KEAM 2025 April 26 Question Paper (Available) :Download Solutions with Answer Key

Evaluate the following expression:
\[ \frac{\cos 75^\circ - \cos 15^\circ}{\cos 75^\circ + \cos 15^\circ} \]

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

In a linear programming problem (L.P.P.), the corner points of the feasible region are \( (5, 0), (10, 0) \) and \( (4, 1) \). Find the maximum value of \( Z = 2x + 3y \).

• (A) \( 20 \)
• (B) \( 25 \)
• (C) \( 30 \)
• (D) \( 35 \)

Evaluate the integral:
\[ \int \frac{1}{x(x^4 + 1)} \, dx \]

• (A) \( \frac{1}{2} \log |x^4 + 1| \)
• (B) \( \frac{1}{x^4 + 1} \)
• (C) \( \frac{1}{2} \log |x| \)
• (D) \( \frac{1}{x} \)

Evaluate the following expression:
\[ \frac{\cos 75^\circ - \cos 15^\circ}{\cos 75^\circ + \cos 15^\circ} \]

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

In a linear programming problem (L.P.P.), the corner points of the feasible region are \( (5, 0), (10, 0) \) and \( (4, 1) \). Find the maximum value of \( Z = 2x + 3y \).

• (A) \( 20 \)
• (B) \( 25 \)
• (C) \( 30 \)
• (D) \( 35 \)

Evaluate the integral:
\[ \int \frac{1}{x(x^4 + 1)} \, dx \]

• (A) \( \frac{1}{2} \log |x^4 + 1| \)
• (B) \( \frac{1}{x^4 + 1} \)
• (C) \( \frac{1}{2} \log |x| \)
• (D) \( \frac{1}{x} \)

Evaluate the following integral:
\[ \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx \]

• (A) \( - \frac{1}{\sec x + \tan x} \)
• (B) \( \frac{1}{\sec x + \tan x} \)
• (C) \( - \ln | \sec x + \tan x | \)
• (D) \( \ln | \sec x + \tan x | \)

Evaluate the following integral:
\[ \int e^{2\theta} \left( 2 \cos^2 \theta - \sin 2\theta \right) \, d\theta \]

• (A) \( \frac{e^{2\theta}}{2} \left( 2 \cos^2 \theta - \sin 2\theta \right) \)
• (B) \( e^{2\theta} \cos^2 \theta - \frac{1}{2} \sin 2\theta \)
• (C) \( e^{2\theta} ( \cos^2 \theta - \sin^2 \theta ) \)
• (D) \( \frac{e^{2\theta}}{2} ( \cos^2 \theta + \sin^2 \theta ) \)

Evaluate the following limit:
\[ \lim_{\theta \to 0} \frac{\theta \sin 2\theta}{1 - \cos 2\theta} \]

• (A) \( 0 \)
• (B) \( 1 \)
• (C) \( 2 \)
• (D) \( 4 \)

Current in a coil changes at the rate of 10 A/s. The induced emf is 120V. Find the inductance \( L \) of the coil.

• (A) \( 12 \, H \)
• (B) \( 10 \, H \)
• (C) \( 15 \, H \)
• (D) \( 18 \, H \)

The initial amount of radioactive element in a sample is \( 6 \times 10^3 \). After 48 years, the number of radioactive elements becomes \( 0.75 \times 10^3 \). Find the half-life.

• (A) \( 24 \, years \)
• (B) \( 48 \, years \)
• (C) \( 72 \, years \)
• (D) \( 96 \, years \)

What is the working principle of a Bunsen Burner?

The effective capacitance when \(n\) identical capacitors are connected in parallel is 10μF, and when connected in series is 0.4μF. Find the value of \(n\).

• (A) \( 25 \)
• (B) \( 20 \)
• (C) \( 15 \)
• (D) \( 10 \)

Find the incorrect pair:

• (A) Isobaric - constant pressure
• (B) Isochoric - constant volume
• (C) Isothermal - constant temperature
• (D) Adiabatic - involves heat exchange

The displacement of a body varies with time \( t \) as \( S = \frac{1}{2} t^2 - 6t \). Find the time at which the velocity becomes zero.

• (A) \( 1 \, s \)
• (B) \( 3 \, s \)
• (C) \( 2 \, s \)
• (D) \( 4 \, s \)

The effective capacitance when \(n\) identical capacitors are connected in parallel is 10μF, and when connected in series is 0.4μF. Find the value of \(n\).

• (A) \( 25 \)
• (B) \( 20 \)
• (C) \( 15 \)
• (D) \( 10 \)

Find the incorrect pair:

• (A) Isobaric - constant pressure
• (B) Isochoric - constant volume
• (C) Isothermal - constant temperature
• (D) Adiabatic - involves heat exchange

The displacement of a body varies with time \( t \) as \( S = \frac{1}{2} t^2 - 6t \). Find the time at which the velocity becomes zero.

• (A) \( 1 \, s \)
• (B) \( 3 \, s \)
• (C) \( 2 \, s \)
• (D) \( 4 \, s \)

Minimum wavelength of Brackett series corresponds to transition from \( n_1 \) to \( n_2 \), where \( n_1 \) and \( n_2 \) are respectively...

• (A) \( 4, 3 \)
• (B) \( 5, 4 \)
• (C) \( 6, 5 \)
• (D) \( 7, 6 \)

When an electron is accelerated through a 480V, the wavelength is \( \lambda \). Find the wavelength in terms of \( \lambda \) if it is accelerated through 120V.

• (A) \( \lambda/2 \)
• (B) \( 2\lambda \)
• (C) \( \lambda/4 \)
• (D) \( \lambda/3 \)

The orbital velocity of a satellite is \( V_0 \), at a height \( h = R \) (where \( R \) is the radius of the Earth) from the surface of the Earth. What is the relationship between \( V_0 \) and the escape velocity \( V_e \)?

• (A) \( V_0 = \frac{V_e}{2} \)
• (B) \( V_0 = \frac{V_e}{\sqrt{2}} \)
• (C) \( V_0 = \frac{V_e}{4} \)
• (D) \( V_0 = \frac{V_e}{3} \)

Two particles of the same mass have charges in the ratio 3 : 1. What is the ratio of their time periods when they enter a constant magnetic field with the same velocity?

• (A) \( 3 : 1 \)
• (B) \( 1 : 3 \)
• (C) \( 9 : 1 \)
• (D) \( 1 : 9 \)

What is the force to be applied on a body of mass 200g to change its velocity by 25 m/s in 5 seconds?

• (A) 5 N
• (B) 10 N
• (C) 20 N
• (D) 25 N

1 torr = ________

• (A) \( 1 \, atm \)
• (B) \( 1 \, Pa \)
• (C) \( 133.322 \, Pa \)
• (D) \( 760 \, Pa \)

What is the ratio of distances travelled by a body in the first two intervals of 5 seconds? (Given the initial velocity \( u = 1 \, m/s \) and the body moves with a constant acceleration of \( 5 \, m/s^2 \))

• (A) 1:2
• (B) 1:4
• (C) 1:3
• (D) 1:1

A body hanged by a rope in a lift which is moving upward with a constant acceleration of \( 0.2 \, m/s^2 \). The tension in the rope is 80 N. Find the mass of the body.

• (A) 10 kg
• (B) 20 kg
• (C) 40 kg
• (D) 80 kg

If \( V \) is the velocity of wave in a rope having tension \( T \), find the velocity when the tension becomes \( 8T \).

• (A) \( 8V \)
• (B) \( \frac{V}{8} \)
• (C) \( \sqrt{8}V \)
• (D) \( V \)

A musician hits a drum 90 times in a minute. The time period of hit is:

• (A) \( \frac{1}{90} \, seconds \)
• (B) \( \frac{1}{60} \, seconds \)
• (C) \( \frac{1}{5400} \, seconds \)
• (D) \( \frac{1}{900} \, seconds \)

Chromatic aberration arises in their lens due to:

• (A) Difference in the focal length for different wavelengths
• (B) Dispersion of light
• (C) Diffraction of light
• (D) Reflection of light

If the mean free path of a gas molecule at 27°C is \( 10 \times 10^{-7} \, m \), then the mean free path at 87°C is:

• (A) \( 10 \times 10^{-6} \, m \)
• (B) \( 12 \times 10^{-7} \, m \)
• (C) \( 15 \times 10^{-7} \, m \)
• (D) \( 8 \times 10^{-7} \, m \)

If an inductor coil of self-inductance 2H stores 25J of magnetic energy, then the current passing through it is:

• (A) 5A
• (B) 10A
• (C) 7A
• (D) 12A

Bernoulli's principle is applicable to:

• (A) Fluid flow
• (B) Thermodynamics
• (C) Electromagnetism
• (D) Quantum Mechanics

If the angular displacement in 10 seconds is \( 150^\circ \), find the number of revolutions in 10 seconds.

• (A) 75
• (B) 100
• (C) 150
• (D) 50

If \( P_0 \) is the atmospheric pressure and \( P \) is the pressure at a depth \( h \), find the Guage pressure.

• (A) \( P - P_0 \)
• (B) \( P_0 - P \)
• (C) \( P_0 \times P \)
• (D) \( P \times h \)

A pn junction diode without bias acts as:

• (A) Transistor
• (B) Resistor
• (C) Voltage regulator
• (D) AC transformer

A light is incident on a surface having refractive index \( \frac{4}{3} \) and reflected light is completely polarised. \( \left( \tan 53^\circ = \frac{4}{3} \right) \). What is the angle of incidence?

• (A) \( 53^\circ \)
• (B) \( 42^\circ \)
• (C) \( 63^\circ \)
• (D) \( 30^\circ \)

A cricmeter hits a ball with an initial velocity of 40 m/s. Calculate the maximum range.

• (A) \( 80 \, m \)
• (B) \( 160 \, m \)
• (C) \( 200 \, m \)
• (D) \( 320 \, m \)

Find \( R \) in the following circuit:

• (A) \( 12 \, \Omega \)
• (B) \( 6 \, \Omega \)
• (C) \( 3 \, \Omega \)
• (D) \( 4 \, \Omega \)

If \( F = 6\pi \eta x \), find the dimension of \( x \).

• (A) \( [M^1 L^1 T^{-1}] \)
• (B) \( [M^0 L^0 T^1] \)
• (C) \( [M^1 L^2 T^{-2}] \)
• (D) \( [M^0 L^1 T^0] \)

If the resistance of a wire is 5 \( \Omega \) and 6 \( \Omega \) at 30°C and 40°C respectively, find the temperature coefficient of resistance.

• (A) \( 0.0015 \, per °C \)
• (B) \( 0.0030 \, per °C \)
• (C) \( 0.0020 \, per °C \)
• (D) \( 0.0005 \, per °C \)

Work done to move a charge of 5C from P to Q is 10J. If the potential at P is 0.5V, then the potential at Q is:

• (A) \( 2 \, V \)
• (B) \( 1.5 \, V \)
• (C) \( 3 \, V \)
• (D) \( 4 \, V \)

Two satellites are revolving at a distance of 2.5R and 7.5R from the center of the Earth. Find the ratio of time period of the satellites.

• (A) \( \frac{1}{3} \)
• (B) \( 1 \)
• (C) \( \frac{1}{9} \)
• (D) \( 1:2 \)

Find the incorrect statement:

A) Isobaric - constant pressure
B) Isochoric - constant volume
C) Isothermal - constant temperature
D) Adiabatic - involves heat exchange

• (A) A
• (B) B
• (C) C
• (D) D

Which of the following cannot form hydrogen bonding?

A) Phenol
B) Diethyl ether
C) Aniline

• (A) A
• (B) B
• (C) C
• (D) None of the above

Entropy decreases in which of the following reactions?
\[ (A) Br_2 (l) \to Br_2 (g) \] \[ (B) Br_2 (g) \to Br_2 (l) \] \[ (C) Br_2 (l) \to Br_2 (g) and Br_2 (g) \to Br_2 (l) \]

• (A) \( Br_2 (l) \to Br_2 (g) \)
• (B) \( Br_2 (g) \to Br_2 (l) \)
• (C) Both A and B
• (D) None of the above

How many bridging complexes are there in \( [Mg(Co)_{10}] \)?

• (A) 3
• (B) 2
• (C) 1
• (D) 0

In which reaction is Benzyl chloride converted to Benzaldehyde?

• (A) Gatterman Koch reaction
• (B) Stephan’s reaction
• (C) Etard reaction
• (D) Rosemend reduction

Given the reaction:
\[ Cr_2O_7^{2-} + x e^- + y H^+ \to 2 Cr^{3+} + z H_2O \]
Find the values of \(x\), \(y\), and \(z\).

• (A) \( x = 6, y = 14, z = 7 \)
• (B) \( x = 3, y = 6, z = 3 \)
• (C) \( x = 6, y = 12, z = 6 \)
• (D) \( x = 4, y = 8, z = 4 \)

Which of the following is an interstitial compound?

• (A) \( SC_2O_3 \)
• (B) \( Mn_4N \)
• (C) \( TiCl_4 \)
• (D) \( TiN \)

Ethanol is made unfit for drinking by adding?

• (A) HCl
• (B) NaOH
• (C) KOH
• (D) Methanol

Which is a Lewis acid?

• (A) HCl
• (B) \( OH^- \)
• (C) \( Co^{3+} \)

Phenol is treated with conc. \( H_2SO_4 \), and then with conc. \( HNO_3 \). The compound A and B are formed.

• (A) A is phenol, B is nitrobenzene
• (B) A is phenol, B is benzene
• (C) A is phenol, B is phenol derivative
• (D) A is phenol, B is benzene derivative

IUPAC name of allyl amine?

• (A) Propanamine
• (B) 2-aminopropene
• (C) 3-aminopropene
• (D) Butanamine

Find the area between the line \( y = x - 1 \), \( y = 0 \), \( -2 \leq y \leq 2 \).

• (A) 6
• (B) 8
• (C) 4
• (D) 10

Find \( \frac{dy}{dx} \) for the equation:
\[ y = \cos x \times \sin y \]

• (A) \( \cos x \)
• (B) \( -\sin x \)
• (C) \( \sin x \)
• (D) \( \cos x \times \cos y \)

In a GP 9, 3, \( \frac{1}{3} \), \( \frac{1}{9} \), \ldots, find the 25th term.

• (A) \( \frac{1}{3^{24}} \)
• (B) \( \frac{9}{3^{24}} \)
• (C) \( \frac{9}{3^{25}} \)
• (D) \( \frac{1}{3^{25}} \)

Evaluate the integral:
\[ \int e^{x + \frac{1}{x}} \frac{x^2 - 1}{x^2} \, dx \]

• (A) \( e^{x + \frac{1}{x}} \)
• (B) \( e^{x + \frac{1}{x}} \left( \frac{x^2 - 1}{x^2} \right) \)
• (C) \( \frac{e^{x + \frac{1}{x}}}{x} \)
• (D) \( e^{x + \frac{1}{x}} \left( \ln |x| - 1 \right) \)

Find the value of:
\[ \sin 75^\circ \times \sin 15^\circ \times \sin 45^\circ \]

• (A) \( \frac{1}{4} \)
• (B) \( \frac{1}{8} \)
• (C) \( \frac{1}{2} \)
• (D) \( \frac{1}{16} \)

If \( \tan^{-1}(x) = \tan^{-1}\left( 3 - \frac{\pi}{4} \right) \), find \( x \).

• (A) \( 3 - \frac{\pi}{4} \)
• (B) \( \frac{\pi}{4} - 3 \)
• (C) \( 3 + \frac{\pi}{4} \)
• (D) \( 3 - \frac{\pi}{2} \)

Find the range of \( f(x) = \sqrt{x^2 + 4x + 4} \).

• (A) \( (-\infty, \infty) \)
• (B) \( [0, \infty) \)
• (C) \( (-2, 2) \)
• (D) \( [2, \infty) \)

Evaluate the integral:
\[ \int_0^{\frac{\pi}{2}} \frac{1}{1 + \sin x} \, dx \]

• (A) \( \frac{\pi}{4} \)
• (B) \( \frac{\pi}{2} \)
• (C) \( \frac{1}{2} \)
• (D) \( \frac{\pi}{3} \)

Evaluate the derivative of
\[ y = \cos x \times \sin y, \quad \frac{dy}{dx} at \left( \frac{\pi}{6}, \frac{\pi}{5} \right) \]

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

If \( f(x) = \log 3 - \sin x \), \( y = f(f(x)) \), find \( y(0) \).

• (A) \( 2 \)
• (B) \( 0 \)
• (C) \( 1 \)
• (D) \( \log 3 \)