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

If a molecule emitting a radiation of frequency $3.100\times 10^9\text{ Hz}$ approaches an observer with a relative speed of $5.000\times 10^6{\text{ m s}}^{-1}$, then the observer detects a frequency of ________ $\times 10^9\text{ Hz}$. (rounded off to three decimal places)
[Given: Speed of light $c=3.000\times 10^8{\text{ m s}}^{-1}$ ]

The mean energy of a molecule having two available energy states at $ϵ=0\text{ J}$ and $ϵ=4.14\times 10^{-21}\text{ J}$ at 300 K is ________ $\times 10^{-21}\text{ J}$ (rounded off to two decimal places).
[Given: Boltzmann constant ($k_B$) = $1.38\times 10^{-23}{\text{ J K}}^{-1}$ ]

For the cell reaction,
$H{g}_{2}C{l}_2(s)+H_2(1\text{ atm})\to 2Hg(l)+2H^{+}(a=1)+2C{l}^{-}(a=1)$
The standard cell potential is ${\mathcal{E}}^{\circ }=0.2676\text{ V}$, and $(\frac{\partial {\mathcal{E}}^{\circ }}{\partial T}{)}_P=-3.19\times 10^{-4}{\text{ V K}}^{-1}$.
The standard enthalpy change of the reaction (${\Delta }_r{H}^{\circ }$) at 298 K is $-x{\text{ kJ mol}}^{-1}$.
The value of $x$ is ________ (rounded off to two decimal places).
[Given: Faraday constant $F=96500{\text{ C mol}}^{-1}$ ]

Consider a Carnot engine with a hot source kept at 500 K. From the hot source, 100 J of energy (heat) is withdrawn at 500 K. The cold sink is kept at 300 K. The efficiency of the Carnot engine is ________ (rounded off to one decimal place).

The Lineweaver-Burk plot for an enzyme obeying the Michaelis-Menten mechanism is given below.


The slope of the line is $0.36\times 10^{-2}\text{ s}$, and the y-intercept is $1.20{\text{ mol}}^{-1}\text{ L s}$. The value of the Michaelis constant ($K_M$) is ________ $\times 10^{-3}{\text{ mol L}}^{-1}$ (in integer).
[Note: $v$ is the initial rate, and $[S{]}_0$ is the substrate concentration]