List of practice Questions

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

Which of the following is NOT suitable as a post-coital contraceptive

If $A$ and $B$ are foot of perpendicular drawn from point $Q (a, b, c)$ to the planes $yz$ and $zx$, then equation of plane through the points $A, B$ and $O$ is ___________

For what value of $k$, the function defined by $ f(x) = \begin{cases} \frac{log(1+2x)sin\,x^\circ}{x^2} & \text{for } x \ge \text {0}\\ k & \text{for } x = \text{ 0} \end{cases}$ is continuous at $x = 0$ ?

Derivative of $log \left(sec\,\theta +tan \,\theta\right) $ with respect to $sec\, \theta$ at $\theta = \pi/4$ is

If $A=\begin{bmatrix}1&1&0\\ 2&1&5\\ 1&2&1\end{bmatrix}$, then $a_{11}A_{21} +a_{12}A_{22}+a_{13}A_{23} $ is equal to

If Rolle�s theorem for $f\left(x\right)= e^{x} \left(sinx - cosx\right)$ is verified on $[\pi/4$, $5 \pi/4]$, then the value of $c$ is

If $ 2 \,tan^{-1} (cos\,x) = tan^{-1} (2 \,cosec\,x)$ then $sin\,x + cos\,x $ is equal to

$\int \frac{1}{\sqrt{8+2x-x^{2}}} dx$ is equal to

The approximate value of $f\left(x\right)= x^{3}+5x^{2}-7x +9$ at $x=1.1 $ is

The joint equation of lines passing through the origin and trisecting the first quadrant is ________

A solution contains 4 g of NaOH and 16.2 g of water. The mole fraction solute and solvent are respectively

For a gas $\frac{R}{C_{v}} = 0.4$, where ?$R$? is the universal gas constant and ?$C_v$? is molar specific heat at constant volume. The gas is made up of molecules which are

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