List of practice Questions

The sum of the order and degree of the differential equation: \[ x \left( \frac{d^2 y}{dx^2} \right)^{1/2} = \left( 1 + \frac{dy}{dx} \right)^{4/3} \] is:

The density of nitric acid solution is 1.5 g mL$^{-1}$. Its weight percentage is 68. What is the approximate concentration (in mol L$^{-1}$) of nitric acid? (N = 14 u; O = 16 u; H = 1 u)

The osmotic pressure of seawater is 1.05 atm. Four experiments were carried out as shown in the table. In which of the following experiments, pure water can be obtained in part-II of the vessel?

The question asks for an example of RNA-containing viruses.

The potential difference across the ends of a conductor is $ (30 \pm 0.3) V $ and the current through the conductor is $ (5 \pm 0.1) A $. The error in the determination of the resistance of the conductor is:

A body thrown vertically upwards reaches a maximum height $ H $. The ratio of the velocities of the body at heights $ \frac{3H}{4} $ and $ \frac{8H}{9} $ from the ground is:

Let R² denote R × R. Let S = {(a, b, c) : a, b, c ∈ R and ax² + 2bxy + cy² > 0 for all (x, y) ∈ R² - {(0, 0}}.
Then which of the following statements is (are) TRUE ?

The equation of projectile motion is given by $ y = 3x - 0.8x^2 $. The time of flight of the projectile is (Acceleration due to gravity $ g = 10 $ m/s²):

A 100 kg gun fires a ball of 1 kg horizontally from a cliff of height 500 m. It falls on the ground at a distance of 400 m from the bottom of the cliff. The recoil velocity of the gun is (Acceleration due to gravity $ g = 10 $ ms$^{-2}$):

Let $S=\left\{a+b\sqrt2:a,b\in Z\right\},T_1=\left\{(-1+\sqrt2)^n:n\in \N\right\},\ \text{and}\ T_2=\left\{(1+\sqrt2)^n:n\in\N\right\}$. Then which of the following statements is (are) TRUE ?

The average power generated by a 90 kg mountain climber who climbs a summit of height 600 m in 90 minutes is (Acceleration due to gravity $ g = 10 $ m/s$^2$):

A boy weighing 50 kg finished a long jump at a distance of 8 m. Considering that he moved along a parabolic path and his angle of jump is $ 45^\circ $, his initial kinetic energy is:

A uniform rod of length $ 2L $ is placed with one end in contact with the earth and is then inclined at an angle $ \alpha $ to the horizontal and allowed to fall without slipping at the contact point. When it becomes horizontal, its angular velocity will be:

Two simple harmonic motions are represented by $y_1 = 5 \left[ \sin 2\pi t + \sqrt{3} \cos 2\pi t \right] $ and $y_2 = 5 \sin \left[ 2\pi t + \frac{\pi}{4} \right] $. The ratio of their amplitudes is:

When a mass $ m $ is connected individually to the springs $ s_1 $ and $ s_2 $, the oscillation frequencies are $ v_1 $ and $ v_2 $. If the same mass is attached to the two springs as shown in the figure, the oscillation frequency would be:

A satellite moving around the Earth in a circular orbit has kinetic energy $ E $. Then, the minimum amount of energy to be added so that it escapes from the Earth is:

The elongation of a copper wire of cross-sectional area $3.5 \text{ mm}^2$, in the figure shown, is $$ Y_{\text{copper}} = 10 \times 10^{10} \, \text{Nm}^{-2} \quad \text{and} \quad g = 10 \, \text{m/s}^2 $$

Consider the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let S (p, q) be a point in the first quadrant such that $\frac{p^2}{9}+\frac{q^2}{4}>1$. Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and Obe the center of the ellipse. If the area of the triangle △ORT is $\frac{3}{2}$ , then which of the following options is correct ?

In the given circuit, if the potential at point B is 24 V, the potential at point A is:

‘X’ is a protecting colloid. The following data is obtained for preventing the coagulation of 10 mL of gold sol to which 1 mL of 10% NaCl is added. What is the gold number of ‘X’?

Water is flowing in a streamline manner in a horizontal pipe. If the pressure at a point where cross-sectional area is $ 10 $ cm$^2$ and velocity $ 1 $ m/s is $ 2000 $ Pa, then the pressure of water at another point where the cross-sectional area is $ 5 $ cm$^2$ is:

A metal ball of mass 100 g at $ 20^\circ C $ is dropped in 200 ml of water at $ 80^\circ C $. If the resultant temperature is $ 70^\circ C $, then the ratio of specific heat of the metal to that of water is:

The efficiency of a heat engine that works between the temperatures where Celsius-Fahrenheit scales coincide and Kelvin-Fahrenheit scales coincide is (approximately):

Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1) = 2 and $\lim\limits_{t→x}\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1$
for each x > 0. Then, for all x > 0, f(x) is equal to