cylindrical rod is having temperatures $\theta_1$ and $\theta_2$ at its ends. The rate of heat flow is $\mathrm{Q} J /

A resistor of $50 \Omega$, inductor of self inductance $\left(\frac{3}{\pi^2}\right) \mathrm{H}$ and a capacitor of unkn

A monoatomic ideal gas, initially at temperature $T_1$ is enclosed in a cylinder fitted with frictionless piston. The ga

Charges of $2 \mu \mathrm{C}$ and $-3 \mu \mathrm{C}$ are placed at two points A and B separated by 1 m . The distance o

For detecting light intensity we use

A ball is released from the top of a tower of height Hm . It takes T second to reach the ground. The height of the ball

1000 small balls, each weighing 1 gram, strike one square cm of area per second with a velocity $50 \mathrm{~m} / \mathr

A wave is given by $Y=3 \sin 2 \pi\left(\frac{t}{0.04}-\frac{x}{0.01}\right)$ where Y is in cm . Frequency of the wave a

Two concentric circular coils A and B having radii 20 cm and 10 cm respectively lie in the same plane. The current in co

The radius of gyration of a circular disc of radius R and mass m rotating about diameter as axis is

What is the current in the following junction diode circuit?

When a certain metallic surface is illuminated with monochromatic light wavelength $\lambda$, the stopping potential for

A plate of refractive index 1.6 is introduced in the path of light from one of the slits in Young's double slit experime

The viscous force between two liquid layers is

In the given reaction $${ }_z \mathrm{X}^A \rightarrow{ }_{z+1} \mathrm{Y}^A \rightarrow{ }_{z-1} \mathrm{~K}^{A-4} \rig

Two point charges $+q_1$ and $q_2$ repel each other with a force of $100 \mathrm{~N} . q_1$ is increased by $10 \%$ and

In an ideal gas at temperature $T$, the average force that a molecule applies on the walls of a closed container depends

Which one of the following graph represent correctly the variation of impedance $(\mathrm{Z})$ of a series LCR circuit w

A galvanometer may be converted into ammeter or a voltmeter. In which of the following cases the resistance of the devic

A glass prism ' A ' deviates the red and blue rays through $10^{\circ}$ and $12^{\circ}$ respectively. A second prism '

In an electric field due to charge $Q$, a charge $q$ moves from point A to B as shown in the figure. The work done is (

An air cored coil has a self inductance 0.1 H . A soft iron core of relative permeability 1000 is introduced and the num

Heat engine operating between temperature $T_1$ and $T_2$ has efficiency $\frac{1}{6}$. When $T_2$ is lowered by 62 K ,

The angle of minimum deviation produced by a thin prism in air is $\delta_1$. If it is immersed in water the angle of mi

An electron is revolving in a circular orbit of radius $r$ in a hydrogen atom. The angular momentum of the electron is L

A p-n junction diode as a rectifier converts

A ball rises to surface at a constant velocity in liquid whose density is 3 times greater than that of the material of t

Three infinite straight wires $\mathrm{A}, \mathrm{B}$ and C carry currents as shown in figure. The resultant force on w

Velocity of sound waves in air is $330 \mathrm{~m} / \mathrm{s}$. For a particular sound wave in air, path difference of

In Young's double slit experiment, the intensity of light at a point on the screen where the path difference is $\lambda

The stopping potential as a function of frequency of incident radiation is plotted for two different photoelectric surfa

The absolute temperature of a gas is determined by

The density of a new planet is twice that of earth. The acceleration due to gravity at the surface of the planet is equa

A particle is moving in a circle with uniform speed. It has constant

A string has mass per unit length of $10^{-6} \mathrm{~kg} / \mathrm{cm}$ The equation of simple harmonic wave produced

A spring has length L and force constant K . It is cut into two springs of length $L_1$ and $L_2$ such that $\mathrm{L}_