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

Wavefunctions and energies for a particle confined in a cubic box are ${\psi }_{n_x,n_y,n_z}$ and $E_{n_x,n_y,n_z}$, respectively. The functions ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3$, and ${\Phi }_4$ are written as linear combinations of ${\psi }_{n_x,n_y,n_z}$. Among these functions, the eigenfunction(s) of the Hamiltonian operator for this particle is/are
${\Phi }_1=\frac{1}{\sqrt{2}}{\psi }_{1,4,1}-\frac{1}{\sqrt{2}}{\psi }_{2,2,3}$
${\Phi }_2=\frac{1}{\sqrt{2}}{\psi }_{1,5,1}+\frac{1}{\sqrt{2}}{\psi }_{3,3,3}$
${\Phi }_3=\frac{1}{\sqrt{2}}{\psi }_{1,3,8}+\frac{1}{\sqrt{2}}{\psi }_{3,8,1}$
${\Phi }_4=\frac{1}{2}{\psi }_{3,3,1}+\frac{\sqrt{3}}{2}{\psi }_{2,4,1}$

If a particle's state function is an eigenfunction of the operator ${\hat{L}}^2$ with eigenvalue $30{\hslash }^2$, then the possible eigenvalue(s) of the operator ${\hat{L}}_z^2$ for the same state function is/are

An archaeological specimen containing ${}^{14}C$ gives 45 counts per gram of carbon in 5 minutes. A specimen of freshly cut wood gives 20 counts per gram of carbon per minute. The counter used recorded a background count of 5 counts per minute in the absence of any ${}^{14}C$ containing sample. The age of the specimen is ________ years (in integer).
[Note: $t_{1/2}$ of ${}^{14}C=5730$ years]

In the following reaction, 13.4 grams of aldehyde P gave a diastereomeric mixture of alcohols Q and R in a ratio of 2:1. If the yield of the reaction is 80%, then the amount of Q (in grams) obtained is ________ (in integer).

The kinetic energies of an electron ($e$) and a proton ($p$) are $E$ and $3E$, respectively. Given that mass of a proton is 1836 times that of an electron, the ratio of their de Broglie wavelengths (${\lambda }_e/{\lambda }_p$) is ________ (rounded off to two decimal places).