Find the distance between the planes \(2x + 3y + 4z = 4\) and \(4x + 6y + 8z = 12\).

• (A) \( \dfrac{1}{\sqrt{29}} \)
• (B) \( \dfrac{2}{\sqrt{29}} \)
• (C) \( \dfrac{3}{\sqrt{29}} \)
• (D) \( \dfrac{4}{\sqrt{29}} \)

If \(A\) is a square matrix of order \(3\) and \(|A| = 5\), find \(|adj(A)|\).

• (A) \(5\)
• (B) \(10\)
• (C) \(25\)
• (D) \(125\)

Evaluate \( \displaystyle \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx \).

• (A) \( \dfrac{\pi}{2} \)
• (B) \( \dfrac{\pi}{3} \)
• (C) \( \dfrac{\pi}{4} \)
• (D) \( \dfrac{\pi}{6} \)

What is the ratio of the de Broglie wavelengths of an electron and a proton moving with the same velocity?

• (A) \( \dfrac{m_e}{m_p} \)
• (B) \( \dfrac{m_p}{m_e} \)
• (C) \(1\)
• (D) \( \sqrt{\dfrac{m_p}{m_e}} \)

A capacitor of \(10\,\mu F\) is charged to \(100V\). Find the energy stored in it.

• (A) \(0.5 \,J\)
• (B) \(0.05 \,J\)
• (C) \(5 \,J\)
• (D) \(0.005 \,J\)

If the current in a coil changes from \(2A\) to \(4A\) in \(0.1\,s\), inducing an EMF of \(20V\), find the self-inductance.

• (A) \(0.5\,H\)
• (B) \(1\,H\)
• (C) \(2\,H\)
• (D) \(4\,H\)

Which of the following is the most acidic: Ethanol, Phenol, or \(o\)-Nitrophenol?

• (A) Ethanol
• (B) Phenol
• (C) \(o\)-Nitrophenol
• (D) All have same acidity

What is the product of the reaction between Benzene and \(CH_3Cl\) in the presence of anhydrous \(AlCl_3\)?

• (A) Toluene
• (B) Phenol
• (C) Chlorobenzene
• (D) Benzaldehyde

What is the oxidation state of Nickel in \(Ni(CO)_4\)?

• (A) \(+2\)
• (B) \(+1\)
• (C) \(0\)
• (D) \(-2\)

Find the missing number: \(2, 6, 12, 20, 30, ?\)

• (A) \(38\)
• (B) \(40\)
• (C) \(42\)
• (D) \(44\)

Pointing to a photograph, a man says, ``I have no brother or sister, but that man's father is my father's son.'' Who is in the photo?

• (A) His father
• (B) His son
• (C) His nephew
• (D) Himself

Choose the correct synonym for ``Diligent.''

• (A) Lazy
• (B) Careless
• (C) Hardworking
• (D) Ignorant