List of practice Questions

A mass of $0.5\, kg$ moving with a speed of $1.5\, m/ s$ on a horizontal smooth surface, collides with a nearly weightless spring of force constant $k = 50\, N/m$. The maximum compression of the spring would be

If the differential equation for a simple harmonic motion is $\frac{d^{2}y}{dt^{2}}+2y=0,$ the time-period of the motion is

The magnetic field at a distance r from a long wire carrying current i is 0.4 tesla. The magnetic field at a distance 2r is

A rain drop of radius $0.3\, mm$ has a terminal velocity in air is $1\, m/s$. The viscosity of air is $8 \times 10^{-5}$ poise. The viscous force on it is

A hollow sphere and a solid sphere, of equal mass and equal radii roll down without slipping on an inclined plane. If the torque experienced by the hollow sphere and solid sphere are $\tau_H$ and $\tau_S$ respectively, then

A circle has radius 3 and its centre lies on the line $y = x -1$ . The equation of the circle, if it passes through $(7, 3)$, is

If $\left(\vec{a}\times\vec{b}\right)^{2}+\left(\vec{a}.\vec{b}\right)^{2}=676$ and $\left|\vec{b}\right|=2,$ then $\left|\vec{a}\right|$ is equal to

The system of linear equations : $x + y + z = 0, 2x + y - z = 0, 3x + 2y = 0$ has :

The area under the curve $y = | cos\, x - sin \,x |, 0 \le x\le\frac{\pi}{2}$, and above x-axis is :

The mean and the median of the following ten numbers in increasing order $10, 22, 26, 29, 34, x 42, 67, 70,\, y$ are $42$ and $35$ respectively, then $\frac{y}{x}$ is equal to :

If vector equation of the line$\frac{x-2}{2}=\frac{2y-5}{-3}=z+1,$ is $\vec{r}=\left(2\hat{i}+\frac{5}{2} \hat{j}-\hat{k}\right)+\lambda\left(2\hat{i}-\frac{3}{2} \hat{j}+p\hat{k}\right)$ then $p$ is equal to

Ramesh and Gautam are among 22 students who write an examination. Ramesh scores 82.5. The average score of the 21 students other than Gautam is 62. The average score of all the 22 students is one more than the average score of the 21 students other than Ramesh. The score of Gautam is

A particle starts from origin $O$ from rest and moves with a uniform acceleration along the positive $x$-axis. Identify all figures that correctly represent the motion qualitatively. (a = acceleration, v = velocity, x = displacement, t = time)

A circular disc $D_1$ of mass $M$ and radius $R$ has two identical discs $D_2$ and $D_3$ of the same mass $M$ and radius $R$ attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis $OO'$, passing through the centre of $D_1$, as shown in the figure, will be :

A rectangular solid box of length $0.3\, m$ is held horizontally, with one of its sides on the edge of a platform of height $5\,m$. When released, it slips off the table in a very short time $\tau = 0.01\,s$, remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to :

A string is wound around a hollow cylinder of mass $5\, kg$ and radius $0.5\, m$. If the string is now pulled with a horizontal force of $40\, N$, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) :

Chlorine on reaction with hot and concentrated sodium hydroxide gives :

Two materials having coefficients of thermal conductivity '3K' and 'K' and thickness 'd' and '3d', respectively, are joined to form a slab as shown in the figure. The temperatures of the outer surfaces are $'\theta_2'$ and $' \theta_1'$ respectively, $(\theta_2 > \theta_1)$. The temperature at the interface is :-

A resonance tube is old and has jagged end. It is still used in the laboratory to determine velocity of sound in air. A tuning fork of frequency $512\, Hz$ produces first resonance when the tube is filled with water to a mark $11\, cm$ below a reference mark, near the open end of the tube. The experiment is repeated with another fork of frequency $256\, Hz$ which produces first resonance when water reaches a mark $27\, cm$ below the reference mark. The velocity of sound in air, obtained in the experiment, is close to:

A $20$ Henry inductor coil is connected to a $10$ ohm resistance in series as shown in figure. The time at which rate of dissipation of energy (joule's heat) across resistance is equal to the rate at which magnetic energy is stored in the inductor is :

A hydrogen atom, initially in the ground state is excited by absorbing a photon of wavelength 980$\mathring A $. The radius of the atom in the excited state, it terms of Bohr radius $a_0$, will be : ($h_c$ = $12500\, eV - \mathring A $)

A metal ball of mass $0.1\, kg$ is heated upto $500^{\circ}C$ and dropped into a vessel of heat capacity $800 \; JK^{-1}$ and containing $0.5\, kg$ water. The initial temperature of water and vessel is $30^{\circ}C$. What is the approximate percentage increment in the temperature of the water ? [Specific Heat Capacities of water and metal are, respectively, $4200 \; Jkg^{-1} K^{-1}$ and $400 \; JKg^{-1} K^{-1}$]

If $x = 3 \,tan \,t $ and $y = 3 \,sec \,t$, then the value of $\frac{d^2 y}{dx^2}$ at $t = \frac{\pi}{4}$, is :

Let $f : (-1,1) \to R$ be a function defined by $f(x) = max \{- | x |,- \sqrt{ 1- x^2} \}$. If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly :