List of practice Questions

Find the compound \( A \) from the following reaction sequences: \[ A \xrightarrow{\text{aqua regia}} B \xrightarrow{(1) \, \text{KNO}_3, \, \text{NH}_4\text{OH}} \text{yellow ppt}. \]

Match List - I with List - II:

A tightly wound long solenoid carries a current of 1.5 A. An electron is executing uniform circular motion inside the solenoid with a time period of 75 ns. The number of turns per meter in the solenoid is …………

The increase in pressure required to decrease the volume of a water sample by 0.2percentage is \( P \times 10^5 \, \text{Nm}^{-2} \). Bulk modulus of water is \( 2.15 \times 10^9 \, \text{Nm}^{-2} \). The value of \( P \) is ………..

A string of length \( L \) is fixed at one end and carries a mass of \( M \) at the other end. The mass makes \( \frac{3}{\pi} \) rotations per second about the vertical axis passing through the end of the string as shown. The tension in the string is \_\_\_\_ ML.

The ratio of the power of a light source \( S_1 \) to that of the light source \( S_2 \) is 2. \( S_1 \) is emitting \( 2 \times 10^{15} \) photons per second at 600 nm. If the wavelength of the source \( S_2 \) is 300 nm, then the number of photons per second emitted by \( S_2 \) is \_\_\_\_\_ \( \times 10^{14} \).

Acceleration due to gravity on the surface of earth is \( g \). If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is \_\_\_\_ g.

A solid sphere and a hollow sphere of the same mass and of the same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be \( t_1 \) and \( t_2 \), respectively, then:

A photograph of a landscape is captured by a drone camera at a height of 18 km. The size of the camera film is 2 cm \( \times \) 2 cm and the area of the landscape photographed is 400 km\(^2\). The focal length of the lens in the drone camera is:

A particle oscillates along the \( x \)-axis according to the law, \( x(t) = x_0 \sin^2 \left( \frac{\pi t}{T} \right) \), where \( x_0 = 1 \, \text{m} \) and \( T \) is the time period of oscillation. The kinetic energy (\( K \)) of the particle as a function of \( x \) is correctly represented by the graph:

Let \( H_1: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and \( H_2: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \) be two hyperbolas having lengths of latus rectums \( 15\sqrt{2} \) and \( 12\sqrt{5} \) respectively. Let their eccentricities be \( e_1 = \frac{5}{\sqrt{2}} \) and \( e_2 \) respectively. If the product of the lengths of their transverse axes is \( 100\sqrt{10} \), then \( 25e_2^2 \) is equal to:

If \[ \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left( \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| \right) + C, \] where \( C \) is the constant of integration, then \( \alpha + 2\beta \) is equal to ……..

Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:

Let \( P \) be the image of the point \( Q(7, -2, 5) \) in the line \( L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \), and let \( R(5, p, q) \) be a point on \( L \). Then the square of the area of \( \triangle PQR \) is:

Let \( y = y(x) \) be the solution of the differential equation \[ 2\cos x \frac{dy}{dx} = \sin 2x - 4y \sin x, \quad x \in \left( 0, \frac{\pi}{2} \right). \] \( y\left( \frac{\pi}{3} \right) = 0 \), then \( y\left( \frac{\pi}{4} \right) + y\left( \frac{\pi}{4} \right) \) is equal to ________.

Let the position vectors of three vertices of a triangle be \( \overrightarrow{p} = 4\hat{i} + \hat{j} - 3\hat{k} \), \( \overrightarrow{q} = -5\hat{i} + 2\hat{j} + 3\hat{k} \), and \( \overrightarrow{r} = -5\hat{i} + 3\hat{j} + 2\hat{k} \). Then \( \alpha + 2\beta + 5\gamma \) is equal to:

For some \( a, b \), let \( f(x) = \left| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} \right| \), where \( x \neq 0 \), \( \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b \).
Then \( (\lambda + \mu + \nu)^2 \) is equal to:

The equation of the chord of the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \), whose mid-point is \( (3, 1) \) is:

Let \( \mathbf{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \mathbf{b} = \mathbf{a} \times (\hat{i} - 2\hat{k}) \) and \( \mathbf{c} = \mathbf{b} \times \hat{k} \). Then the projection of \( \mathbf{c} - 2\hat{j} \) on \( \mathbf{a} \) is:

The function \( f: (-\infty, \infty) \to (-\infty, 1) \), defined by \[ f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}, \] is:

Let \( f: (0, \infty) \to \mathbb{R} \) be a function which is differentiable at all points of its domain and satisfies the condition \( x^2 f'(x) = 2f(x) + 3 \), with \( f(1) = 4 \). Then \( 2f(2) \) is equal to:

If \( \alpha>\beta>\gamma>0 \), then the expression \[ \cot^{-1} \beta + \left( \frac{1 + \beta^2}{\alpha - \beta} \right) + \cot^{-1} \gamma + \left( \frac{1 + \gamma^2}{\beta - \gamma} \right) + \cot^{-1} \alpha + \left( \frac{1 + \alpha^2}{\gamma - \alpha} \right) \] is equal to:

The number of real solution(s) of the equation \( x^2 + 3x + 2 = \min \left( |x - 3|, |x + 2| \right) \) is:

The molar mass of the water insoluble product formed from the fusion of chromite ore \( {FeCr}_2{O}_4 \) with \( {Na}_2{CO}_3 \) in presence of \( {O}_2 \) is ....... g mol\(^{-1}\).

0.1 mole of compound S will weigh ....... g,
(given the molar mass in g mol\(^{-1}\) \( {C} = 12, \, {H} = 1, \, {O} = 16 )\):