Find the area of the region bounded by the curve \(y^2 = 4x\) and the line \(x = 3\).

• (A) \(8\sqrt{3}\)
• (B) \(6\sqrt{3}\)
• (C) \(4\sqrt{3}\)
• (D) \(12\sqrt{3}\)

If the vectors \(2i - j + k\), \(i + 2j - 3k\) and \(3i + aj + 5k\) are coplanar, find the value of \(a\).

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

Find the general solution of the differential equation \(\dfrac{dy}{dx} + y\cot x = \csc x\).

• (A) \(y\sin x = x + c\)
• (B) \(y\cos x = x + c\)
• (C) \(y\sin x = c - x\)
• (D) \(y\cos x = c + x\)

The probability of a shooter hitting a target is \( \frac{3}{4} \). Find the probability of hitting the target exactly 4 times in 5 shots.

• (A) \(\frac{405}{1024}\)
• (B) \(\frac{243}{1024}\)
• (C) \(\frac{405}{512}\)
• (D) \(\frac{81}{256}\)

Find the value of \(k\) if the function \(f(x)=\dfrac{k\sin x}{x}\) for \(x\neq0\) and \(f(0)=3\) is continuous at \(x=0\).

• (A) \(1\)
• (B) \(2\)
• (C) \(3\)
• (D) \(4\)

A stone is dropped from a height of \(20\,m\); calculate its velocity just before it hits the ground \((g = 10\,m/s^2)\).

• (A) \(10\,m/s\)
• (B) \(15\,m/s\)
• (C) \(20\,m/s\)
• (D) \(25\,m/s\)

Calculate the de Broglie wavelength of an electron accelerated through a potential difference of \(100\,V\).

• (A) \(0.1227\,\AA\)
• (B) \(1.227\,\AA\)
• (C) \(12.27\,\AA\)
• (D) \(2.27\,\AA\)

What is the ratio of the surface area of two soap bubbles if their radii are in the ratio \(2:3\)?

• (A) \(2:3\)
• (B) \(4:9\)
• (C) \(3:2\)
• (D) \(9:4\)

If the current in a coil changes from \(5\,A\) to \(2\,A\) in \(0.1\,s\) and an average emf of \(30\,V\) is induced, find the self-inductance.

• (A) \(0.5\,H\)
• (B) \(1\,H\)
• (C) \(2\,H\)
• (D) \(3\,H\)

Find the energy stored in a capacitor of \(10\,\mu F\) charged to a potential of \(50\,V\).

• (A) \(0.0125\,J\)
• (B) \(0.025\,J\)
• (C) \(0.125\,J\)
• (D) \(0.25\,J\)

Calculate the number of atoms per unit cell in a Face-Centered Cubic (FCC) crystal structure.

• (A) \(2\)
• (B) \(4\)
• (C) \(6\)
• (D) \(8\)

Identify the major product formed when Ethyl Bromide reacts with alcoholic KOH.

• (A) Ethanol
• (B) Ethene (Ethylene)
• (C) Diethyl ether
• (D) Acetaldehyde

What is the hybridization of the central atom in \(SF_6\) molecule?

• (A) \(sp^3\)
• (B) \(sp^2\)
• (C) \(sp^3d\)
• (D) \(sp^3d^2\)

Find the osmotic pressure of a \(0.1\,M\) glucose solution at \(27^\circ C\) \((R = 0.0821\,L\,atm\,mol^{-1}K^{-1})\).

• (A) \(1.23\,atm\)
• (B) \(2.46\,atm\)
• (C) \(3.46\,atm\)
• (D) \(4.10\,atm\)

Name the catalyst used in the manufacture of ammonia by Haber’s process.

• (A) Nickel
• (B) Platinum
• (C) Finely divided Iron (Fe) with Molybdenum (Mo) as a promoter
  (D) Vanadium pentoxide