List of practice Questions

A straight rod of length \( L \) extends from \( x = a \) to \( x = L + a \). Find the gravitational force it exerts on a point mass \( m \) at \( x = 0 \) if the mass per unit length of the rod is \( \mu = A + Bx^2 \):
Two identical coins of mass 8 g are 50 cm apart on a tabletop. How many times larger is the weight of one coin than the gravitational attraction of the other coin for it? (G = \( 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \), g = 9.81 m/s\(^2\)):
An electrical appliance is rated 1500 W, 250 V. This appliance is connected to a 250 V supply mains. The current drawn by the appliance (assuming unity power factor) is:
If \( Z_1, Z_2, Z_3 \) are three complex numbers with unit modulus such that \[ |Z_1 - Z_2|^2 + |Z_1 - Z_3|^2 = 4 \] then \[ Z_1 Z_2 + \overline{Z_1} Z_2 + Z_1 Z_3 + \overline{Z_1} Z_3 = \]
A car goes around a curve of radius 48 m. If the road is banked at an angle of 15° with the horizontal, the maximum speed in kilometers per hour at which the car can travel if there is to be no tendency to skid even on very slippery pavement (tan 15° = 0.27 approximately) is:
Rain, pouring down at an angle \( \alpha \) with the vertical, has a constant speed of 10 m/s. A woman runs against the rain with a speed of 8 m/s and sees that the rain makes an angle \( \beta \) with the vertical. The relation between \( \alpha \) and \( \beta \) is given by:

The motion of a particle in the XY plane is given by \( x(t) = 25 + 6t^2 \, \text{m} \); \( y(t) = -50 - 20t + 8t^2 \, \text{m} \). The magnitude of the initial velocity of the particle, \( v_0 \), is given by:

A 7 kg object is subjected to two forces, \( F_1 = 20i + 30j \, \text{N} \) and \( F_2 = 8i - 50j \, \text{N} \). Find the acceleration of the object.
If the ratio of the radii of nuclei \( \mathbf{^{52}X_A} \) and \( \mathbf{^{13}Al^{27}} \) is 5:3, then the number of neutrons in the nucleus \( X \) is:
Choose the correct answer from the options given below:
A body of mass \( m \) moves along the X-axis such that at time \( t \), its position is \( x(t) = \alpha t^4 - \beta t^3 + \gamma t \), where \( \alpha \), \( \beta \), and \( \gamma \) are constants. The acceleration of the body is:
A particle executes SHM with a period of 6 s and amplitude of 3 cm. Its maximum speed in cm/s is:
A stone is dropped from the top of a tower. The height through which it falls in the first 3 seconds of its motion equals the height through which it falls in the last second of its motion. To reach the ground, the stone takes time equal to (g = 10 m/s\(^2\)):
The speed \( v \) of a wave on a string depends on the tension \( F \) in the string and the mass per unit length \( m/L \) of the string. If it is known that [F] = [ML][T]–2, the values of the constants \( a \) and \( b \) in the following equation for the speed of a wave on a string are: \[ v = (\text{constant}) F^a \left( \frac{m}{L} \right)^b \]
Sodium atoms emit a spectral line with a wavelength in the yellow, 589.6 nm. What is the difference in energy between the two energy levels involved in the emission of this spectral line?

\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]

Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \). 
 

The source of energy in the Sun is:
To generate energy, nuclear reactors use the principle of:
What back emf is induced in a coil of self-inductance 0.008 H when the current in the coil is changing at the rate of 110 A/s?
If a copper wire carries a current of 80.0 mA, how many electrons flow past a given cross-section of the wire in 10.0 minutes?
A parallel-plate capacitor has plates of dimensions 2.0 cm by 3.0 cm separated by a 1.0 mm thickness of paper. The relative dielectric constant of paper is 3.7. Find its capacitance.
A triode valve has an anode resistance of 20,000 and an amplification factor of 20. The mutual conductance is:
The deviation produced by an equilateral prism, when a ray of light is incident on it, does not depend on:

Complete the following nuclear equation: \( \^{30}_{15}Si + ? \rightarrow ? + 1e^0 \)

Two coherent monochromatic light beams of intensity \( I \) and \( 4I \) are superimposed. The maximum and minimum possible intensities in the resulting beam are: