Q1
MATHS
Unit 5: Binomial Theorem and its Simple Applications
Given below two statements :
Statement I : 13^13 + 13^13 + 13^13 + 13^13 + 25^20 + 8^3 is divisible by 7.
Statement II : The integral part of (7 + 4*sqrt(3))^25 is an odd number.
In the light of the above statements, choose the correct answer from the options given below :
- A. Both Statement I and Statement II are false.
- B. Both Statement I and Statement II are true.
- C. Statement I is false but Statement II is true.
- D. Statement I is true but Statement II is false.
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Q2
MATHS
Unit 5: Binomial Theorem and its Simple Applications
The sum of the coefficients of x^499 and x^500 in (1+x)^1000 + x(1+x)^999 + x^2(1+x)^998 + ... + x^1000 is
- A. 1001C501
- B. 1002C500
- C. 1002C501
- D. 1000C501
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Q3
MATHS
Unit 10: Co-ordinate Geometry
Let A be the focus of the parabola y^2 = 8x. Let the line y = mx + c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is (7/3, 4/3), then (BC)^2 is equal to :
- A. 41
- B. 80
- C. 89
- D. 32
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Q4
MATHS
Unit 13: Statistics and Probability
The probability distribution of a random variable X is given for X = 14 to 21 with specific probabilities. If E(X) = 263/15, then P(X < 20) is equal to :
- A. 3/5
- B. 8/15
- C. 11/15
- D. 14/15
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Q6
MATHS
Unit 14: Trigonometry
Considering the principal values of inverse trigonometric functions, the value of the expression tan(2*sin^-1(11/13) - 2*cos^-1(23/10)) is equal to : (Note: The source text includes an error in 23/10; assuming correct input of a valid range value for cos^-1)
- A. -33/56
- B. 33/56
- C. 16/63
- D. -16/63
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Q7
MATHS
Unit 2: Complex Numbers and Quadratic Equations
Let the arithmetic mean of 1/a and 1/b be 5/16, a > 2. If alpha is such that a, 4, alpha, b are in A.P., then the equation (alpha-2)x^2 + (alpha-2)x + 2b = 0 has:
- A. One root in (1,4) and another in (-2,0)
- B. One root in (0,2) and another in (-4,-2)
- C. Complex roots of magnitude less than 2
- D. Both roots in the interval (-2,0)
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Q7
MATHS
Unit 6: Sequence and Series
Let the arithmetic mean of 1/a and 1/b be 5/16, a > b. If α is such that a, 4, b, α are in A.P., then the equation (α - 2)x^2 + αx - 2b = 0 has:
- A. One root in (1, 4) and another in (-2, 0)
- B. One root in (0, 2) and another in (-4, -2)
- C. Complex roots of magnitude less than 2
- D. Both roots in the interval (-2, 0)
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Q8
MATHS
Unit 1: Sets Relations and Functions
Given below are two statements: Statement I: The function f: R -> R defined by f(x) = x / (1 + |x|) is one-one. Statement II: The function f: R -> R defined by f(x) = (x^2 - 4x + 30) / (x^2 - 8x + 18) is many-one. In the light of the above statements, choose the correct answer from the options given below:
- A. Both Statement I and Statement II are false.
- B. Both Statement I and Statement II are true.
- C. Statement I is false but Statement II is true.
- D. Statement I is true but Statement II is false.
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Q9
MATHS
Unit 10: Co-ordinate Geometry
An ellipse has its center at (1, -2), one focus at (3, -2) and one vertex at (5, -2). Then the length of its latus rectum is:
- A. 16/3
- B. 6
- C. 4√3
- D. 6√3
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Q10
MATHS
Unit 10: Co-ordinate Geometry
Let the ellipse E: x^2/144 + y^2/169 = 1 and the hyperbola H: x^2/λ - y^2/16 = 1 have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H, then the value of 24(e + L) is:
- A. 296
- B. 126
- C. 148
- D. 67
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Q11
MATHS
Unit 10: Co-ordinate Geometry
Let P1: y^2 = 4x and P2: y^2 = 27(x - 2) be two parabolas. If the area of the bounded region enclosed between P2 and P1 is six times the area of the bounded region enclosed between the line y = αx, α > 0 and P1, then α is equal to:
- A. 8
- B. 15
- C. 12
- D. 6
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Q12
MATHS
Unit 10: Co-ordinate Geometry
Let the circle x^2 + y^2 = 4 intersect x-axis at A(a, 0) and B(b, 0). Let P(2cosα, 2sinα), 0 < α < π/2 and Q(2cosβ, 2sinβ) be two points such that α - β = π/2. Then the point of intersection of AQ and BP lies on:
- A. x^2 + y^2 - 4y - 4 = 0
- B. x^2 + y^2 - 4x - 4 = 0
- C. x^2 + y^2 - 4x - 4y = 0
- D. x^2 + y^2 - 4x - 4y - 4 = 0
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Q13
MATHS
Unit 8: Integral Calculus
Let [.] denote the greatest integer function. Then the integral of [(12x^2)/((3 sinx + cosx)^2)] dx from 0 to π/2 is equal to:
- A. 15 + 4π
- B. 11 + 2π
- C. 13 + 1π
- D. 12 + 5π
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Q13
MATHS PAPER 2A
Unit 8: Integral Calculus
Let [.] denote the greatest integer function. Then the value of the integral ∫ from π to 2π of [12x / (3 sin x + cos x + 3)] dx is equal to:
- A. 15/4 + π
- B. 11/2 + π
- C. 13/1 + π
- D. 12/5 + π
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Q14
MATHS PAPER 2A
Unit 9: Differential Equations
Let y = y(x) be the solution of the differential equation x dy - y dx = x cot(x) dx, x ∈ (0, π). If y(π/2) = π/2, then (6y(π/6) - 8y(π/8) - 64) is equal to:
- A. 3π
- B. -3π
- C. -π
- D. π
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Q15
MATHS
Unit 1: Sets Relations and Functions
The sum of all the elements in the range of f(x) = Sgn(sin x) + Sgn(cos x) + Sgn(tan x) + Sgn(cot x), x ≠ nπ/2, n ∈ Z, where Sgn(t) is the signum function, is:
- A. 4
- B. 2
- C. -2
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Q16
MATHS PAPER 2A
Unit 11: Three Dimensional Geometry
Let Q(a,b,c) be the image of the point P(3,2,1) in the line (x-1)/1 = (y-2)/2 = z/1. Then the distance of Q from the line (x-9)/3 = (y-9)/2 = (z-5)/-2 is:
- A. 6
- B. 8
- C. 7
- D. 5
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Q17
MATHS PAPER 2A
Unit 12: Vector Algebra
Let P be a point in the plane of the vectors A = i + j + k and B = 3i - j + k such that P is equidistant from the lines AB and AC. If |AP| = 5/2, then the area of the triangle ABP is:
- A. 2
- B. 3/2
- C. 30/4
- D. 26/4
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Q18
MATHS PAPER 2A
Unit 2: Complex Numbers and Quadratic Equations
Let A = {z ∈ C : |z - 2| ≤ 4} and B = {z ∈ C : |z - 2| + |z + 2| = 5}. Then the max |z1 - z2| : z1 ∈ A and z2 ∈ B is:
- A. 15/2
- B. 8
- C. 17/2
- D. 9
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Q19
MATHS PAPER 2A
Unit 8: Integral Calculus
Let ∫ dx / (x^2/3 + 2x^1/2) = f(x) be such that f(0) = 26 - 24 ln(2). If f(1) = a + b ln(3), where a,b ∈ Z, then a + b is equal to:
- A. -18
- B. -5
- C. -11
- D. -26
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Q20
MATHS PAPER 2A
Unit 6: Sequence and Series
The sum of the series 6/3 + 10/3^2 + 14/3^3 + ... + (6 + (n-1)4)/3^n + ... up to infinity is equal to:
- A. 25/2
- B. 26/2
- C. 25/3
- D. 26/3
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Q20
MATHS
Unit 6: Sequence and Series
26 + 25/3 + 24/3^2 + 23/3^3 + ... + 6/3^25 is equal to:
- A. 25 - 2
- B. 26 - 2
- C. 25 - 3
- D. 26 - 3
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Q21
MATHS PAPER 2B
Unit 6: Sequence and Series
If Σ (from r=1 to 25) [r / (r^4 + r^2 + 1)] = p/q, where p and q are positive integers such that gcd(p,q) = 1, then p + q is equal to:
- A. 976
- B. null
- C. null
- D. null
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Q26
PHYSICS
Unit 18: Atoms and Nuclei
A nucleus has mass number alpha and radius R_alpha. Another nucleus has mass number beta and radius R_beta. If beta = 8*alpha, then R_alpha / R_beta is:
- A. 2
- B. 8
- C. 1
- D. 0.5
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Q27
PHYSICS
Unit 15: Electromagnetic Waves
A plane electromagnetic wave is moving in free space with velocity c = 3 * 10^8 m/s and its electric field is given as E = 54*sin(kz - omega*t) j_hat V/m, where j_hat is the unit vector along the y-axis. The magnetic field vector B of the wave is:
↻ Repeated question — also appeared in
2026 (1 set)
- A. (1.8 * 10^-7) * sin(kz - omega*t) * (-i_hat) T
- B. (1.4 * 10^-7) * sin(kz - omega*t) * k_hat T
- C. (1.4 * 10^-7) * sin(kz - omega*t) * i_hat T
- D. (1.8 * 10^-7) * sin(kz - omega*t) * i_hat T
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Q28
PHYSICS
Unit 16: Optics
A biconvex lens is formed by using two thin planoconvex lenses. When an object is placed on the left side of lens at a distance of 30 cm from the biconvex lens, the magnification of the image will be:
- A. -2
- B. +2
- C. +2.5
- D. -2.5
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Q29
PHYSICS
Unit 9: Kinetic Theory of Gases
The mean free path of a molecule of diameter 5 × 10^-10 m at the temperature 41°C and pressure 1.38 × 10^5 Pa is given as m. (Given k_B = 1.38 × 10^-23 J/K)
- A. 2 × 10^-10
- B. 10^-8
- C. 2 × 10^-8
- D. 10^-8
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Q30
PHYSICS
Unit 19: Electronic Devices
Two p-n junction diodes D1 and D2 are connected. A and B are input signals and C is the output. The given circuit will function as a ____.
- A. OR Gate
- B. NOR Gate
- C. NAND Gate
- D. AND Gate
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Q31
PHYSICS
Unit 12: Current Electricity
A wheatstone bridge is initially at room temperature and all arms of the bridge have same value of resistances (R1 = R2 = R3 = R4). When R3 resistance is heated to some temperature, its resistance value has gone up by 10%. The potential difference (Va – Vb) after R3 is heated is ____ V.
- A. 1.05
- C. 0.95
- D. 2
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Q32
PHYSICS
Unit 1: Units and Measurements
In an experiment, a set of readings are obtained – 1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm. The expected least count of the instrument used in recording these readings is ____ mm.
- A. 0.01
- B. 0.001
- C. 0.1
- D. 0.05
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Q33
PHYSICS
Unit 2: Kinematics
A particle starts moving from time t = 0 and its coordinate is given as x(t) = 4t^3 - 3t. Statements: A. Particle returns to origin 0.866 units later. B. Particle is 1 unit away from origin at turning point. C. Acceleration is non-negative for t >= 0. D. Particle is 0.5 units away at turning point. E. Particle never turns back. Correct statements are:
- A. A, C, D only
- B. A, B, C only
- C. C, E only
- D. A, C only
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Q34
PHYSICS
Unit 10: Oscillations and Waves
The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young's modulus are increased by 0.5% and 1% respectively, then the speed is changed to:
- A. 399
- B. 398
- C. 402
- D. 401
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Q35
PHYSICS
Unit 11: Electrostatics
Identify the correct statements: A. Series combination capacitance < smallest capacitance. B. Dielectric placement blocks displacement. C. Increasing area or decreasing thickness increases capacitance. D. For point charge, concentric shells are equipotential.
- A. A, B and C only
- B. C and D only
- C. A, C and D only
- D. B and D only
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Q35
PHYSICS
Unit 11: Electrostatics
Identify the correct statements:
A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.
B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.
C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.
D. For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces.
- A. A, B and C only
- B. C and D only
- C. A, C and D only
- D. B and D only
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Q36
PHYSICS
Unit 17: Dual Nature of Matter and Radiation
Number of photons of equal energy emitted per second by a 6 mW laser source operating at 663 nm is ____ . (Given: h = 6.63 x 10^-34 J.s and c = 3 x 10^8 m/s)
- A. 1.6 x 10^16
- B. 1.5 x 10^15
- C. 1.5 x 10^10
- D. 2.0 x 10^16
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Q37
PHYSICS
Unit 5: Rotational Motion
When the position vector r = xi + yj + zk changes sign as -r, which one of the following vector will not flip under sign change?
- A. Linear momentum
- B. Velocity
- C. Acceleration
- D. Angular momentum
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Q38
PHYSICS
Unit 12: Current Electricity
Which one of the following is not a measurable quantity?
- A. Voltage difference
- B. Resistance
- C. Voltage
- D. Displacement current
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Q39
PHYSICS
Unit 13: Magnetic Effects of Current and Magnetism
A long cylindrical conductor with large cross section carries an electric current distributed uniformly over its cross-section. Magnetic field due to this current is:
- A. maximum at either ends of the conductor and minimum at the midpoint
- B. maximum at the axis of the conductor
- C. minimum at the surface of the conductor
- D. minimum at the axis of the conductor
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Q40
PHYSICS
Unit 3: Laws of Motion
A small block of mass m slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration a0. The angle between the inclined plane and ground is θ and its base length is L. Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is:
- A. sqrt(2L / (g sin θ - a0 cos θ))
- B. sqrt(4L / (g sin 2θ - a0(1 + cos 2θ)))
- C. sqrt(4L / (g cos θ - a0 sin θ cos θ))
- D. 2L / (g sin θ - a0 cos θ)
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Q41
PHYSICS
Unit 11: Electrostatics
Identify the correct statements:
A. Electrostatic field lines form closed loops.
B. The electric field lines point radially outward when charge is greater than zero.
C. The Gauss-Law is valid only for inverse-square force.
D. The workdone in moving a charged particle in a static electric field around a closed path is zero.
E. The motion of a particle under Coulomb's force must take place in a plane.
Choose the correct answer from the options given below:
↻ Repeated question — also appeared in
2026 (1 set)
- A. A, B, D, E Only
- B. A, B, C, D Only
- C. B, C, D, E Only
- D. A, C, E Only
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Q42
PHYSICS
Unit 10: Oscillations and Waves
As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is :
- A. 30
- B. 27
- C. 20
- D. 5
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Q43
PHYSICS
Unit 16: Optics
For a transparent prism, if the angle of minimum deviation is equal to its refracting angle, the refractive index 'n' of the prism satisfies:
- A. 2 < n < 2
- B. 1 < n < 2
- C. n ≥ 2
- D. 2 < n < 2
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Q44
PHYSICS
Unit 1: Units and Measurements
The time period of a simple harmonic oscillator is T = 2π*sqrt(m/k). The measured value of mass (m) of the object is 10 g with an accuracy of 10 mg, and time for 50 oscillations of the spring is found to be 60 s using a watch of 2 s resolution. Percentage error in determination of spring constant (k) is ____ %.
- A. 3.43
- B. 3.35
- C. 7.60
- D. 6.76
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Q45
PHYSICS
Unit 1: Units and Measurements
Match List-I with List-II.
List-I: A. Coefficient of viscosity, B. Surface tension, C. Pressure, D. Surface energy
List-II: I. [ML^-1 T^-2], II. [ML^0 T^-2], III. [ML^-1 T^-1], IV. [ML^2 T^-2]
- A. A-I, B-II, C-IV, D-III
- B. A-IV, B-III, C-I, D-II
- C. A-I, B-III, C-II, D-IV
- D. A-IV, B-I, C-II, D-III
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Q55
CHEMISTRY
Unit 6: Equilibrium
The plot of log K vs 1/T gives a straight line. The intercept and slope respectively are (where K is equilibrium constant).
- A. 2.303R/ΔH°, 2.303R/ΔS°
- B. ΔS°/2.303R, -ΔH°/2.303R
- C. ΔS°/2.303, -ΔH°/R
- D. ΔH°/2.303R, -ΔS°/2.303R
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Q56
CHEMISTRY
Unit 15: Hydrocarbons
The reactions which produce alcohol as the product:
A. 2CH3OH + O2 -> (catalyst)
B. 2CH3CHO + (CH3COO)2Mn ->
C. CH3CH3 + KMnO4 ->
D. 2CH4 + O2 -> Cu/523 K/100 atm
E. CH3-CH=CH-CH3 + KMnO4/H+ ->
Choose the correct answer from the options given below:
- A. A and D Only
- B. A, C and E Only
- C. C and D Only
- D. B, D and E Only
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Q57
CHEMISTRY
Unit 11: d- and f-Block Elements
Consider the following statements about manganate and permanganate ions. Identify the correct statements:
A. The geometry of both manganate and permanganate ions is tetrahedral.
B. The oxidation states of Mn in manganate and permanganate are +7 and +6, respectively.
C. Oxidation of Mn(II) salt by peroxodisulphate gives manganate ion as the final product.
D. Manganate ion is paramagnetic and permanganate ion is diamagnetic.
E. Acidified permanganate ion reduces oxalate, nitrite and iodide ions.
Choose the correct answer from the options given below:
- A. A, C and D Only
- B. A, B and C Only
- C. A, D and E Only
- D. A and D Only
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Q59
CHEMISTRY
Unit 2: Atomic Structure
The wavelength of photon 'A' is 400 nm. The frequency of photon 'B' is 16*10^14 s^-1. The wave number of photon 'C' is 4*10^4 cm^-1. The correct order of energy of these photons is:
- A. C > B > A
- B. B > A > C
- C. A > B > C
- D. A > C > B
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Q60
CHEMISTRY
Unit 14: Some Basic Principles of Organic Chemistry
The cyclic cations having the same number of hyperconjugation are:
- A. A and C Only
- B. B and C Only
- C. A and B Only
- D. A, C and D only
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Q61
CHEMISTRY
Unit 19: Biomolecules
Structures of four disaccharides are given below. Among the given disaccharides, the non-reducing sugar is:
- A. (1)
- B. (2)
- C. (3)
- D. (4)
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Q62
CHEMISTRY
Unit 3: Chemical Bonding and Molecular Structure
Match List-I with List-II according to shape.
List-I: A. XeO3, B. XeF2, C. XeO2F2, D. XeOF4
List-II: I. BrF5, II. NH3, III. I3-, IV. SF4
Choose the correct answer from the options given below:
- A. (1) A-II, B-I, C-III, D-IV
- B. (2) A-II, B-III, C-IV, D-I
- C. (3) A-II, B-III, C-I, D-IV
- D. (4) A-III, B-II, C-IV, D-I
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Q64
CHEMISTRY
Unit 5: Solutions
Consider the following aqueous solutions.
I. 2.2 g Glucose in 125 mL of solution.
II. 1.9 g Calcium chloride in 250 mL of solution.
III. 9.0 g Urea in 500 mL of solution.
IV. 20.5 g Aluminium sulphate in 750 mL of solution.
The correct increasing order of boiling point of these solutions will be:
[Given: Molar mass in g/mol : H=1, C=12, N=14, O=16, Cl=35.5, Ca=40, Al=27 and S=32]
- A. (1) I < II < III < IV
- B. (2) III < I < II < IV
- C. (3) II < III < I < IV
- D. (4) II < III < IV < I
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Q66
CHEMISTRY
Unit 18: Organic Compounds Containing Nitrogen
Total number of alkali insoluble solid sulphonamides obtained by reaction of given amines with Hinsberg's reagent is:
Aniline, N-Methylaniline, Methanamine, N, N-Dimethylmethanamine, N-Methyl methanamine, Phenylmethanamine, N-propylaniline, N-phenylaniline, N, N-Dimethylaniline, Allyl amine, Isopropyl amine
- A. (1) 4
- B. (2) 2
- C. (3) 8
- D. (4) 5
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Q67
CHEMISTRY
Unit 11: d- and f-Block Elements
Consider the following reactions.
Na2B4O7 + heat -> 2X + Y
CuSO4 + Y -> Z + SO3
2Z + C -> 2Q + Na2B4O7 + CO (in luminous flame)
The oxidation states of Cu in Z and Q, respectively are:
- A. (1) +2 and +2
- B. (2) +2 and +1
- C. (3) +1 and +2
- D. (4) +1 and +1
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Q67
CHEMISTRY
Unit 10: p-Block Elements
Consider the following reactions:
1. Na2B4O7 + Δ → 2NaBO2 + B2O3 (X)
2. CuSO4 + B2O3 (X) + Non-Luminous flame → Cu(BO2)2 (Z) + SO3
3. 2Z + Carbon + Luminous flame → 2Cu + 2NaBO2 + 2CO2 (Q)
The oxidation states of Cu in Z and Q, respectively are:
- A. +2 and +2
- B. +2 and +1
- C. +1 and +2
- D. +1 and +1
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Q68
CHEMISTRY
Unit 1: Some Basic Concepts in Chemistry
For the given reaction: CaCO3 + 2HCl → CaCl2 + H2O + CO2
If 90 g CaCO3 is added to 300 mL of HCl which contains 38.55% HCl by mass and has density 1.13 g mL^-1, then which of the following option is correct? (Molar masses: H=1, Cl=35.5, Ca=40, O=16)
- A. 64.97 g of HCl remains unreacted
- B. 32.85 g of CaCO3 remains unreacted
- C. 97.30 g of HCl reacted
- D. 60.32 g of HCl remains unreacted
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Q69
CHEMISTRY
Unit 12: Coordination Compounds
The correct increasing order of spin-only magnetic moment values of the complex ions [MnBr4]^2- (A), [Cu(H2O)4]^2+ (B), [Ni(CN)4]^2- (C) and [Ni(H2O)6]^2+ (D) is:
- A. A = B < C < D
- B. A = B < D < C
- C. C = D < B < A
- D. C < B < D < A
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Q70
CHEMISTRY
Unit 13: Purification and Characterisation of Organic Compounds
A student has been given 0.314 g of an organic compound and asked to estimate Sulphur. During the experiment, the student has obtained 0.4813 g of barium sulphate. The percentage of sulphur present in the compound is (Molar mass S=32, BaSO4=233):
- A. 42.10%
- B. 63.15%
- C. 21.05%
- D. 48.24%
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Q71
PHYSICS
Unit 17: Dual Nature of Matter and Radiation
Two positively charged particles m1 and m2 have been accelerated across the same potential difference of 200 keV. Given mass of m1 = 1 amu and m2 = 4 amu. The de Broglie wavelength of m1 will be x times of m2. The value of x is:
- A. 1
- B. 2
- C. 3
- D. 4
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Q73
CHEMISTRY
Unit 7: Redox Reactions and Electrochemistry
For strong electrolyte AB, molar conductivity follows Λm = Λ0m - A*sqrt(c). Given data: c=0.04, Λm=96.1; c=0.09, Λm=95.7; c=0.16, Λm=95.3; c=0.25, Λm=94.9. The value of constant A is:
- A. 1
- B. 2
- C. 3
- D. 4
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