List of practice Questions

Which of the following is a self adjusting force?

An alternating voltage of 220 V, 50 Hz frequency is applied across a capacitor of capacitance 2 $\mu$F. The impedence of the circuit is

Two capacitors $C_{1}$ and $C_{2}$ in a circuit are joined as shown in figure. The potentials of points A and B are $V_{1}$ and $V_{2}$ respectively. Then the potential of point D will be

If $z=\frac{7-i}{3-4i} \, then\, z^{14}=$

The value of x obtained from the equation $\begin{vmatrix}x+\alpha&\beta&\gamma\\ \gamma&x+\beta&\alpha\\ \alpha&\beta&x+\gamma\end{vmatrix}=0$ will be

Which graph corresponds to an object moving with a constant negative acceleration and a positive velocity ?

In an experiment a sphere of aluminium of mass 0.20 kg is heated upto $150^{\circ}C$. Immediately, it is put into water of volume 150 cc at $27^{\circ}C$ kept in a calorimeter of water equivalent to 0.025 kg. Final temperature of the system is $40^{\circ}C$. The specific heat of aluminium is : (take 4.2 Joule=1 calorie)

1 gram of a carbonate $(M_2CO_3)$ on treatment with excess $HCl$ produces $0.01186$ mole of $CO_2$. The molar mass of $M_2CO_3$ in g $mol^{-1}$ is :

Excess of $NaOH_{(aq)}$ was added to $100\, mL$ of ${ FeCl_3}$ (aq) resulting into $2.14\, g$ of ${Fe(OH)_3}$. The molarity of ${FeCl_3}$ (aq) is : (Given molar mass of ${Fe = 56 \, g \, mol^{-1}}$ and molar mass of ${Cl = 35.5 \, g \, mol^{-1}}$)

Let $l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :

An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency $10\, GHz$. What is the frequency of the microwave measured by the observer ? (speed of light $= 3 \times 10^8 \, ms^{-1}$)

A slender uniform rod of mass $M$ and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical is :

An external pressure \(P\) is applied on a cube at \(0^{\circ}C\) so that it is equally compressed from all sides.\(K\) is the bulk modulus of the material of the cube and \(\alpha\) is its coefficient of linear expansion.Suppose we want to bring the cube to its original size by heating it. The temperature should be raised by :

The integral $\int\limits^{\frac{3\, \pi}{4}}_{\frac{\pi}{4}} \frac{dx}{ 1 + \cos \, x}$ is equal to :

For a reaction, $ {A_(g) \to A_(l); \Delta H = - 3RT}$. The correct statement for the reaction is :

If \[ y(x) = \int_{\sqrt{x}}^x e^t \, dt, \quad x>0 \] then \( y'(1) = \) .............

The radius of convergence of the power series \[ \sum_{n=0}^{\infty} n! x^{n^2} \] is ...........

Let \( \alpha, \beta, \gamma, \delta \) be the eigenvalues of the matrix

\[ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 2 & 0 \end{pmatrix} \] Then \( \alpha^2 + \beta^2 + \gamma^2 + \delta^2 = \) ..........

For \( x>1 \), let \[ f(x) = \int_1^x \left( \sqrt{\log t - \frac{1}{2} \log \sqrt{t}} \right) dt. \] The number of tangents to the curve \( y = f(x) \) parallel to the line \( x + y = 0 \) is ..............

For a real number \( x \), define \( \left\lceil x \right\rceil \) to be the smallest integer greater than or equal to \( x \). Then \[ \int_0^1 \int_0^1 \left( \left\lceil x \right\rceil + \left\lceil y \right\rceil + \left| z \right| \right) dx \, dy \, dz = \text{.............}. \]

Let \( a_n = \sqrt{n}, n \geq 1 \), and let \[ s_n = a_1 + a_2 + \cdots + a_n. \text{ Then} \] \[ \lim_{n \to \infty} \left( \frac{a_n}{s_n} \right) \left( -\ln \left( 1 - \frac{a_n}{s_n} \right) \right) = \text{............}. \]

The maximum order of a permutation \( \sigma \) in the symmetric group \( S_{10} \) is ................

Let \( f(x) = \frac{\sin \left( \frac{n\pi x}{\pi \sin x} \right)}{ \sin x} \), Let \( x_0 \in (0, \pi) \) and let \( f'(x_0) = 0 \). Then \[ (f(x_0))^2 \left( 1 + (\pi^2 - 1)\sin^2 x_0 \right) = \text{.........}. \]

Let \( T \) be the smallest positive real number such that the tangent to the helix \[ x = \cos t, \quad y = \sin t, \quad z = \frac{t}{\sqrt{2}} \] at \( t = T \) is orthogonal to the tangent at \( t = 0 \). Then the line integral of \( \mathbf{F} = xj - y\mathbf{i} \) along the section of the helix from \( t = 0 \) to \( t = T \) is ..........

Let \( y(x), x>0 \) be the solution of the differential equation \[ x^2 \frac{d^2y}{dx^2} + 5x \frac{dy}{dx} + 4y = 0 \] satisfying the conditions \( y(1) = 1 \) and \( y'(1) = 0 \). Then the value of \( e^2y(e) \) is .............