List of top Questions asked in GATE CY

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

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

Consider a Carnot engine with a hot source kept at 500 K. From the hot source, 100 J of energy (heat) is withdrawn at 500 K. The cold sink is kept at 300 K. The efficiency of the Carnot engine is ___________ (rounded off to one decimal place).

For the cell reaction, \[ Hg_2Cl_2 (s) + H_2 (1 \, {atm}) \rightarrow 2Hg (l) + 2H^+ (a=1) + 2Cl^- (a=1) \] The standard cell potential is \( \mathcal{E}^0 = 0.2676 \) V, and \( \left(\frac{\partial \mathcal{E}^0}{\partial T}\right)_P = -3.19 \times 10^{-4} \) V K\(^{-1}\). The standard enthalpy change of the reaction (\( \Delta_r H^0 \)) at 298 K is \( -x \) kJ mol\(^{-1}\). The value of \( x \) is ___________ (rounded off to two decimal places). [Given: Faraday constant \( F = 96500 \) C mol\(^{-1}\)]

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

The kinetic energies of an electron (\(e\)) and a proton (\(p\)) are \(E\) and \(3E\), respectively. Given that the 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).

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 percent, then the amount of Q (in grams) obtained is ___________ (in integer). 

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]

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

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 \[ \phi_1 = \frac{1}{\sqrt{2}} \psi_{1,4,1} - \frac{1}{\sqrt{2}} \psi_{2,2,3} \] \[ \phi_2 = \frac{1}{\sqrt{2}} \psi_{1,5,1} + \frac{1}{\sqrt{2}} \psi_{3,3,3} \] \[ \phi_3 = \frac{1}{\sqrt{2}} \psi_{1,3,8} + \frac{1}{\sqrt{2}} \psi_{3,8,1} \] \[ \phi_4 = \frac{1}{2} \psi_{3,3,1} + \frac{\sqrt{3}}{2} \psi_{2,4,1} \]

For the following reaction, the possible product(s) is/are 

The process(es) and/or intermediate(s) through which the following transformation proceeds is/are 

The correct option(s) of reagents and reaction sequences suitable for carrying out the following transformation is/are 

Consider the following two reactions and their corresponding Hammett plots

The complex(es) having metal-metal bond order \( \ge 3.5 \) is/are [Given: The atomic numbers of Mo, Cr, Mn, and Re are 42, 24, 25, and 75, respectively.]

Cytochrome P450 (CYP) enzymes catalyze stereoselective C-H hydroxylation of hydrocarbons in the presence of O\( _2 \). The correct statement(s) about the structure and activity of CYP is/are

The UV-visible spectrum of [Ni(en)\(_3\)]\(^{2+}\) (en = ethylenediamine) shows absorbance maxima at 11200 cm\(^{-1}\), 18350 cm\(^{-1}\), and 29000 cm\(^{-1}\). 

[Given: Atomic number of Ni = 28] The correct match(es) between absorbance maximum and electronic transition is/are

The compound(s) having structure similar to that of B\( _2 \)H\( _6 \) is/are

An approximate partition function \(Q(N, V, T)\) of a gas is given below. \[ Q(N, V, T) = \frac{1}{N!} \left( \frac{2 \pi m k_B T}{h^2} \right)^{3N/2} (V - Nb)^N \] The equation of state(s) for this gas is/are [Note: \(b\) is a parameter independent of volume.]

The correct option with regard to the following statements is 
(a) Time-independent Schrödinger equation can be exactly solved for Be\(^{2+}\). 
(b) For a particle confined in a one-dimensional box of length \( l \) with infinite potential barriers, the trial variation function \( \phi = \left[ \left( \frac{3}{l^3} \right)^{1/2} x \right] \) is not an acceptable trial wavefunction for \( 0 \le x \le l \). 
(c) Wavefunctions for system of Fermions must be anti-symmetric with respect to exchange of any two Fermions in the system. 
(d) Born-Oppenheimer approximation can be used to separate the vibrational and rotational motion of a molecule.

In the following asymmetric transformation, the key aldol reaction involves the attack of 

The major product formed in the following reaction sequences is: 

The major products X and Y formed in the following reaction sequences are:

The ¹H NMR spectrum of the given iridium complex at room temperature gave a single signal at 2.6 ppm, and its ³¹P NMR spectrum gave a single signal at 23.0 ppm. When the spectra were recorded at lower temperatures, both these signals split into a complex pattern. The intra-molecular dynamic processes shown by this molecule are: