GATE CY 2025 Question Paper (15-Feb-2025) (Shift-1)

In the ${}^{1}H$ -NMR spectrum of the following molecule, the signal of proton $H_a$ appears as

A disaccharide X does NOT show mutarotation in aqueous solution. Acidic hydrolysis of X affords two different monosaccharides. The disaccharide X is

The symmetry element that does NOT belong to $C_{4v}$ point group is

Rigid rotor wavefunctions are given by $Y_{l,m}(\theta ,\varphi )$. The wavefunctions $Y_{1,0}(\theta ,\Phi )$ and $Y_{2,0}(\theta ,\varphi )$ are given below

$$Y_{1,0}(\theta ,\varphi )=\sqrt{\frac{3}{4\pi }}\cos \theta$$

$$Y_{2,0}(\theta ,\varphi )=\sqrt{\frac{5}{16\pi }}(3{\cos }^2\theta -1)$$
For a non-polar diatomic molecule, the value of transition dipole moment integral for transition between $Y_{1,0}(\theta ,\Phi )$ and $Y_{2,0}(\theta ,\varphi )$ is equal to

The translational, vibrational, and rotational molecular partition functions for a system containing ideal diatomic gas molecules in the canonical ensemble (N, V, T) are written as, $q_{trans}$, $q_{vib}$, and $q_{rot}$, respectively. The option that correctly defines their thermodynamic variable(s) dependency is