If the statement \( (p \land q) \rightarrow (r \lor \neg s) \) is False (F), what are the truth values of \(p, q, r,\) and \(s\) respectively?

• (A) \(T, T, F, T\)
• (B) \(T, F, F, T\)
• (C) \(T, T, T, F\)
• (D) \(F, T, F, T\)

Evaluate the integral: \( \displaystyle \int \frac{4x^2\cot^{-1}(x^3)}{1+x^6}\,dx \) (where \(C\) is a constant of integration).

• (A) \( \dfrac{2}{3}\big(\cot^{-1}(x^3)\big)^2 + C \)
• (B) \( -\dfrac{2}{3}\big(\cot^{-1}(x^3)\big)^2 + C \)
• (C) \( \dfrac{1}{3}\big(\cot^{-1}(x^3)\big)^2 + C \)
• (D) \( -\dfrac{1}{3}\big(\cot^{-1}(x^3)\big)^2 + C \)

If \( y = \sin^{-1}(3x - 4x^3) \), find the derivative \( \dfrac{dy}{dx} \) in its standard form.

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

Find the angle between non-zero vectors \( \mathbf{a} \) and \( \mathbf{b} \) if their dot product \( \mathbf{a}\cdot\mathbf{b} = 0 \).

• (A) \(0^\circ\)
• (B) \(45^\circ\)
• (C) \(90^\circ\)
• (D) \(180^\circ\)

Evaluate the definite integral: \( \displaystyle \int_{3}^{5} |x-4|\,dx \).

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

Calculate the potential energy of a \(1.5\,kg\) block attached to a spring with \(k = 100\,N/m\) displaced by \(0.2\,m\).

• (A) \(1\,J\)
• (B) \(2\,J\)
• (C) \(4\,J\)
• (D) \(6\,J\)

At what temperature will the r.m.s. velocity of a hydrogen molecule be equal to that of an oxygen molecule at \(47^\circC\)?

• (A) \(40\,K\)
• (B) \(20\,K\)
• (C) \(10\,K\)
• (D) \(5\,K\)

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

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

A wire of length \(L\) and resistance \(R\) is stretched to twice its length; what is the new resistance?

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

Find the ratio of the de Broglie wavelengths of an alpha particle and a proton accelerated through the same potential.

• (A) \(1:1\)
• (B) \(1:\sqrt{2}\)
• (C) \(1:2\)
• (D) \(1:4\)

Which of the following compounds will undergo a Cannizzaro reaction:
\( CH_3CHO,\ C_6H_5CHO,\ CH_3COCH_3 \)?

• (A) \( CH_3CHO \)
• (B) \( C_6H_5CHO \)
• (C) \( CH_3COCH_3 \)
• (D) Both \( CH_3CHO \) and \( C_6H_5CHO \)

Identify the strongest acid among: acetic acid, chloroacetic acid, dichloroacetic acid, and trichloroacetic acid.

• (A) Acetic acid
• (B) Chloroacetic acid
• (C) Dichloroacetic acid
• (D) Trichloroacetic acid

What is the difference in the oxidation number of Manganese (Mn) between \( KMnO_4 \) and \( MnO_2 \)?

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

What product is formed when a ketone reacts with hydrazine (\( NH_2NH_2 \))?

• (A) Hydrazone
• (B) Alcohol
• (C) Ester
• (D) Carboxylic acid

Predict the major product formed when an alkene reacts with a hydrogen halide (e.g., HCl or HBr).

• (A) Alkane
• (B) Alkyl halide
• (C) Alcohol
• (D) Ether