KEAM 2025 April 27 Question Paper (Available) :Download Solutions with Answer Key

Solution set of
\( -12x > 38 for x \in \mathbb{N} \)\[ -12x > 38 \]

• (A) \( x \leq -\frac{38}{12} \)
• (B) \( x > -\frac{38}{12} \)
• (C) \( x > \frac{38}{12} \)
• (D) \( x \leq \frac{38}{12} \)

Evaluate the following integral: \[ \int e^x \left[\frac{1}{1+x(1+x)^2}\right] dx \]

• (A) \( \frac{e^x}{1+x(1+x)^2} \)
• (B) \( \int \frac{e^x}{(1+x(1+x)^2)} dx \)
• (C) \( \frac{e^x}{(1+x(1+x))^2} \)
• (D) None of the above

Evaluate the following sum: \[ \sum_{n=1}^{2025} i^n (1+i) \]

• (A) \( 2025i \)
• (B) \( 2025(1+i) \)
• (C) \( 2025 \)
• (D) None of the above

Evaluate the following statement: \[ f(x) = \sin(|x|) - |x| is not differentiable at x = \hspace{2cm} \]

• (A) \( x = 0 \)
• (B) \( x = 1 \)
• (C) \( x = -1 \)
• (D) \( x = 2 \)

Evaluate the derivative of the function: \[ f(x) = x |x|, \quad find \quad f'(-10) \]

• (A) \( -20 \)
• (B) \( 0 \)
• (C) \( 20 \)
• (D) \( -10 \)

Given the following probabilities: \[ P(A) = 0.7, P(B) = 0.5, P(A \cup B) = 0.9, \quad find \quad P(A/B) \]

• (A) \( 0.2 \)
• (B) \( 0.4 \)
• (C) \( 0.6 \)
• (D) \( 0.9 \)

Find the mean deviation of the following data:

• (A) 2.5
• (B) 3
• (C) 4
• (D) 5

Evaluate the following integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^4 + x^3 + x \right) \cos x \, dx \]

• (A) 0
• (B) \( \frac{2\pi}{3} \)
• (C) \( \frac{\pi}{3} \)
• (D) \( 1 \)

Find the minimum of the function \[ f(x) = |x+2| \]

• (A) \( x = -2 \)
• (B) \( x = 0 \)
• (C) \( x = 2 \)
• (D) \( x = -4 \)

Find the center of the ellipse given by the equation \[ 4x^2 + 24x + 9y^2 - 18y + 9 = 0 \]

• (A) \( (h, k) = (-3, 1) \)
• (B) \( (h, k) = (3, -1) \)
• (C) \( (h, k) = (-2, 3) \)
• (D) \( (h, k) = (0, 0) \)

Water flows through a pipe with velocity \( V_1 = 3 \, m/s \) where the area of the pipe is \( A_1 \). What is the velocity \( V_2 \), where the diameter of the pipe is half of that at area \( A_1 \)?

• (A) \( V_2 = 6 \, m/s \)
• (B) \( V_2 = 1.5 \, m/s \)
• (C) \( V_2 = 3 \, m/s \)
• (D) \( V_2 = 9 \, m/s \)

When a rotating disc suddenly shrinks in radius, what happens to the angular velocity \( \omega \)?

• (A) \( \omega \) decreases
• (B) \( \omega \) increases
• (C) \( \omega \) remains constant
• (D) None of the above

The center of mass of a rod of length \( l \) is at a distance ______ from one end of the rod.

• (A) \( \frac{l}{4} \)
• (B) \( \frac{l}{2} \)
• (C) \( \frac{l}{3} \)
• (D) \( l \)

The acceleration of a particle varies with time as \( a = 6t \). What is the displacement and velocity of the particle at \( t = 4s \)?

• (A) Displacement = 48 m, Velocity = 24 m/s
• (B) Displacement = 72 m, Velocity = 24 m/s
• (C) Displacement = 72 m, Velocity = 48 m/s
• (D) Displacement = 48 m, Velocity = 72 m/s

What is the potential energy of an object of mass \( m \) placed on the surface of the Earth? (Mass of Earth = \( M \), radius of Earth = \( R \))

• (A) \( -\frac{GMm}{R} \)
• (B) \( \frac{GMm}{R^2} \)
• (C) \( -\frac{GMm}{R^2} \)
• (D) \( \frac{GMm}{R} \)

A block is tied to a string of length 98 cm and is rotated in a horizontal circle. Find the angular velocity if the centripetal acceleration \( a = g \).

• (A) \( \omega = \sqrt{g} \)
• (B) \( \omega = \frac{g}{R} \)
• (C) \( \omega = \sqrt{g/R} \)
• (D) \( \omega = \frac{g}{\sqrt{R}} \)

If \( n \) cells of emf \( E \) and internal resistance \( r \) are connected in parallel, what is the equivalent emf and internal resistance of the combination?

• (A) \( E_{eq} = E \), \( r_{eq} = \frac{r}{n} \)
• (B) \( E_{eq} = nE \), \( r_{eq} = \frac{r}{n} \)
• (C) \( E_{eq} = E \), \( r_{eq} = nr \)
• (D) \( E_{eq} = nE \), \( r_{eq} = nr \)

A guitar string is vibrating in the third harmonic. The number of nodes and antinodes present are:

• (A) 3 nodes, 2 antinodes
• (B) 4 nodes, 3 antinodes
• (C) 2 nodes, 3 antinodes
• (D) 3 nodes, 4 antinodes

A rod of length 0.5 m moves with a velocity of 3 m/s in a perpendicular magnetic field of strength 2 T. Find the emf induced across its ends.

• (A) 1.5 V
• (B) 0.5 V
• (C) 3 V
• (D) 2 V

According to Huygen’s principle, secondary wavelets move:

• (A) Forward
• (B) In all directions
• (C) In a circular motion
• (D) Backward

Coefficient of friction is the ratio of?

• (A) Force of friction to the normal force
• (B) Normal force to the force of friction
• (C) Force of friction to the weight of the object
• (D) Weight of the object to the normal force

Electric field lines around a positive charge are directed:

• (A) Outward
• (B) Inward
• (C) Radially inward and outward
• (D) None of the above

The temperature of the source and sink of a heat engine are 127°C and 27°C respectively. By how much should the temperature of the source be increased so as to double the efficiency?

• (A) 100°C
• (B) 200°C
• (C) 150°C
• (D) 50°C

Which of the following is an ambidentate ligand?

• (A) \( C_2 O_4^{2-} \)
• (B) \( CN^- \)
• (C) \( NH_3 \)
• (D) None of the above

10 g of (90 percent pure) \( CaCO_3 \), treated with excess of HCl, gives what mass of \( CO_2 \)?

• (A) 4.4 g
• (B) 5.6 g
• (C) 3.6 g
• (D) 6.4 g

Pressure of pure benzene is 0.75. When 0.2g of a non-volatile solute is added to 39g of benzene, pressure changes to 0.745. The molar mass of solute is?

• (A) 40 g/mol
• (B) 100 g/mol
• (C) 60 g/mol
• (D) 80 g/mol

In which of the following \( K_c = K_p \)?

a) \( PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g) \)
b) \( H_2(g) + I_2(g) \rightleftharpoons 2HI(g) \)

• (A) Reaction a)
• (B) Reaction b)
• (C) Both reactions
• (D) None of the above

Given the following concentrations and rates, which one corresponds to the rate law for the reaction?
\[ [A] = 0.10 \quad [D] = 0.2 \quad Rate = 0.2 \] \[ [A] = 0.2 \quad [D] = 0.2 \quad Rate = 0.4 \] \[ [A] = 0.10 \quad [D] = 0.1 \quad Rate = 0.05 \]

• (A) Rate law is \( Rate = k[A][D] \)
• (B) Rate law is \( Rate = k[A]^2[D] \)
• (C) Rate law is \( Rate = k[A][D]^2 \)
• (D) Rate law is \( Rate = k[A]^2[D]^2 \)

Friedel Craft's reaction is an example of:

• (A) Electrophilic substitution
• (B) Nucleophilic substitution
• (C) Free radical substitution
• (D) Addition reaction

Find the \( \Delta G \) during the reaction: \[ H_2O(l) \rightleftharpoons H_2O(g) \quad \Delta S = +1 \, kJ/mol \, at \, 100^\circ C \]

• (A) -10 kJ/mol
• (B) +10 kJ/mol
• (C) 0 kJ/mol
• (D) +20 kJ/mol

\( E^\circ_{Cell} = 1.1 \, V. \, Find \, E_{Cell} \, for the reaction: \, Zn + Cu^{2+} \, (0.1M) \rightleftharpoons Zn^{2+} \, (0.001M) + Cu \)

• (A) 1.05 V
• (B) 1.2 V
• (C) 1.1 V
• (D) 1.0 V

The most basic oxide \[ a) Cr_2O_3 \quad b) CrO \quad c) Mn_2O_7 \quad d) V_2O_5 \]

Length of latus rectum of ellipse \[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]

Evaluate the integral: \[ \int \frac{\log(1+x)}{1+x} \, dx \]

Evaluate the integral: \[ \int_{-3}^{3} \left\lfloor x \right\rfloor \, dx \]

Variance of 240, 260, 270, 280

Evaluate the limit: \[ \lim_{x \to 2} \frac{(x^3 - 8) \sin(x - 2)}{x^2 - 4x + 4} \]

Find the range of \( f(x) = \log_e(4x^2 - 4x + 1) \)

Evaluate the integral: \[ \int \frac{\sin(2x)}{\sin(x)} \, dx \]

If \[ (a - 2)^2 + (b - 2)^2 + (c - 2)^2 = 0, then a + b + c = \]

Equation of line through (0, 0, 1) and (1, 1, 0)

Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b}, |\mathbf{a} + \mathbf{b}| = \sqrt{29}, \mathbf{a} \cdot \mathbf{b} = ?

What is the ratio of de Broglie wavelength of the particles if their kinetic energy are 0.002 eV and 2 eV respectively?

If \[ (a - 2)^2 + (b - 2)^2 + (c - 2)^2 = 0, then a + b + c = \]

Equation of line through (0, 0, 1) and (1, 1, 0)

Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b}, |\mathbf{a} + \mathbf{b}| = \sqrt{29}, \mathbf{a} \cdot \mathbf{b} = ?

What is the ratio of de Broglie wavelength of the particles if their kinetic energy are 0.002 eV and 2 eV respectively?

The position of body varies as \[ S = 9t^2 - 6t. What is the time at which the body becomes at rest? \]

Given: \[ \frac{197}{96} X \to \frac{197}{95} Y + Z + V \quad what is Z? \]

Galvanometer can be converted into a voltmeter by connecting...

What is the dimension of \[ \frac{M B}{K T} \]
where \( M \) represents magnetic moment, \( K \) represents the Boltzmann constant, \( B \) represents the magnetic field, and \( T \) represents temperature?

If force is 500 N and instantaneous velocity is 20 m/s, what is the instantaneous power?

The tennis ball of mass 120 g moving with a velocity of 20 m/s towards gets by a rocket of velocity and the final speed becomes 30 m/s with direction reversed. If the time of contact is 0.01 s, what is the average force acting on the ball?

For an ideal gas of molar mass \( M \), the slope of the graph between the rms speed \( V \) and \( \sqrt{T} \) where \( T \) represents the temperature is...

Relation between Azimuthal quantum number and magnetic quantum number

Maximum magnetic moment shown by:

Molecule with 2\( \sigma \) and 2\( \pi \) bonds:

Number of valence electrons in Germanium?

Which of the following contains 2 primary –OH groups in its structure?

Name of the reaction in which benzoyl chloride is converted to benzaldehyde is

Benzene when treated with Br\(_2\) in the presence of FeBr\(_3\), gives 1,4-dibromobenzene and 1,2-dibromobenzene. Which type of reaction is this?

Alkali metal with highest first ionization enthalpy is:

A first-order reaction is 75 percent completed in 6000s. What is the half-life?

Total number of orbitals when \( n = 3 \)?

Nitrogen base not present in DNA:

Given that \( a, \frac{3}{4} a r^2, a r^3 \) are in G.P. Product of first four terms = \( \frac{3^6}{4^3} \), then find \( a \):

Coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \)

Integrating factor of \[ \frac{dy}{dx} - 2y = 2x - 3 \]

If \[ \frac{x}{4} + \frac{x}{3} < 13, then x \in \]

Find the limit: \[ \lim_{x \to 0} \frac{\sin[x]}{[x]}, where [x] represents greatest integer function \]

Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]

Find the probability of getting at most 2 heads when 4 coins are tossed.

Find the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \sin^2 x)}{x^2} \]

Find the values of: \[ \cos 75^\circ, \cos 15^\circ, \cos 45^\circ \]

Evaluate the expression: \[ \frac{\sin \frac{\pi}{7} + \sin \frac{3\pi}{7}}{1 + \cos \frac{\pi}{7} + \cos \frac{2\pi}{7}} \]

Axis of parabola is \( x = 0 \). If the vertex is at a distance of 3 from the origin above the x-axis, find the vertex.

Find the integral: \[ \int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx \]

Evaluate: \[ \sec \left( \cos^{-1} \left( \frac{2024}{2025} \right) \right) \]