MHT CET 2026 May 18 Shift 2 PCM Question Paper- Download PDF with Solutions

Concept: A complex number \(z\) is typically represented in the form \(z = x + iy\), where \(x\) and \(y\) are real numbers representing the real and imaginary parts respectively, and \(i = \sqrt{-1}\). The modulus (or absolute value) of a complex number, denoted by \(|z|\), represents its distance from the origin in the complex plane and is calculated as: \[ |z| = \sqrt{x^2 + y^2} \]

When solving equations involving complex numbers, a fundamental principle is that two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.



Step 1: Substituting the standard form of a complex number.

Let the complex number be \[ z = x + iy \]
Then, \[ |z| = \sqrt{x^2 + y^2} \]
Substituting into the given equation: \[ \sqrt{x^2 + y^2} + (x + iy) = 2 + i \]



Step 2: Equating the imaginary parts.
\[ y = 1 \]



Step 3: Equating the real parts and solving for \(x\).
\[ \sqrt{x^2 + 1} + x = 2 \] \[ \sqrt{x^2 + 1} = 2 - x \]
Squaring both sides: \[ x^2 + 1 = 4 + x^2 - 4x \] \[ 4x = 3 \] \[ x = \frac{3}{4} \]



Step 4: Calculating the value of \(|z|\).
\[ |z| = \sqrt{\left(\frac{3}{4}\right)^2 + 1} \] \[ |z| = \sqrt{\frac{9}{16} + \frac{16}{16}} \] \[ |z| = \sqrt{\frac{25}{16}} \] \[ |z| = \frac{5}{4} \] Quick Tip: For equations involving complex numbers, compare real and imaginary parts separately.

Concept: Two vectors are perpendicular if their dot product is zero: \[ \vec{a} \cdot \vec{b} = 0 \]



Step 1: Applying the perpendicularity condition.
\[ (2\hat{i}+\hat{j}+3\hat{k}) \cdot (p\hat{i}+2\hat{j}+2\hat{k}) = 0 \]



Step 2: Expanding the dot product.
\[ 2p + 2 + 6 = 0 \]



Step 3: Solving for \(p\).
\[ 2p + 8 = 0 \] \[ 2p = -8 \] \[ p = -4 \] Quick Tip: Perpendicular vectors always satisfy: \[ \vec{a} \cdot \vec{b} = 0 \]

Concept: For a 1:1 electrolyte: \[ K_{sp} = s^2 \]
where \(s\) is molar solubility.



Step 1: Writing the dissociation equation.
\[ AgBr(s) \rightleftharpoons Ag^+ + Br^- \]



Step 2: Writing the \(K_{sp}\) expression.
\[ K_{sp} = [Ag^+][Br^-] \] \[ K_{sp} = s^2 \]



Step 3: Calculating the solubility.
\[ s^2 = 4.9 \times 10^{-13} \] \[ s = \sqrt{4.9 \times 10^{-13}} \] \[ s = 7 \times 10^{-7}\ mol dm^{-3} \] Quick Tip: For salts of type \(AB\), \[ K_{sp} = s^2 \]

Concept: Henderson-Hasselbalch equation: \[ pH = pK_a + \log \left(\frac{[Salt]}{[Acid]}\right) \]



Step 1: Substituting the values.
\[ 5.5 = 4.5 + \log \left(\frac{[Salt]}{0.1}\right) \]



Step 2: Simplifying the equation.
\[ 1 = \log \left(\frac{[Salt]}{0.1}\right) \]



Step 3: Solving for concentration.
\[ 10 = \frac{[Salt]}{0.1} \] \[ [Salt] = 1.0\ M \] Quick Tip: If \(pH - pK_a = 1\), then: \[ \frac{[Salt]}{[Acid]} = 10 \]

Concept: Combined Gas Law: \[ \frac{PV}{T} = constant \]



Step 1: Writing the combined gas law.
\[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]



Step 2: Analyzing the options.

Option (1): Incorrect equation
Option (2): Incorrect ideal gas law form
Option (3): Correct combined gas law
Option (4): Incorrect ideal gas law form Quick Tip: Always use temperature in Kelvin while applying gas laws.

Concept: Intermolecular forces are the forces of attraction and repulsion between interacting particles (atoms and molecules). To determine the type of force present in a substance, we must first analyze the molecular structure and polarity of its constituent molecules.


Dipole-dipole interactions occur between molecules with permanent dipoles (polar molecules).
Dipole-induced dipole interactions occur between a polar molecule and a non-polar molecule.
London dispersion forces (or van der Waals forces) are present in all molecules but are the \textit{only intermolecular forces in non-polar molecules and noble gases.
Hydrogen bonding occurs when hydrogen is bonded to highly electronegative atoms like F, O, or N.




Step 1: Analyzing the molecular structure of dinitrogen.

Dinitrogen is represented by \(N_2\). It is a homonuclear diatomic molecule in which both atoms are identical and have the same electronegativity.



Step 2: Determining the polarity of the molecule.

Since both atoms are identical, electron sharing is equal and the molecule is non-polar.



Step 3: Identifying the intermolecular force.

As \(N_2\) is non-polar:

It does not exhibit dipole-dipole interaction.
It does not exhibit hydrogen bonding.
The only intermolecular force present is London dispersion force. Quick Tip: All homonuclear diatomic molecules such as \(H_2\), \(N_2\), \(O_2\), and \(Cl_2\) are non-polar and mainly exhibit London dispersion forces.

Concept: Quantum numbers describe the position and state of an electron in an atom.


Principal quantum number \(n\): Specifies shell.
Azimuthal quantum number \(l\): Specifies orbital shape.
Magnetic quantum number \(m_l\): Specifies orientation.
Spin quantum number \(m_s\): Specifies spin direction.


For orbital notation: \[ s \rightarrow 0, \quad p \rightarrow 1, \quad d \rightarrow 2, \quad f \rightarrow 3 \]



Step 1: Determining \(n\) and \(l\) for 4d orbital.

For a 4d orbital: \[ n = 4, \qquad l = 2 \]



Step 2: Determining allowed values of \(m_l\) and \(m_s\).

For \(l = 2\): \[ m_l = -2, -1, 0, +1, +2 \]
and \[ m_s = \pm \frac{1}{2} \]



Step 3: Checking the options.


\( (4,3,2,1/2) \): Represents 4f orbital.
\( (4,2,1,1/2) \): All values are valid.
\( (4,1,2,1/2) \): Invalid because for \(l=1\), \(m_l\) cannot be 2.


Hence, the correct answer is option (2). Quick Tip: Always remember: \[ |m_l| \leq l \] If \(m_l\) is greater than \(l\), the quantum number set is invalid.

Concept: Water gas is a fuel gas consisting mainly of carbon monoxide and hydrogen. It is formed when steam reacts with hot carbon.



Step 1: Writing the formation reaction of water gas.
\[ C + H_2O \xrightarrow{high temperature} CO + H_2 \]



Step 2: Identifying the products.

The gases produced are: \[ CO \quad and \quad H_2 \]



Step 3: Checking the options.


\(CO + H_2\): Correct composition of water gas.
\(CO_2 + H_2\): Incorrect.
\(H_2O + CH_4\): Reactants used in steam reforming.
\(H_2 + O_2\): Explosive mixture. Quick Tip: Do not confuse: Water Gas \(=\ CO + H_2\) Producer Gas \(=\ CO + N_2\)

Concept: Electrolytic conductivity (\(\kappa\)) depends on the number of ions present per unit volume of solution. Greater concentration means greater conductivity.



Step 1: Calculating molarity of each solution.

\[ M = \frac{moles}{volume in liters} \]


Solution (1):
\[ \frac{0.2}{0.250} = 0.8\,M \]

Solution (2):
\[ \frac{0.3}{0.600} = 0.5\,M \]

Solution (3):
\[ \frac{0.6}{1.000} = 0.6\,M \]

Solution (4):
\[ \frac{0.8}{2.000} = 0.4\,M \]




Step 2: Comparing concentrations.
\[ 0.8M > 0.6M > 0.5M > 0.4M \]

Hence, solution (1) has maximum concentration.



Step 3: Relating concentration to conductivity.

Higher concentration means more ions per unit volume, hence higher conductivity. Quick Tip: Conductivity increases with concentration because the number of ions available to carry electric current increases.

Concept: According to the First Law of Thermodynamics: \[ \Delta U = q - w \]
For an ideal gas, internal energy depends only on temperature.



Step 1: Understanding the meaning of isothermal process.

In an isothermal process: \[ T = constant \]
Therefore, \[ \Delta T = 0 \]



Step 2: Relating temperature to internal energy.

For an ideal gas: \[ \Delta U \propto \Delta T \]
Since \(\Delta T = 0\): \[ \Delta U = 0 \]



Step 3: Applying First Law of Thermodynamics.
\[ \Delta U = q - w \]
Substituting \(\Delta U = 0\): \[ 0 = q - w \] \[ q = w \]

This means the entire heat supplied is converted into work done by the gas. Quick Tip: For an ideal gas: \[ \Delta U = 0 \] whenever temperature remains constant. Hence, in isothermal expansion, all supplied heat is used to perform work.