GATE 2025 CY Question Paper (Available)- Download Solution PDF with Answer Key

Courage : Bravery :: Yearning : ___________
Select the most appropriate option to complete the analogy.

• (A) \( Longing \)
• (B) \( Yelling \)
• (C) \( Yawning \)
• (D) \( Glaring \)

We _____________ tennis in the lawn when it suddenly started to rain.
Select the most appropriate option to complete the above sentence.

• (A) \( have been playing \)
• (B) \( had been playing \)
• (C) \( would have been playing \)
• (D) \( could be playing \)

A \( 4 \times 4 \) digital image has pixel intensities (\( U \)) as shown in the figure. The number of pixels with \( U \leq 4 \) is:

0  1   0   2
4  7   3   3
5  5   4   4
6  7   3   2

• (A) \( 3 \)
• (B) \( 8 \)
• (C) \( 11 \)
• (D) \( 9 \)

In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of P, Q, and R?

• (A) \( P = 6; Q = 5; R = 3 \)
• (B) \( P = 5; Q = 6; R = 3 \)
• (C) \( P = 3; Q = 6; R = 6 \)
• (D) \( P = 5; Q = 3; R = 6 \)

A rectangle has a length \( L \) and a width \( W \), where \( L > W \). If the width, \( W \), is increased by 10%, which one of the following statements is correct for all values of \( L \) and \( W \)?

• (A) \( Perimeter increases by 10% \)
• (B) \( Length of the diagonals increases by 10% \)
• (C) \( Area increases by 10% \)
• (D) \( The rectangle becomes a square \)

Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk. Identify the option that has the correct match between Column-I and Column-II.


Column-I                                                                           Column-II
P. This house is in a mess.                                    1.  Alright, I won't bring it up during our conversations.
Q. I am not happy with the marks given to me.   2. Well, you can easily look it up.
R. Politics is a subject I avoid talking about.       3. No problem, let me clear it up for you.
S. I don't know what this word means.                 4. Don't worry, I will take it up with your teacher.

• (A) \( P – 2; Q – 3; R – 1; S – 4 \)
• (B) \( P – 3; Q – 4; R – 1; S – 2 \)
• (C) \( P – 4; Q – 1; R – 2; S – 3 \)
• (D) \( P – 1; Q – 2; R – 4; S – 3 \)

Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monotonically increases till the age of 50 years and then it monotonically decreases. Let \( a_1 \) and \( a_2 \) (in years) denote the ages of the brothers and \( a_1 < a_2 \). Which one of the following statements is correct about their age on the day when they attain the same weight?

• (A) \( a_1 < a_2 < 50 \)
• (B) \( a_1 < 50 < a_2 \)
• (C) \( 50 < a_1 < a_2 \)
• (D) Either \( a_1 = 50 \) or \( a_2 = 50 \)

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side \( r \) cm and each numbered triangle is used only once to form a square.

The number of squares that can be formed and the number of triangles required to form each square, respectively, are:

Note: The figure shown is representative.

• (A) \( 3;\ 4 \)
• (B) \( 4;\ 3 \)
• (C) \( 3;\ 3 \)
• (D) \( 3;\ 2 \)

If a real variable \( x \) satisfies \( 3^{x^2} = 27 \times 9^x \), then the value of \( \dfrac{2^{x^2}}{(2^x)^2} \) is:

• (A) \( 2^{-1} \)
• (B) \( 2^0 \)
• (C) \( 2^3 \)
• (D) \( 2^{15} \)

The number of patients per shift (\( X \)) consulting Dr. Gita in her past 100 shifts is shown in the figure. If the amount she earns is Rupees \( 1000(X - 0.2) \), what is the average amount (in Rupees) she has earned per shift in the past 100 shifts?

Note: The figure shown is representative.

• (A) Rupees 6,100
• (B) Rupees 6,300
• (C) Rupees 6,000
• (D) Rupees 6,500

The phosphazene compound that acts as a superbase is

• (A)
  [1em]
• (B)
  [1em]
• (C)
  [1em]
• (D)
  [1em]

The reaction for the synthesis of Me\textsubscript{2}SiCl\textsubscript{2} through Rochow–Müller process is:

• (A) SiCl\textsubscript{4} + Me\textsubscript{2}Zn \quad \xrightarrow{0^\circC}
• (B) Si:Fe (9:1) + 2 MeCl \quad \xrightarrow{300^\circC}
• (C) Si:Cu (9:1) + 2 MeCl \quad \xrightarrow{300^\circC}
• (D) SiCl\textsubscript{4} + 2 MeMgBr \quad \xrightarrow{0^\circC}

Upon cooling from room temperature, the magnetic susceptibility of MnO slowly increases until 118 K, and then it decreases. This phenomenon is known as:

• (A) ferromagnetism
• (B) paramagnetism
• (C) antiferromagnetism
• (D) ferrimagnetism

An aqueous solution of Co(ClO\textsubscript{4})\textsubscript{2}·6H\textsubscript{2}O is light pink in colour. Addition of conc. HCl results in an intense blue coloured solution due to the formation of a new species. The new species among the following is:





[Given: Atomic number of Co = 27]

• (A) I
• (B) II
• (C) III
• (D) IV

For an unambiguous single step synthesis of the following target molecule (TM), the best bond disconnection in its retrosynthetic analysis is:

• (A)
• (B)
• (C)
• (D)

In the \(^1\)H-NMR spectrum of the following molecule, the signal of proton H\textsubscript{a appears as:

• (A) singlet
• (B) triplet
• (C) quintet
• (D) quartet

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

• (A)
• (B)
• (C)
• (D)

The symmetry element that does NOT belong to the \( C_{4v} \) point group is:

• (A) \( C_4 \)
• (B) \( C_2 \)
• (C) \( i \)
• (D) \( \sigma_v \)

Rigid rotor wavefunctions are given by \( Y_{l,m}(\theta, \phi) \). The wavefunctions \( Y_{1,0}(\theta, \phi) \) and \( Y_{2,0}(\theta, \phi) \) are given below:
\[ Y_{1,0}(\theta, \phi) = \sqrt{\frac{3}{4\pi}} \cos \theta \qquad Y_{2,0}(\theta, \phi) = \sqrt{\frac{5}{16\pi}} (3\cos^2 \theta - 1) \]

For a non-polar diatomic molecule, the value of transition dipole moment integral for the transition between \( Y_{1,0}(\theta, \phi) \) and \( Y_{2,0}(\theta, \phi) \) is equal to:

• (A) \( \frac{1}{\sqrt{2\pi}} \)
• (B) \( 0 \)
• (C) \( 2 \)
• (D) \( \frac{1}{\sqrt{4\pi}} \)

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:

• (A) \( q_{trans}(T, V), \quad q_{vib}(T, V), \quad q_{rot}(T, V) \)
• (B) \( q_{trans}(T, V), \quad q_{vib}(T), \quad q_{rot}(T) \)
• (C) \( q_{trans}(T), \quad q_{vib}(T, V), \quad q_{rot}(T) \)
• (D) \( q_{trans}(T, V), \quad q_{vib}(T), \quad q_{rot}(T, V) \)

The Vaska’s complex trans-IrCl(CO)(PPh\textsubscript{3})\textsubscript{2} shows a band at 1967 cm\textsuperscript{-1} for the \( \nu_{CO} \) stretching vibration in its infrared spectrum. The complex(es) that will show an increase in the \( \nu_{CO} \) stretching vibration from 1967 cm\textsuperscript{-1 is/are:

• (A)
• (B)
• (C)
• (D)

Under the conditions mentioned for each reaction, the reaction(s) that would give borazine (B\textsubscript{3}N\textsubscript{3}H\textsubscript{6}) as the major product is/are:

• (A) LiBH\textsubscript{4} + NH\textsubscript{4}Cl \quad \xrightarrow{230\,^\circC}
• (B) B\textsubscript{2}H\textsubscript{6} + 2 NH\textsubscript{3} \quad \xrightarrow{180\,^\circC}
• (C) NaBH\textsubscript{4} + (NH\textsubscript{4})\textsubscript{2}SO\textsubscript{4} \quad \xrightarrow{THF,\,40\,^\circC}
• (D) BCl\textsubscript{3} + NH\textsubscript{4}Cl \quad \xrightarrow{chlorobenzene,\,135\,^\circC}

The essential symmetry(ies) for a monoclinic crystal system is/are the presence of:

• (A) one \( C_3 \) axis
• (B) one \( C_2 \) axis
• (C) one \( C_4 \) axis
• (D) one \( C_6 \) axis

Compound(s) that show(s) an intense peak at \( m/z = 120 \) in the EI mass spectrum is/are:

• (A)
• (B)
• (C)
• (D)

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

• (A) (i) NBS, (PhCOO)\textsubscript{2};\quad (ii) aq.\ NaOH;\quad (iii) active MnO\textsubscript{2};\quad (iv) Li/liq.\ NH\textsubscript{3}, \emph{t}-BuOH
• (B) (i) m-CPBA;\quad (ii) BF_3\cdotEt_2O
• (C) (i) SeO\textsubscript{2};\quad (ii) Dess-Martin periodinane;\quad (iii) K[BH(\emph{s}-Bu)\textsubscript{3}] (K-selectride)
• (D) (i) dil.\ KMnO\textsubscript{4};\quad (ii) NaIO\textsubscript{4}

Among the given options, the possible product(s) that can be obtained from the following reaction is/are:

• (A)
• (B)
• (C)
• (D)

Choose the correct option(s) with regard to mechanism of the following transformation.

• (A) It proceeds through divinyl cyclopropane rearrangement
• (B) It involves a diradical intermediate
• (C) It proceeds through di-\(\pi\)-methane rearrangement
• (D) It proceeds through [2+2+2] cycloaddition reaction

Consider two non-interacting particles confined to a one-dimensional box with infinite potential barriers. Their wavefunctions are \( \psi_1 \) and \( \psi_2 \), and energies are \( E_1 \) and \( E_2 \), respectively. The INCORRECT statement(s) about this system is/are:

• (A) The total energy is \( E_1 + E_2 \)
• (B) The total wavefunction is \( \psi_1 + \psi_2 \)
• (C) The total energy is \( E_1 E_2 \)
• (D) The total wavefunction is \( \psi_1 \psi_2 \)

The thermodynamic criterion/criteria for a spontaneous process is/are

• (A) \( \Delta U > 0 \) at constant \( S \) and \( V \)
• (B) \( \Delta S > 0 \) at constant \( U \) and \( V \)
• (C) \( \Delta(H - TS) > 0 \) at constant \( T \) and \( P \)
• (D) \( \Delta(U - TS) < 0 \) at constant \( T \) and \( V \)

Xe and F\textsubscript{2} in 1:1 molar ratio when mixed in a closed flask and kept in sunlight for a day, gave white crystals of a compound Q. Two equivalents of Q on reaction with one equivalent of AsF\textsubscript{5} gave an ionic compound X\textsuperscript{+}Y\textsuperscript{–} with the cation having two Xe atoms. The total number of lone pairs present on the cation X\textsuperscript{+} is \hspace{2cm} (in integer).

The total number of hyperfine lines expected in the EPR spectrum of \(\cdot\)CH\textsubscript{2OH (radical) is \underline{\hspace{2cm (in integer).

Partial hydrolysis of a pentapeptide yields all possible tripeptides and dipeptides. The dipeptides that are obtained upon hydrolysis are given below:


Val-Ala
Gln-His
Phe-Val
Ala-Gln


The total number of tripeptides obtained that contain ‘Ala’ as one of the amino acids is \hspace{2cm} (in integer).

The specific rotation of enantiomerically pure (S)-2-butanol is \( +14^\circ \). The specific rotation of the enantiomeric mixture of 2-butanol obtained from an asymmetric reduction of 2-butanone is found to be \( +7^\circ \).

The percentage of (R)-2-butanol present in the reaction mixture is \hspace{2cm} (in integer).

The ratio of the fundamental vibrational frequencies \( \left( \nu_{^{13}C^{16}O} / \nu_{^{12}C^{16}O} \right) \) of two diatomic molecules \( ^{13}C^{16}O \) and \( ^{12}C^{16}O \), considering their force constants to be the same, is \underline{\hspace{2cm (rounded off to two decimal places).

The expressions for the vapour pressure of solid (\( p_1 \)) and vapour pressure of liquid (\( p_2 \)) phases of a pure substance, respectively, are
\[ \ln p_1 = \frac{-2000}{T} + 5 \quad and \quad \ln p_2 = \frac{-4000}{T} + 10 \]

The triple point temperature of this substance is \hspace{2cm} K (in integer).

The reaction that proceeds through an oxidative addition followed by a reductive elimination is

\textit{[Given: Atomic numbers Ni = 28, Ta = 73, Zr = 40, Pt = 78]

• (A)
• (B)
• (C)
• (D)

The homogeneous catalyst whose metal ion does NOT undergo either oxidation or reduction in any of the steps during the hydrogenation of terminal olefins is:

• (A) RhCl(PPh\(_3\))\(_3\)
• (B) HRuCl(PPh\(_3\))\(_3\)
• (C) [Ir(COD)(PCy\(_3\))(Py)]\(^+\) PF\(_6^-\) \quad (COD = cyclooctadiene)
• (D) [Rh(COD)(PPh\(_3\))\(_2\)]\(^+\) PF\(_6^-\) \quad (COD = cyclooctadiene)

The given zirconocene compound, (\(\eta^5\)-Cp)\(_2\)ZrEt\(_2\), when heated in the presence of an equimolar amount of PMe\(_3\) results in the formation of a compound X which obeys the 18-electron rule. The reaction also resulted in the release of a saturated hydrocarbon.





[Given: Atomic number of Zr = 40]

\medskip

The structure of compound \textbf{X is:

• (A)
• (B)
• (C)
• (D)

The \(^1\)H NMR spectrum of the given iridium complex at room temperature gave a single signal at 2.6 ppm, and its \(^{31}\)P 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





\medskip

• (A) Berry pseudo-rotation and rotation of the ethylene units along the C=C axis
• (B) Berry pseudo-rotation and propeller type rotation of the ethylene units along the Ir-alkene axis
• (C) Ray-Dutt twist and rotation of the ethylene units along the C=C axis
• (D) Ray-Dutt twist and propeller type rotation of the ethylene units along the Ir-alkene axis

The effective magnetic moment, \(\mu_{eff}\) value for [Cr(H\(_2\)O)\(_6\)]\(^{3+}\) taking into account spin-orbit coupling is closest to:

\medskip

[Given: Atomic number of Cr = 24, spin-orbit coupling constant \(\lambda = 92\) cm\(^{-1\), and \(\Delta_o = 17400\) cm\(^{-1}\)]

• (A) 3.79 \(\mu_B\)
• (B) 3.87 \(\mu_B\)
• (C) 4.05 \(\mu_B\)
• (D) 3.60 \(\mu_B\)

The major products \textbf{X} and \textbf{Y} formed in the following reaction sequences are:

\medskip

• (A)
• (B)
• (C)
• (D)

Compound \textbf{K} displayed a strong band at 1680 cm\(^{-1}\) in its IR spectrum. Its \(^1\)H-NMR spectral data are as follows:
\[ \delta~(ppm):
7.30~(d,~J = 7.2~Hz,~2H),
6.8~(d,~J = 7.2~Hz,~2H),
3.8~(septet,~J = 7.0~Hz,~1H),
2.2~(s,~3H),
1.9~(d,~J = 7.0~Hz,~6H)
\]

The correct structure of compound K is:

• (A)
• (B)
• (C)
• (D)

The major product formed in the following reaction sequences is:

• (A)
• (B)
• (C)
• (D)

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

• (A) \(Si\) face of enolate on the \(Re\) face of aldehyde
• (B) \(Si\) face of enolate on the \(Si\) face of aldehyde
• (C) \(Re\) face of enolate on the \(Re\) face of aldehyde
• (D) \(Re\) face of enolate on the \(Si\) face of aldehyde

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.

• (A) (a) True (b) False (c) False (d) True
• (B) (a) True (b) True (c) False (d) False
• (C) (a) False (b) True (c) True (d) False
• (D) (a) False (b) True (c) True (d) True

The phase diagram of a single component system is given below.





The option with the correct number of degrees of freedom corresponding to the labelled points i, j, and k, respectively, is

• (A) 0, 1, 2
• (B) 3, 2, 1
• (C) 2, 0, 1
• (D) 0, 2, 1

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.]

• (A) \(P(V - Nb) = Nk_BT\)
• (B) \(PV(N - b) = k_BT\)
• (C) \(PV = Nk_BT\)
• (D) \(P(V - Nb) = Nk_B\)

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

• (A) I\( _2 \)Cl\( _6 \)
• (B) Si\( _2 \)Cl\( _6 \)
• (C) Al\( _2 \)Cl\( _6 \)
• (D) Cl\( _2 \)O\( _6 \)

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



 Absorbance maximum                Electronic transition
(a)  11200 cm\(^{-1}\)                    (i) \(^{3}A_{2g} \rightarrow {}^{3}T_{1g} (F)\)
(b) & 18350 cm\(^{-1}\)                 (ii) \(^{3}A_{2g} \rightarrow {}^{3}T_{2g}\)
(c) & 29000 cm\(^{-1}\)                 (iii) \(^{3}A_{2g} \rightarrow {}^{3}T_{1g} (P)\)
[Given: Atomic number of Ni = 28]

The correct match(es) between absorbance maximum and electronic transition is/are

• (A) (a) \(\rightarrow\) (ii)
• (B) (b) \(\rightarrow\) (i)
• (C) (a) \(\rightarrow\) (iii)
• (D) (c) \(\rightarrow\) (iii)

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

• (A) A thiolate group is coordinated to the Fe center at one of the axial positions around Fe.
• (B) While one of the oxygen atoms of O\( _2 \) is inserted into a C-H bond of a hydrocarbon, the other oxygen atom gets reduced to water.
• (C) An imidazole group is coordinated to the Fe center at one of the axial positions around Fe.
• (D) An iron-oxo species acts as a key oxidant in the catalytic cycle of CYP.

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.]

• (A) [Mo\( _2 \)(\(\mu\)-SO\( _4 \)\)\(_4\)(H\( _2 \)O)\(_2\)]\(^{3-}\)
• (B) [Mn\( _2 \)(CO)\(_{10}\)]
• (C) [Cr\( _2 \)(\(\mu\)-O\( _2 \)CCH\( _3 \))\(_4\)]
• (D) [Mo\( _2 \)(\(\mu\)-HPO\( _4 \))\(_4\)(H\( _2 \)O)\(_2\)]\(^{2-}\)

Consider the following two reactions and their corresponding Hammett plots





Choose the option(s) that correctly match(es) the points on the graph given in Column-I with substituents X given in Column-II in accordance with their substituent constants \( \sigma_X \) given in Column-II.

Column-I (points on the graph)         Column-II (substituent X)
p                                                            NH\( _2 \)
q                                                            NO\( _2 \)
r                                                            OMe
s                                                           Cl
t                                                            Me
u                                                           CN

• (A) s \(\rightarrow\) \( \sigma_X \)(X=Cl); t \(\rightarrow\) \( \sigma_X \)(X=OMe); u \(\rightarrow\) \( \sigma_X \)(X=NH\( _2 \)); r \(\rightarrow\) \( \sigma_X \)(X=NO\( _2 \))
• (B) s \(\rightarrow\) \( \sigma_X \)(X=Me); u \(\rightarrow\) \( \sigma_X \)(X=NH\( _2 \)); t \(\rightarrow\) \( \sigma_X \)(X=OMe); r \(\rightarrow\) \( \sigma_X \)(X=Br)
• (C) p \(\rightarrow\) \( \sigma_X \)(X=Me); q \(\rightarrow\) \( \sigma_X \)(X=CN); r \(\rightarrow\) \( \sigma_X \)(X=NO\( _2 \)); t \(\rightarrow\) \( \sigma_X \)(X=OMe)
• (D) p \(\rightarrow\) \( \sigma_X \)(X=Cl); q \(\rightarrow\) \( \sigma_X \)(X=NO\( _2 \)); r \(\rightarrow\) \( \sigma_X \)(X=CN); t \(\rightarrow\) \( \sigma_X \)(X=Me)

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

• (A) (i) Li-C\( \equiv \)C-H, THF, -70 \(^\circ \)C; (ii) cat. HgSO\( _4 \), H\( _2 \)SO\( _4 \), H\( _2 \)O; (iii) aqueous acid, \( \Delta \)
• (B) (i) , NaH; (ii) aqueous acid, \( \Delta \)
• (C) (i) LDA, TfNPh\( _2 \); (ii) cat. [(dppe)Pd(0)], OBu; (iii) aqueous acid, \( \Delta \) (dppe = diphenylphosphinoethane)
• (D) (i) H\( _3 \)C-NO\( _2 \), NaOCH\( _3 \); (ii) sat. NaCl; (iii) TiCl\( _3 \), H\( _2 \)O; (iv) aqueous acid, \( \Delta \)

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

• (A) 1,2-methide shift
• (B) 1,3-methide shift
• (C) non-classical carbocation
• (D) tertiary carbocation

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

• (A)
• (B)
• (C)
• (D)

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

• (A) \( \phi_2 \)
• (B) \( \phi_4 \)
• (C) \( \phi_3 \)
• (D) \( \phi_1 \)

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

• (A) \( 10 \hbar^2 \)
• (B) \( 16 \hbar^2 \)
• (C) \( 25 \hbar^2 \)
• (D) \( 0 \)

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 \hspace{2cm 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 percent, then the amount of Q (in grams) obtained is \underline{\hspace{2cm (in integer).

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 \underline{\hspace{2cm (rounded off to two decimal places).

If a molecule emitting a radiation of frequency \(3.100 \times 10^9\) Hz approaches an observer with a relative speed of \(5.000 \times 10^6\) m s\(^{-1}\), then the observer detects a frequency of \hspace{2cm \( \times 10^9 \) Hz. (rounded off to three decimal places)
[Given: Speed of light \(c = 3.000 \times 10^8\) m s\(^{-1\)]

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 \hspace{2cm \( \times 10^{-21 \) J (rounded off to two decimal places).
[Given: Boltzmann constant \( k_B = 1.38 \times 10^{-23} \) J K\(^{-1}\)]

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 \hspace{2cm (rounded off to two decimal places).
[Given: Faraday constant \( F = 96500 \) 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 \underline{\hspace{2cm (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 \(0.36 \times 10^2\) 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]