If \( z \) be a complex number such that \( |z| + z = 2 + i \), then find the value of \( |z| \).

• (A) \( \frac{1}{2} \)
• (B) \( \frac{3}{4} \)
• (C) \( \frac{5}{4} \)
• (D) 1

If \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \) and \( \vec{b} = p\hat{i} + 2\hat{j} + 2\hat{k} \) are perpendicular to each other, find the value of \( p \).

• (A) 4
• (B) \( -4 \)
• (C) 2
• (D) \( -2 \)

The solubility product of \( AgBr \) is \( 4.9 \times 10^{-13} \) at a certain temperature. Calculate the molar solubility of \( AgBr \).

• (A) \( 4 \times 10^{-6} mol dm^{-3} \)
• (B) \( 4 \times 10^{-7} mol dm^{-3} \)
• (C) \( 7 \times 10^{-7} mol dm^{-3} \)
• (D) \( 3 \times 10^{-8} mol dm^{-3} \)

Find the concentration of sodium acetate when added to 0.1 M solution of acetic acid to form a buffer solution of \( pH = 5.5 \). (Given: \( pK_a \) of \( CH_3COOH = 4.5 \))

• (A) 0.1 M
• (B) 0.01 M
• (C) 1.0 M
• (D) None of these

Which of the following equations gives the combined relationship of Boyle’s law and Charles’s law?

• (A) \( \frac{P_1V_2}{T_1} = \frac{P_2V_1}{T_2} \)
• (B) \( n = \frac{RT}{PV} \)
• (C) \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \)
• (D) \( P = \frac{RT}{nV} \)

Which of the following forces is involved in dinitrogen?

• (A) Dipole-dipole interaction
• (B) Dipole-induced dipole interaction
• (C) London dispersion force
• (D) Hydrogen bonding

Which of the following is a correct set of four quantum numbers for a 4d orbital?

• (A) \( (4, 3, 2, 1/2) \)
• (B) \( (4, 2, 1, 1/2) \)
• (C) \( (4, 1, 2, 1/2) \)
• (D) None of these

Which of the following mixtures of gases represent water gas?

• (A) \( CO + H_2 \)
• (B) \( CO_2 + H_2 \)
• (C) \( H_2O + CH_4 \)
• (D) \( H_2 + O_2 \)

Which of the following electrolytic solutions of the same electrolyte has maximum electrolytic conductivity at the same temperature?

• (A) 0.2 mole of solute is dissolved in 250 ml
• (B) 0.3 mol in 600 ml
• (C) 0.6 mol in 1000 ml
• (D) 0.8 mole in 2000 ml

In an isothermal expansion of an ideal gas, the heat supplied to the gas is:

• (A) entirely used to increase internal energy
• (B) entirely used to do work
• (C) partly used to do work and partly to increase internal energy
• (D) none of the above