List of practice Questions

A satellite of mass $m$ is orbiting the earth (of radius $R$) at a height $h$ from its surface. The total energy of the satellite in terms of $g_0$, the value of acceleration due to gravity at the earth's surface, is -
A light rod of length $l$ has two masses $m_1$ and $m_2$ attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is -
Two rotating bodies $A$ and $B$ of masses $m$ and $2m$ with momenta of inertia $I_A$ and $I_B (I_B > I_A)$ have equal kinetic energy of rotation. If $L_A$ and $L_B$ be their angular momenta respectively, then -
A particle moves from a point $(-2 \hat{i} + 5 \hat{j})$ to $( 4 \hat{j} + 3 \hat{k})$ when a force of $(4 \hat{i} + 3 \hat{j} ) N$ is applied. How much work has been done by the force ?
A bullet of mass $10\,g$ moving horizontally with a velocity of $400 \, ms^{-1}$ strikes a wooden block of mass $2 \,kg$ which is suspended by a light inextensible string of length $5\, m$. As a result, the centre of gravity of the block is found to rise a vertical distance of $10\,cm$. The speed of the bullet after it emerges out horizontally from the block will be -
In the given figure, $a =15\, m / s ^{2}$ represents the total acceleration of a particle moving in the clockwise direction in a circle of radius $R =2.5\, m$ at a given instant of time. The speed of the particle is -
A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $(E_{\text{sphere}} / E_{\text{cylinder}})$ Will be
A $100\, \Omega$ resistance and a capacitor of $100\, \Omega$ reactance are connected in series across a $220\, V$ source. When the capacitor is $50\%$ charged, the peak value of the displacement current is
Four rods with different radii $r$ and length $l$ are used to connect two heat reservoirs at different temperature. Which one will conduct most heat ?
If $f(x) = x^2 , g(x) = 2x,0 \leq x \leq 2$ then the value of $I(x) = \int\limits_0^2 max (f(x), g(x))$ is
Derivative of $log \left(sec\,\theta +tan \,\theta\right) $ with respect to $sec\, \theta$ at $\theta = \pi/4$ is
A solution contains 4 g of NaOH and 16.2 g of water. The mole fraction solute and solvent are respectively
A compound 'A'with molecular formula \({C_5H_{10}O}\)gives a positive \({2,4-DNP}\) test but a negative Tollen's test. On treatment with sodium hypochlorite, it gives \({CHCl_3}\) and compound 'B'. Compound 'B' is
Sphagnum is slowly carbonised, compressed and fossilised over thousands of years to produce a dark spongy mass called peat. Peat helps to keep soil porous and it also improves water holding capacity of the soil.
The number of surjective functions from $A$ to $B$ where $A = \{1, 2, 3, 4 \}$ and $B = \{a, b\}$ is
If the tangent to the function $y = f(x) $ at $(3,4)$ makes an angle of $\frac{3 \pi }{4}$ with the positive direction of x-axis in anticlockwise direction then $f ' (3)$ is
If $|\vec{a} | = 3, |\vec{b}| = 2, |\vec{c}| = 1$ then the value of $|\vec{a}. \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}| $ is (given that $\vec{a} + \vec{b} + \vec{c} = 0$)
The value of $ (1 + \omega - \omega^2)^7$ is
The equation of one of the common tangents to the parabola $y^2 = 8x$ and $x^2 + y^2 - 12x + 4 = 0$ is
What is the area of a loop of the curve $r = a \sin^3 \theta$ ?
A tetrahedron has vertices at $O (0, 0, 0), A (1, 2, 1) B (2, 1, 3) $ and $C (-1, 1, 2)$. Then the angle between the faces $OAB$ and $ABC$ will be
If $ A$ and $B$ are matrices and $B = ABA^{-1}$ then the value of $(A + B) (A - B)$ is
In a culture the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000 if the rate of growth of bacteria is proportional to the number present.
If $e^x = y + \sqrt{ 1 + y^2}$ , then the value of y is
$CO$ is a stronger ligand than $Cl^-$ because