Evaluate the integral: \( \displaystyle \int \frac{x}{x+2}\,dx \)

• (A) \( x - 2\log|x+2| + C \)
• (B) \( x + 2\log|x+2| + C \)
• (C) \( \log|x+2| + C \)
• (D) \( \frac{x^2}{2(x+2)} + C \)

Find the value of \( k \) if the function \( f(x) = k\sin x + 2\cos x \) is increasing for all \( x \)

• (A) Possible for some \( k \)
• (B) \( k > 2 \)
• (C) Impossible for all \( x \in \mathbb{R} \)
• (D) \( k = 2 \)

Calculate \( y(\log 2) \) if \( \dfrac{dy}{dx} = y + 5 \) and \( y(0) = 4 \).

• (A) \(12\)
• (B) \(13\)
• (C) \(14\)
• (D) \(15\)

Identify the co-ordinates of the point where the line joining \( (1,1,1) \) and \( (2,2,2) \) intersects the plane \( x+y+z=9 \).

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

A ball is thrown upwards with a velocity of \(20\ m/s\); find the maximum height reached \((g = 10\ m/s^2)\).

• (A) \(10\) metres
• (B) \(20\) metres
• (C) \(30\) metres
• (D) \(40\) metres

Calculate the ratio of potential energy to kinetic energy at time \(t=\dfrac{T}{6}\) for a particle starting SHM from its mean position.

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

Find the work done by a gas that expands from \(0.2\,dm^3\) to \(0.8\,dm^3\) against a constant pressure of \(2\) bar.

• (A) \(-120\) J
• (B) \(120\) J
• (C) \(-60\) J
• (D) \(60\) J

Calculate the radius of a circular path for an electron moving through a magnetic field \(B = 12 \times 10^{-4}\,T\) with \(v = 3.2 \times 10^{7}\,m/s\).

• (A) \(0.05\) m
• (B) \(0.10\) m
• (C) \(0.15\) m
• (D) \(0.20\) m

Identify the reagent used in the Lucas test to distinguish between primary, secondary, and tertiary alcohols.

• (A) Dilute \(HCl\) and \(ZnCl_2\)
• (B) Conc. \(HCl\) and anhydrous \(ZnCl_2\)
• (C) \(H_2SO_4\) and \(ZnCl_2\)
• (D) Conc. \(HNO_3\) and \(ZnCl_2\)

Calculate the number of carbon atoms present in \(0.35\) mole of glucose \((C_6H_{12}O_6)\).

• (A) \(1.26 \times 10^{23}\)
• (B) \(1.26 \times 10^{24}\)
• (C) \(2.10 \times 10^{24}\)
• (D) \(3.50 \times 10^{23}\)

Determine the hybridization and geometry of carbon in ethyne \((C_2H_2)\).

• (A) \(sp^3\), Tetrahedral
• (B) \(sp^2\), Trigonal planar
• (C) \(sp\), Linear
• (D) \(sp^3d\), Trigonal bipyramidal

Find the volume occupied by \(8\) g of a gas at STP if its vapour density is \(16\).

• (A) \(2.8\) L
• (B) \(5.6\) L
• (C) \(11.2\) L
• (D) \(22.4\) L

Identify the most stable carbocation among primary, secondary, and tertiary.

• (A) Primary carbocation
• (B) Secondary carbocation
• (C) Tertiary carbocation
• (D) All are equally stable