List of practice Questions

The sum of coefficients of integral powers of $x$ in the binomial expansion $(1-2\sqrt x)^{50}$ is

Addition of phosphate fertilisers to water bodies causes :

If the mean and the variance of a binomial variate $X$ are $2$ and $1$ respectively, then the probability that $X$ takes a value greater than or equal to one is :

$A +2 B +3 C \rightleftharpoons AB _{2} C _{3}$ Reaction of $6.0\, g$ of $A , 6.0 \times 10^{23}$ atoms of $B$, and $0.036\, mol$ of $C$ yields $4.8\, g$ of compound $AB _{2} C _{3}$. If the atomic mass of $A$ and $C$ are 60 and 80 amu respectively, the atomic mass of $B$ is (Avogadro no. $=6 \times 10^{23}$ ):

A simple harmonic oscillator of angular frequency $2 \,rad \,s^{-1}$ is acted upon by an external force $F= \sin\, t \,N$. If the oscillator is at rest in its equilibrium position at $t\, =\, 0$, its position at later times is proportional to :

An ellipse passes through the foci of the hyperbola, $9x^2? 4y^2 = 36$ an and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\frac{1}{2},$ then which of the following points does not lie on the ellipse ?

If $y + 3x = 0$ is the equation of a chord of the circle, $x^2 + y^2 - 30x = 0$, then the equation of the circle with this chord as diameter is :

Let k be a non-zero real number. If

\( f(x) = \begin{cases} \dfrac{(e^x - 1)^2}{\sin\left(\frac{x}{k}\right)\log\left(1+\frac{x}{4}\right)}, & x \ne 0 \\[2ex] 12, & x = 0 \end{cases} \)

is a continuous function, then the value of \( k \) is:

Let $O$ be the vertex and $Q$ be any point on the parabola, $x^2$ = 8y. If the point $P$ divides the line segment $OQ$ internally in the ratio $1 : 3$, then the locus of $P$ is

Let $PQ$ be a double ordinate of the parabola, $y^2= - 4x$, where P lies in the second quadrant. If R divides $PQ$ in the ratio $2 : 1$, then the locus of R is :

Locus of the image of the point $(2, 3)$ in the line $\left(2x - 3y + 4\right) + k\left(x - 2y + 3\right) = 0, k \in R,$ is a

Which of the following compounds is not an antacid?

In Carius method of estimation of halogens $250\, mg$ of an organic compound gave $141\, mg$ of $AgBr$. The percentage of bromine in the compound is (atomic mass $Ag\, =\, 108, Br \,= \,80$)

An experiment takes $10\, min$ to raise the temperature of water in a container from $0^{\circ} C$ to $100^{\circ} C$ and another $55\, min$ to convert it totally into steam by a heater supplying heat at a uniform rate. Neglecting the specific heat of the container and taking specific heat of water to be $1\, cal / g { }^{\circ} C$, the heat of vaporisation according to this experiment will come out to be

When $5\,V$ potential difference is applied across a wire of length $0.1\,m$, the drift speed of electrons is $2.5\times10^{-4}$ $ms^{-1}$. If the electron density in the wire is $8\times10^{28}\, m^{-3}$ the resistivity of the material is close to

For plane electromagnetic waves propagating in the $z$ direction, which one of the following combination gives the correct possible direction for $\vec{E}$ and $\vec{B}$ field respectively ?

A solution at $20^{\circ}$C is composed of $1.5\, mol$ of benzene and $3.5\, mol$ of toluene. If the vapour pressure of pure benzene and pure toluene at this temperature are $74.7$ torr and $22.3$ torr, respectively, then the total vapour pressure of the solution and the benzene mole fraction in equilibrium with it will be, respectively :

Choose the incorrect formula out of the four compounds for an element X below :

A $25\, cm$ long solenoid has radius $2\, cm$ and $500$ total number of turns. It carries a current of $15\, A$. If it is equivalent to a magnet of the same size and magnetization $|\bar{M}|$ (magnetic moment/volume), then $|\bar{M}|$ is :

Consider a spherical shell of radius $R$ at temperature $T$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u=\frac{U}{V} ? T^4$ and pressure $p=\frac{1}{3}\bigg(\frac{U}{V}\bigg)$ If the shell now undergoes an adiabatic expansion, the relation between $T$ and $R$ is

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as $V^q$ , where $V$ is the volume of the gas. The value of $q$ is $\bigg(\gamma =\frac{C_p}{C_v}\bigg)$

A solid body of constant heat capacity $1 \, J/^{\circ}C$ is being heated by keeping it in contact with reservoirs in two ways (i) Sequentially keeping in contact with $2$ reservoirs such that each reservoir supplies same amount of heat. (ii) Sequentially keeping in contact with $8$ reservoirs such that each reservoir supplies same amount of heat. In both the cases, body is brought from initial temperature $100^{\circ}C$ to final temperature $200^{\circ}C$. Entropy change of the body in the two cases respectively, is

Monochromatic light is incident on a glass prism of angle $A$. If the refractive index of the material of the prism is $u$, a ray incident at an angle $\theta$, on the face $AB$ would get transmitted through the face $AC$ of the prism provided

In the given circuit, charge $Q_2 $ on the $2 \mu F $ capacitor changes as $C$ is varied from $1 \mu F $ to $3 \mu F. $ $Q_2 $ as a function of $'C'$ is given properly by (figures are drawn schematically and are not to scale)

Assuming human pupil to have a radius of $0.25\, cm$ and a comfortable viewing distance of $25\, cm$, the minimum separation between two objects that human eye can resolve at $500\, nm$ wavelength is