A particle moves along a straight line such that its position is given by \( x(t) = 3t^2 - t^3 \). What is its velocity at \( t = 2 \) seconds?

• (A) \( 0 m/s \)
• (B) \( 4 m/s \)
• (C) \( -4 m/s \)
• (D) \( 12 m/s \)

A force of \( 20 N \) is applied to a \( 5 kg \) object at an angle of \( 60^{\circ} \) to the horizontal. What is the horizontal acceleration of the object, ignoring friction?

• (A) \( 2 m/s^2 \)
• (B) \( 4 m/s^2 \)
• (C) \( 1.73 m/s^2 \)
• (D) \( 2.5 m/s^2 \)

A ball is projected horizontally from the top of a tower with a velocity of \( 10 m/s \). If it hits the ground \( 2 seconds \) later, what is the height of the tower? (Take \( g = 10 m/s^2 \))

• (A) \( 20 m \)
• (B) \( 10 m \)
• (C) \( 40 m \)
• (D) \( 25 m \)

An ideal gas is compressed isothermally. During this process:

• (A) Internal energy remains constant
• (B) Temperature increases
• (C) No work is done on the gas
• (D) Pressure remains constant

Which of the following compounds exhibits the highest boiling point due to hydrogen bonding?

• (A) \( H_2O \)
• (B) \( H_2S \)
• (C) \( CH_4 \)
• (D) \( HCl \)

For a first-order reaction \( A \rightarrow B \), the rate constant is \( 0.1 s^{-1} \). What is the time required for \( 50% \) completion?

• (A) \( 6.93 s \)
• (B) \( 5 s \)
• (C) \( 10 s \)
• (D) \( 0.693 s \)

In a galvanic cell, oxidation always occurs at:

• (A) The anode
• (B) The cathode
• (C) The salt bridge
• (D) The electrolyte

Which of the following is an example of an intensive property?

• (A) Temperature
• (B) Mass
• (C) Volume
• (D) Total energy

If \[ f(x)=\ln\left(\frac{\sin x}{1+\cos x}\right), \]
then \(f'(x)\) is equal to:

• (A) \( \csc x \)
• (B) \( \cot x \)
• (C) \( \tan x \)
• (D) \( \sec x \)

The sum of the first \(20\) terms of an arithmetic progression is \(640\), and the difference between the \(15^th\) and \(5^th\) terms is \(30\). Find the first term of the A.P.

• (A) \( 12 \)
• (B) \( 14 \)
• (C) \( 17 \)
• (D) \( 19 \)

If a real matrix \(A\) satisfies \[ A^T=A \quad and \quad A^2=I, \]
then the eigenvalues of \(A\) must be:

• (A) Only \(1\)
• (B) Only \(-1\)
• (C) Either \(1\) or \(-1\)
• (D) Zero only

Evaluate: \[ \int_0^1 x^3\ln(1+x)\,dx \]

• (A) \( \frac{7}{48} \)
• (B) \( \frac{25}{48} \)
• (C) \( \frac{1}{8} \)
• (D) \( \frac{5}{24} \)