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

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

Find the area of the region bounded by the curve \(y^2 = 8x\) and the line \(x = 2\).

• (A) \( \frac{28}{3} \) sq. units
• (B) \( \frac{32}{3} \) sq. units
• (C) \( \frac{16}{3} \) sq. units
• (D) \( \frac{20}{3} \) sq. units

Solve the differential equation \( \frac{dy}{dx} = \frac{1+y^2}{1+x^2} \) given \(y(0)=1\).

• (A) \( \tan^{-1}y = \tan^{-1}x + \frac{\pi}{6} \)
• (B) \( \tan^{-1}y = \tan^{-1}x + \frac{\pi}{4} \)
• (C) \( \tan^{-1}y = \tan^{-1}x + \frac{\pi}{3} \)
• (D) \( \tan^{-1}y = \tan^{-1}x + \frac{\pi}{2} \)

In a Binomial distribution with \(n=10\) and \(p=0.4\), find the variance.

• (A) \(2.0\)
• (B) \(2.4\)
• (C) \(4.0\)
• (D) \(3.2\)

Find the direction cosines of a line that makes equal angles with the coordinate axes.

• (A) \( \pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{2}}, 0 \)
• (B) \( \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}} \)
• (C) \( \pm \frac{1}{2}, \pm \frac{1}{2}, \pm \frac{1}{2} \)
• (D) \( \pm 1, 0, 0 \)

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

• (A) \(1.227\,\AA\)
• (B) \(0.612\,\AA\)
• (C) \(2.45\,\AA\)
• (D) \(3.12\,\AA\)

A Carnot engine works between \(600\,K\) and \(300\,K\). Find its efficiency.

• (A) \(25%\)
• (B) \(40%\)
• (C) \(50%\)
• (D) \(60%\)

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

• (A) \(0.025\,J\)
• (B) \(0.0125\,J\)
• (C) \(0.05\,J\)
• (D) \(0.1\,J\)

Find the moment of inertia of a uniform ring of mass \(M\) and radius \(R\) about a tangent perpendicular to its plane.

• (A) \(MR^2\)
• (B) \(2MR^2\)
• (C) \(3MR^2\)
• (D) \(4MR^2\)

What is the major product when 2-bromopentane reacts with alcoholic KOH?

• (A) Pent-1-ene
• (B) Pent-2-ene
• (C) Pentanol
• (D) Pentanal

How many atoms are present in a unit cell of a body-centered cubic (BCC) lattice?

• (A) 1
• (B) 2
• (C) 3
• (D) 4

Which reagent is used in the Lucas test to distinguish alcohols?

• (A) \( ZnCl_2 + HCl \)
• (B) \( KMnO_4 \)
• (C) \( NaOH \)
• (D) \( H_2SO_4 \)

Calculate the osmotic pressure of \(0.1\,M\) glucose at \(27^\circ C\).

• (A) \(1.23\ atm\)
• (B) \(2.46\ atm\)
• (C) \(4.92\ atm\)
• (D) \(0.82\ atm\)

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

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