If the statement \( (p \land q) \rightarrow (r \lor \neg s) \) is False, find the truth values of \(p, q, r,\) and \(s\).

• (A) \(p=T, q=T, r=F, s=T\)
• (B) \(p=F, q=T, r=F, s=T\)
• (C) \(p=T, q=F, r=T, s=F\)
• (D) \(p=F, q=F, r=T, s=T\)

Find the value of \( k \) if the function \( f(x) = \dfrac{k\cos x}{\pi - 2x} \) is continuous at \( x = \dfrac{\pi}{2} \).

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

Find the unit vector perpendicular to both \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 3\hat{k} \).

• (A) \( \dfrac{-\hat{i} + 7\hat{j} + 5\hat{k}}{5\sqrt{3}} \)
• (B) \( \dfrac{\hat{i} - 7\hat{j} - 5\hat{k}}{5\sqrt{3}} \)
• (C) \( \dfrac{-\hat{i} + 7\hat{j} + 5\hat{k}}{\sqrt{75}} \)
• (D) \( \dfrac{\hat{i} + 7\hat{j} + 5\hat{k}}{5\sqrt{3}} \)

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

• (A) \(0\)
• (B) \( \dfrac{\pi}{4} \)
• (C) \( \dfrac{\pi}{2} \)
• (D) \( \pi \)

If \( y = \sin^{-1}(3x - 4x^3) \), find \( \dfrac{dy}{dx}. \)

• (A) \( \dfrac{3 - 12x^2}{\sqrt{1-(3x-4x^3)^2}} \)
• (B) \( \dfrac{3 - 12x^2}{\sqrt{1-(3x-4x^3)}} \)
• (C) \( \dfrac{12x^2 - 3}{\sqrt{1-(3x-4x^3)^2}} \)
• (D) \( \dfrac{3}{\sqrt{1-(3x-4x^3)^2}} \)

A particle moves with a constant velocity of \(5\ m/s\) in a circular path of radius \(2\ m\); calculate its centripetal acceleration.

• (A) \(5\ m/s^2\)
• (B) \(10\ m/s^2\)
• (C) \(12.5\ m/s^2\)
• (D) \(25\ m/s^2\)

A block of mass \(5\ kg\) is on a frictionless surface; find its acceleration when a force of \(10\ N\) is applied.

• (A) \(1\ m/s^2\)
• (B) \(2\ m/s^2\)
• (C) \(5\ m/s^2\)
• (D) \(10\ m/s^2\)

What is the time period of a simple pendulum of length \(1\ m\) if \(g = 9.8\ m/s^2\)?

• (A) \(1\ s\)
• (B) \(2\ s\)
• (C) \(2.01\ s\)
• (D) \(3\ s\)

Determine the minimum speed at the topmost point of a vertical circle of radius \(L\) for the string to remain taut.

• (A) \( \sqrt{gL} \)
• (B) \( \sqrt{2gL} \)
• (C) \( \sqrt{3gL} \)
• (D) \( gL \)

Calculate the equivalent capacitance of an infinite circuit formed by repeating identical capacitors of capacitance \(C\).

• (A) \(0\)
• (B) \(C\)
• (C) \(2C\)
• (D) \( \dfrac{C}{2} \)

Identify the product formed when phenol reacts with bromine water.

• (A) Bromobenzene
• (B) 2,4,6-Tribromophenol
• (C) Phenyl bromide
• (D) Benzoic acid

What is the IUPAC name for the compound \( CH_3 - CH(OH) - CH_2 - CHO \)?

• (A) 3-Hydroxybutanal
• (B) 2-Hydroxybutanal
• (C) 4-Hydroxybutanal
• (D) Butanol

Which reagent is used in the Lucas test to distinguish between primary, secondary, and tertiary alcohols?

• (A) Dilute HCl
• (B) Concentrated HCl and \(ZnCl_2\)
• (C) \(NaOH\)
• (D) \(KMnO_4\)

Define the coordination number and oxidation state of the central metal ion in \([Co(NH_3)_6]Cl_3\).

• (A) Coordination number = 4, Oxidation state = +2
• (B) Coordination number = 6, Oxidation state = +3
• (C) Coordination number = 6, Oxidation state = +2
• (D) Coordination number = 3, Oxidation state = +3

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

• (A) \(2.46\ atm\)
• (B) \(1.23\ atm\)
• (C) \(0.82\ atm\)
• (D) \(4.10\ atm\)