Two point charges \(+q\) and \(-q\) are held fixed at a distance \(d\) apart. The net electric potential at a point midway between the two charges and its net field are?

• (A) \( E=0 \quad V=0 \)
• (B) \( E \neq 0 \quad V \neq 0 \)
• (C) \( E=0 \quad V \neq 0 \)
• (D) \( E \neq 0 \quad V=0 \)

A charge \(Q\) is placed at the center of an uncharged conducting spherical shell of inner radius \(R_1\) and outer radius \(R_2\). The surface charge density on the outer surface of the shell is:

• (A) \( \frac{Q}{(4\pi R_1^2)} \)
• (B) \( \frac{Q}{(4\pi R_2^2)} \)
• (C) \( \frac{-Q}{(4\pi R_1^2)} \)
• (D) \( Zero \)

A parallel plate capacitor is charged and then disconnected from the charging battery. If the plates are now pulled further apart using insulating handles:

• (A) The charge increases
• (B) The voltage decreases
• (C) The capacitance increases
• (D) The electrostatic energy stored increases

The magnetic field at the center of a circular current-carrying loop of radius \(R\) is \(B_0\). The magnetic field at an axial point at a distance \(x = R\) from the center of the loop is:

• (A) \( B_0/2 \)
• (B) \( B_0/2\sqrt{2} \)
• (C) \( B_0/4 \)
• (D) \( B_0\sqrt{2} \)

The value of enthalpy change (\(\Delta H\)) for the reaction \(C_2H_5OH(l) + 3O_2(g) \rightarrow 2CO_2(g) + 3H_2O(l)\), at \(27^\circC\) is \(-1366.5 kJ mol^{-1}\). The value of internal energy change for the above reaction at this temperature will be

• (A) \( -1371.5 kJ mol^{-1} \)
• (B) \( -1369.0 kJ mol^{-1} \)
• (C) \( -1364.0 kJ mol^{-1} \)
• (D) \( -1361.5 kJ mol^{-1} \)

For the complete combustion of ethanol, \(C_2H_5OH(l) + 3O_2(g) \rightarrow 2CO_2(g) + 3H_2O(l)\), the amount of heat produced as measured in bomb calorimeter is \(1364.47 kJ mol^{-1}\) at \(25^\circC\). Assuming ideality the enthalpy of combustion, \(\Delta H_C\), for the reaction will be (\(R = 8.314 JK^{-1}mol^{-1}\))

• (A) \( -1366.95 kJ mol^{-1} \)
• (B) \( -1361.95 kJ mol^{-1} \)
• (C) \( -1460.50 kJ mol^{-1} \)
• (D) \( -1350.50 kJ mol^{-1} \)

What is \([NH_4^+]\) in a solution that is \(0.02M NH_3\) and \(0.01 M KOH\)? \([K_b(NH_3) = 1.8 \times 10^{-5}]\)

• (A) \( 3.6 \times 10^{-5} M \)
• (B) \( 1.8 \times 10^{-5} M \)
• (C) \( 0.9 \times 10^{-5} M \)
• (D) \( 7.2 \times 10^{-5} M \)

1.1 mole of A mixed with 2.2 moles of B and the mixture is kept in a 1 L flask and the equilibrium, \(A + 2B \rightleftharpoons 2C + D\) is reached. If at equilibrium 0.2 mole of C is formed then the value of \(K_c\) will be

• (A) \( 0.1 \)
• (B) \( 0.01 \)
• (C) \( 0.001 \)
• (D) \( 1 \)

If \(y^x = e^{y - x}\), then \(\frac{dy}{dx}\) is equal to

• (A) \( \frac{1 + \log y}{y \log y} \)
• (B) \( \frac{(1 + \log y)^2}{y \log y} \)
• (C) \( \frac{1 + \log y}{(\log y)^2} \)
• (D) \( \frac{(1 + \log y)^2}{\log y} \)

The domain of the function \(f(x) = \sqrt{x - \sqrt{1 - x^2}}\) is

• (A) \( \left[-1, -\frac{1}{\sqrt{2}}\right] \cup \left[\frac{1}{\sqrt{2}}, 1\right] \)
• (B) \( [-1, 1] \)
• (C) \( \left(-\infty, -\frac{1}{\sqrt{2}}\right] \cup \left[\frac{1}{\sqrt{2}}, +\infty\right] \)
• (D) \( \left[\frac{1}{\sqrt{2}}, 1\right] \)

Let \(A = \{1, 2, 3, 4, 5\}\) and \(R\) be a relation defined by \(R = \{(x, y) : x, y \in A, x + y = 5\}\). Then, \(R\) is

• (A) reflexive and symmetric but not transitive
• (B) an equivalence relation
• (C) neither reflexive nor transitive
• (D) neither reflexive nor symmetric but transitive

If A and B are symmetric matrices of same order such that \(AB + BA = X\) and \(AB - BA = Y\), then \((XY)^T =\)

• (A) \( XY \)
• (B) \( X^TY^T \)
• (C) \( -YX \)
• (D) \( -Y^TX^T \)