List of practice Questions

A box contains 12 red and 8 blue balls. Two balls are drawn randomly from the box without replacement. The probability of drawing a pair of balls having the same color is ________
A ship of length 200 m has a beam of 25 m. She floats in seawater with an even keel draught of 5 m. The prismatic coefficient of the ship is 0.9; the mass displacement is 20500 tonne and the density of seawater is 1025 kg/m3. The midship section coefficient is ________ (rounded off to two decimal places).
Consider \( f(t) = \cos(at) \), where \( a \) is a real constant. The Laplace transform of \( f(t) \) is _________.
For a butt weld joint of two plates, which one of the following loading scenarios has the least permissible stress?
Which of the following boiler arrangements will ensure that there is NO contamination of the primary feed system?
Consider a Planar Motion Mechanism (PMM) test of a ship model in a towing tank. The transverse motion of the model from the centerline of the tank is described by \( y = a_0 \cos(\omega t) \), where \( \omega \) is the angular frequency. The carriage speed is 3 m/s, \( \omega = \frac{\sqrt{3}}{2} \) rad/s, and the maximum drift angle during the test is \( 30^\circ \). The amplitude of oscillation \( a_0 \) lies between ________.
What is the moment of inertia of a solid sphere of mass \( M \) and radius \( R \) about its diameter?
A force of 20 N is applied to a body at an angle of \( 30^\circ \) to the horizontal, moving the body a distance of 5 m. What is the work done by the force?
A body of mass 10 kg is at a height of 5 m above the surface of the Earth. What is the gravitational potential energy of the body? (Take \( g = 10 \, \text{m/s}^2 \))
A gas expands from a volume of 2 m³ to 4 m³ against a constant pressure of 5 atm. How much work is done by the gas during expansion? (1 atm = \( 1.01 \times 10^5 \, \text{Pa} \))
A concave mirror has a focal length of 15 cm. An object is placed 30 cm from the mirror. What is the image distance?
A water tank is open at the top and has a hole of area \( 10^{-4} \, \text{m}^2 \) at the bottom. The height of the water column is 5 m. What is the speed of the water flowing out of the hole? (Take \( g = 10 \, \text{m/s}^2 \))
If \( \mathbf{a} = \frac{1}{\sqrt{10}} (4\hat{i} - 3\hat{j} + \hat{k}) \) and \( \mathbf{b} = \frac{1}{5} (\hat{i} + 2\hat{j} + 2\hat{k}) \), then the value of \[ (2\mathbf{a} - \mathbf{b}) \cdot \left[ (\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} + 2\mathbf{b}) \right] \]
Evaluate the integral: \[ \int \frac{x^2 + 2x}{\sqrt{x^2 + 1}} \, dx \]
If two dice are rolled, what is the probability of getting a sum of 7?
In the reaction \( 2\text{H}_2 + \text{O}_2 \rightarrow 2\text{H}_2\text{O} \), if 4 moles of hydrogen react with excess oxygen, how many moles of water are produced?
What is the pH of a solution with a hydrogen ion concentration of \( 1 \times 10^{-5} \, \text{mol/L} \)?
Which of the following elements has the highest electronegativity?
The enthalpy change for the reaction \( \text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g) \) is \( -92.4 \, \text{kJ/mol} \). What is the enthalpy change when 4 moles of nitrogen react?
What is the volume occupied by 2 moles of an ideal gas at standard temperature and pressure (STP)?
In the reaction \( \text{Zn} + 2\text{Ag}^+ \rightarrow \text{Zn}^{2+} + 2\text{Ag} \), what is the oxidation state of zinc in \( \text{Zn} \) and \( \text{Zn}^{2+} \)?
Find the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \).
A bag contains 5 red balls, 7 green balls, and 8 blue balls. One ball is drawn at random. What is the probability that the ball is either red or green?
If \( P(A \cap B) = \frac{2}{25} \) and \( P(A \cup B) = \frac{8}{25} \), then find the value of \( P(A) \).
Which one of the following about an electron occupying the 1 s orbital in a hydrogen atom is incorrect ? (Bohr's radius is represented by $\mathrm{a}_{0}$ )