A wire of length \(L\) and cross-sectional area \(A\) is made of a material of Young's modulus \(Y\). If it is stretched by an amount \(x\), the elastic potential energy stored in the wire is:

• (A) \( \frac{YAx^2}{L} \)
• (B) \( \frac{YAx^2}{2L} \)
• (C) \( \frac{2YAx^2}{L} \)
• (D) \( \frac{YAx}{L} \)

A particle moves in a circle of radius \(R\) such that its linear speed varies with time \(t\) as \(v = kt\), where \(k\) is a positive constant. The angle \(\theta\) between the net acceleration vector and the velocity vector at time \(t\) is given by:

• (A) \( \tan^{-1}\left(\frac{k^2 t^2}{R}\right) \)
• (B) \( \tan^{-1}\left(\frac{kt^2}{R}\right) \)
• (C) \( \tan^{-1}\left(\frac{k t}{R}\right) \)
• (D) \( \tan^{-1}\left(\frac{R}{k^2 t^2}\right) \)

Two wires \(X\) and \(Y\) of the same material have lengths in the ratio \(1:2\) and diameters in the ratio \(2:1\). If they are subjected to the same stretching force, the ratio of the elongation produced in wire \(X\) to that in wire \(Y\) (\(\Delta L_X : \Delta L_Y\)) is:

• (A) \( 1 : 4 \)
• (B) \( 1 : 8 \)
• (C) \( 1 : 2 \)
• (D) \( 8 : 1 \)

The elastic potential energy stored per unit volume (energy density) in a stretched string under a longitudinal tension stress \(\sigma\) and material Young's modulus \(Y\) is expressed as:

• (A) \( \frac{\sigma^2}{2Y} \)
• (B) \( \frac{2Y}{\sigma^2} \)
• (C) \( \frac{Y\sigma^2}{2} \)
• (D) \( \frac{\sigma^2}{Y} \)

An octahedral coordination complex with the electronic configuration \(t_{2g}^4 e_g^0\) is expected to exhibit which of the following magnetic properties and d-d transition characteristics?

• (A) Paramagnetic with 4 unpaired electrons; spin-allowed transitions
• (B) Paramagnetic with 2 unpaired electrons; spin-allowed transitions
• (C) Diamagnetic; spin-forbidden transitions
• (D) Paramagnetic with 2 unpaired electrons; spin-forbidden transitions

During the structural analysis of an unknown aldohexose, a chemist treats a sample with periodic acid (\(HIO_4\)). If the carbohydrate is completely cleaved to yield five molecules of formic acid (\(HCOOH\)) and one molecule of formaldehyde (\(HCHO\)), this diagnostic breakdown directly proves the presence of:

• (A) A ketohexose structure with a carbonyl at C-2
• (B) A cyclic pyranose ring configuration
• (C) A continuous straight-chain structure containing five \(-CHOH\) groups and one \(-CH_2OH\) group
• (D) Three isolated, non-adjacent primary alcohol branches

In an analytical laboratory, a \(20.0 mL\) sample of an aqueous solution containing oxalic acid (\(H_2C_2O_4\)) requires exactly \(16.0 mL\) of a \(0.05 M\) potassium permanganate (\(KMnO_4\)) solution for complete oxidation in a hot, acidic medium (\(H_2SO_4\)). Calculate the molarity of the oxalic acid solution.

• (A) \( 0.010 M \)
• (B) \( 0.040 M \)
• (C) \( 0.100 M \)
• (D) \( 0.250 M \)

A current of \(2.0 A\) is passed for 5 hours through an electrolytic cell containing an aqueous solution of a metal salt, depositing \(12.0 g\) of the metal at the cathode. If the atomic mass of the metal is \(193 g mol^{-1}\), find the oxidation state of the metal ion in the solution. (Take Faraday's constant \(F = 96500 C mol^{-1}\)).

• (A) \( +1 \)
• (B) \( +2 \)
• (C) \( +3 \)
• (D) \( +6 \)

In how many ways can the letters of the word COCHIN be arranged such that the two 'C's are never separated by any other letter?

• (A) \( 360 \)
• (B) \( 120 \)
• (C) \( 240 \)
• (D) \( 720 \)

Evaluate the definite integral: \(\int_{0}^{2026} \frac{x^5}{x^5 + (2026 - x)^5} \, dx\)

• (A) \( 2026 \)
• (B) \( 1013 \)
• (C) \( 506.5 \)
• (D) \( 0 \)

If the vectors \(\vec{a} = 2\hat{i} - \hat{j} + \hat{k}\), \(\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}\), and \(\vec{c} = 3\hat{i} + \lambda\hat{j} + 5\hat{k}\) represent the concurrent coterminous edges of a parallelopiped whose volume is \(0\) (i.e., the vectors are coplanar), find the value of the scalar parameter \(\lambda\).

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

A pair of fair dice is thrown simultaneously. What is the probability that the sum of the numbers appearing on the top faces is at least 10?

• (A) \( \frac{1}{6} \)
• (B) \( \frac{1}{12} \)
• (C) \( \frac{5}{36} \)
• (D) \( \frac{1}{4} \)