Evaluate the integral: \[ \int \frac{2x}{x^2 - 5x + 4}\, dx \]

• (A) \( \frac{8}{3}\ln|x-4| - \frac{2}{3}\ln|x-1| + C \)
• (B) \( \frac{2}{3}\ln|x-1| - \frac{8}{3}\ln|x-4| + C \)
• (C) \( 2\ln|x-1| + \ln|x-4| + C \)
• (D) \( \ln|x^2-5x+4| + C \)

For the curve \( y = 3x^3 - 3x^2 + 1 \) at \( x = 1 \), find the equation of the tangent.

• (A) \( y = 3x - 2 \)
• (B) \( y = 3x - 3 \)
• (C) \( y = 3x - 1 \)
• (D) \( y = x + 1 \)

Find the general solution for \(x\) if \( \cos 4x = \cos 3x \).

• (A) \(x = 2n\pi \)
• (B) \(x = \frac{2n\pi}{7} \)
• (C) \(x = \frac{2n\pi}{7}, \; 2n\pi \)
• (D) \(x = \frac{n\pi}{7} \)

Evaluate the integral: \[ \int \frac{x}{x+2}\, dx \]

• (A) \(x - 2\ln|x+2| + C \)
• (B) \(x + 2\ln|x+2| + C \)
• (C) \(x - \ln|x+2| + C \)
• (D) \( \ln|x+2| + C \)

If \(A, B, C\) are vertices of a triangle with position vectors \(\vec a, \vec b, \vec c\), find the position vector of the point \(D\) where the angle bisector from vertex \(A\) meets \(BC\).

• (A) \( \dfrac{\vec b + \vec c}{2} \)
• (B) \( \dfrac{|\vec{AC}|\,\vec b + |\vec{AB}|\,\vec c}{|\vec{AB}| + |\vec{AC}|} \)
• (C) \( \dfrac{|\vec{AB}|\,\vec b + |\vec{AC}|\,\vec c}{|\vec{AB}| + |\vec{AC}|} \)
• (D) \( \dfrac{\vec a + \vec b + \vec c}{3} \)

A particle starts oscillating simple harmonically from its mean position with time period \(T\); find the ratio of potential energy to kinetic energy at time \(t = \frac{T}{6}\).

• (A) \( \frac{1}{3} \)
• (B) \( 3 \)
• (C) \( \frac{1}{2} \)
• (D) \( 1 \)

Calculate the percentage increase in apparent frequency for an observer moving towards a stationary sound source with \( \frac{1}{5} \)th the velocity of sound.

• (A) \(10%\)
• (B) \(20%\)
• (C) \(25%\)
• (D) \(30%\)

Find the ratio of fundamental frequencies \( f_1/f_2 \) for a pipe open at both ends when \( \frac{1}{3} \) of its length is later submerged in water.

• (A) \( \frac{3}{2} \)
• (B) \( 2 \)
• (C) \( \frac{4}{3} \)
• (D) \( \frac{3}{4} \)

A car starts from rest and accelerates uniformly at \(3\ m/s^2\); determine its velocity after \(5\) seconds.

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

Find the acceleration of a \(2\) kg block acted upon by a net force of \(10\) N.

• (A) \(2\ m/s^2\)
• (B) \(5\ m/s^2\)
• (C) \(10\ m/s^2\)
• (D) \(20\ m/s^2\)

Identify the product formed when \( CH_3{-}CH_2{-}Br \) reacts with alcoholic \(KOH\).

• (A) \(CH_3CH_2OH\)
• (B) \(CH_2=CH_2\)
• (C) \(CH_3CHO\)
• (D) \(CH_3COOH\)

What is the oxidation state of oxygen in peroxides?

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

Identify the specific reagent used in the Swarts reaction (e.g., \(AgF\) or \(SbF_3\)).

• (A) \(AgCl\)
• (B) \(SbF_3\)
• (C) \(FeCl_3\)
• (D) \(HCl\)

Which of the following is a disproportionation reaction: Clemmensen, Wolff–Kishner, or Cannizzaro?

• (A) Clemmensen reaction
• (B) Wolff–Kishner reaction
• (C) Cannizzaro reaction
• (D) None of these

Predict the product when an alcohol reacts with sodium metal.

• (A) Alkane
• (B) Sodium alkoxide + Hydrogen gas
• (C) Ether
• (D) Aldehyde