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91. Evaluate the definite integral ∫0π |cos x| dx.

Evaluate the Integral ∫₀^π |cos x| dx

90. A circle is given by the equation 2x² + 2y² + 8x − 20y + 10 = 0. The area of a square whose side equals the radius of the circle is ______.

Radius of a Circle from the General Equation

89. The value of the complex number (1 + i)150 + (1 − i)150 is ______.

Find the Value of the Complex Number

$88. The total number of mappings from the set {1, 2} to the set {3, 4, 5, 6, 7} is _____.$

Total Number of Mappings

87. Let N be the set of all natural numbers. Consider the relation R on N given by R = {(m, n) : m − n is divisible by 2}. Then (A) R is symmetric and transitive (B) R is symmetric but NOT transitive (C) R is reflexive but NOT symmetric (D) R is reflexive and transitive

Relation R = {(m, n) : m − n

86. Let a and b be two non-zero vectors such that |a + b| = |a − b| Then (A) a and b are parallel to each other (B) a and b are perpendicular to each other (C) a is NOT a scalar multiple of b (D) a × b = 0

Condition for Equal Magnitudes of ∣a+b∣

85. If A and B are two skew-symmetric matrices, the matrix AB + BA must be (A) skew-symmetric (B) symmetric (C) invertible (D) NOT invertible

If A and B are Two Skew-Symmetric Matrices

84. The lengths of two sides of a triangle are 2 units and 3 units and the angle included by these two sides is 60°. The length of the third side of the triangle will be (A) √5 units (B) √7 units (C) 4 units (D) 5 units

The Lengths of Two Sides of a Triangle

83. Let nCr denote the binomial coefficient nCr = n! / [r!(n-r)!]. For n = 100, find the sum of the series 1 − nC1 + nC2 − nC3 + ··· + (−1)nnCn. (A) 0 (B) 1 (C) 2 (D) 1024

Sum of the Binomial Coefficient Series

82. Let R be the set of all real numbers. Consider the sets P = {x ∈ R : (x − 1)(x² + 1) = 0} Q = {x ∈ R : x² − 9x + 2 = 0} S = {x ∈ R : x = 5y for some y ∈ R} Then the set (P ∩ S) ∪ Q contains (A) exactly two elements (B) exactly three elements (C) exactly four elements (D) infinitely many elements

Set Theory MCQ on Intersection and Union

81. If [x] denotes the greatest integer function (for example, [1.16] = 1 and [1.8] = 1), then ∫₁√3 (1 + [x])/(1 + x²) dx = L, then L = ______ degrees.

Evaluate ∫₁√3 (1 + [x])/(1 + x²) dx

80. From the database of a clinic it was found that out of 2000 patients who had visited the clinic in a year, 900 had high BP, 900 had high Sugar and 400 had neither high BP nor high Sugar. On a given day, if 20 patients visit the clinic, the expected number of patients who have both high BP and high Sugar is ______.

Expected Number of Patients with Both High BP and High Sugar

79. The plane x + y + z = 0 intersects the sphere x2 + y2 + z2 = 9 along a circle. If (2, y, z) is a point on the circle, then the value of |y + z| is _____.

The Plane x + y + z = 0 Intersects the Sphere x² + y² + z² = 9

78. In ΔPQR, ∠Q = 60° and S is the midpoint of QR. If QS = PS and PR = 5, then (A) PQ = 5√3/3 (B) PS = 5/2 (C) Area of ΔPSR = (25 − 5√3)/(2√3) (D) S is the circumcenter of ΔPQR

Triangle Geometry

77. If f(x)= ax²+b, 0≤x≤1 cx+sin(πx/2), 1≤x≤2 is continuous and differentiable at all points in the interval [0,2], and f(2)=π/4, then determine the correct option. (A) a = π/16 and b = π/16 + 1 (B) b = π/16 + 1 and c = π/8 (C) a = π/8 and c = π/8 (D) a = π/8 and b = 3π/16

Piecewise Function Continuity and Differentiability

76. Consider two vectors P and Q of equal magnitude. If the magnitude of P + Q is two-times larger than that of P - Q, then the angle between them is (A) 107° (B) 117° (C) 127° (D) 137°

Consider Two Vectors P and Q of Equal Magnitude

75. Let Z be the set of all integers and let f and g be one-one mappings from Z into itself such that For even n, f(g(n)) = g(n + 1) + 1 For odd n, g(f(n)) = f(n − 1) − 1 and f(1)=3. Then (A) g(2)=0 (B) f(3)=2 (C) g(2)=1 (D) f(3)=1

Functional Equation MCQ on One-One Mapping

74. Let P(t) denote the population of a species at time t. If dP/dt = P(1 − P) and the initial population is P(0)=0.1 million, then the population at t = 1 is (A) 9 + e / e (B) e / (9 − e) (C) 9e / (e − 1) (D) 9e / (9 + e)

Solve the Logistic Differential Equation dt/dP =P(1−P)

73. A line parallel to the vector passes through the point and meets the xy-plane at a point . The distance between the origin and is (A) 10 (B) 11 (C) 12 (D) 13

Line Parallel to Vector i + j + k

72. From the set of 10 numbers {1, 2,...,10} three numbers are selected at random without replacement. The probability that the sum of these selected numbers is 9, is (A) 1/40 (B) 1/20 (C) 3/10 (D) 3/80

Probability That the Sum of Three Selected Numbers

71. Using the letters in the word TRICK a new word containing five distinct letters is formed such that T appears in the middle. The number of distinct arrangements is ______.

Number of Arrangements of the Word TRICK

69. Evaluate ∫₀¹ x dx + ∫₁² (2 − x) dx

Evaluate ∫₀¹ x dx + ∫₁² (2 − x) dx

68. The positive root of the equation x4 + x2 - 2 = 0 is .

Positive Root of x 4 +x 2 −2=0

67. Which of the following point(s) lies(lie) on the plane 2x + 3y + z = 6? (A) (0, 0, 6) (B) (0, 2, 0) (C) (1, 1, 1) (D) (3, 0, 0)

Which Points Lie on the Plane 2x+3y+z=6?

66. lf P = {1, 2, —1, 3}, Q = (0, 4, 1, 3} and R = {1, 6, 7}, then PM(QUR)= (A) {1, 2} (B) ( 1, 3} (C) (2, 1} (D) (2, 3}

Find P∩(Q∪R) – Set Theory

65. If one of the diameters of a circle has end points (2, 0) and (4, 0), then the equation of that circle is (A) x2 — 3x + y2 + 5 = 0 (B) x2 — 4x + y2 + 6 = 0 (C) x2 — 5x + y2 + 7 = 0 (D) x2 — 6x + y2 + 8 = 0

Equation of a Circle from Diameter Endpoints

64. Two dice are thrown simultaneously. The probability that the sum of the numbers obtained is divisible by 7 is (A) 1/ 6 (B) 1/36 (C) 0 (D) 1/18

Probability of Sum Divisible by 7

63. If u(x) and v(x) are differentiable at x = 0, and u(0) = 5, u'(0) = -3, v(0) = -1, v'(0) = 2, then find d/dx (uv + u/v) evaluated at x = 0. (A) −20 (B) −7 (C) 6 (D) 13

Derivative of uv+ v/u at x=0

62. If [ x y p q u v ] R = [ 0 0 0 0 0 0 0 0 1 ] then the order of R is (A) 2 × 3 (B) 3 × 2 (C) 2 × 2 (D) 3 × 3

Order of a Matrix MCQ

61. cos(x + yx) = (A) cos (x) cos(yx) — sin (x) sin (yx) (C) cos (x) sin (yx) — sin(x) cos (yx) (B) cos (x) cos(yx) + sin (x) sin (yx) (D) cos (x) sin (yx) + sin (x) cos (yx)

cos(x + yx) Formula

60. If A and B are events such that P(A) = 0.30 P(B) = 0.20 P(A ∪ B) = 0.45 Find the value of P(A ∩ B̅).

How to Calculate P(A∩ B ) Using Probability Formula

59. For a = ____, the following simultaneous equations have an infinite number of solutions: 10x + 13y = 6 ax + 32.5y = 15

Find the Value of a for Infinite Solutions of Simultaneous Equations

58. Find the determinant of the following matrix: | 1 3 0 | | 2 6 4 | | −1 −1 2 |

Determinant of a 3×3 Matrix Using Cofactor Expansion

57. The value of log𝑛 4−16 is −32. The value of 𝑛 is .

Find the Value of n if logₙ(4⁻¹⁶) = −32

56. Consider the equation x3 − 1 = 0. If one of the solutions to this equation is 1, the other solution(s) is/are ______.

Solve x3−1=0x^3 – 1 = 0x3−1=0

55. The area of an equilateral triangle with sides of length a is: (A) (√3/4)a2 (B) (√3/2)a2 (C) (1/2)a2 (D) (1/√2)a2

Find the Area of an Equilateral Triangle

54. What is the solution of ∫ 𝑥2 ln 𝑥𝑑𝑥 ? Given C is an arbitrary constant.

Evaluate ∫ x² ln x dx Using Integration

53. The number of three letter words, with or without meaning, which can be formed using letters of the word ‘VIRUS’ without repetition of letters is (A) 30 (B) 40 (C) 60 (D) 120

Number of Three-Letter Words Formed From VIRUS

52. If φ(x) = x² and ψ(x) = 2ˣ, then ψ(φ(x)) is: (A) 2x² (B) x² (C) 22x (D) x2x

Find ψ(φ(x)) When φ(x) = x² and ψ(x) = 2ˣ

51. The value of: limx→2 (x3 − 8)/(x − 2) is _______. (in integer)

Evaluate the Limit of (x³ − 8)/(x − 2)

50. Three vectors are as follows: a⃗ = 3î − 10ĵ + 7k̂ b⃗ = −9î + 6ĵ − 47k̂ c⃗ = 11ĵ − 17k̂ The value of (a⃗ + b⃗) · c⃗ is _______. (A) 614 (B) 746 (C) 2 (D) 134

Find the Value of (a⃗ + b⃗) · c⃗ Using Vector

49. The value of: limn→∞ (3n + 5n + 4)/(4 + 2n²) is _______. (A) 0 (B) 0.75 (C) 1.5 (D) 3

Evaluate the Limit of (3n + 5n + 4)/(4 + 2n²)

48. Let a⃗ = 4î − 2ĵ + 6k̂ and b⃗ = 7î + ĵ − 12k̂. If: a⃗ × b⃗ = αî + βĵ + γk̂ then the value of α + β + γ equals _______.

Find α + β + γ Using the Cross Product

47. Select the value(s) of x for which the determinants of the two matrices are equal. (A) √3 (B) 3 (C) −1 (D) −√3

Reconstructed Determinant Equality Problem

46. Let N be the set of natural numbers and f : N → N be defined by: f(x) = x/2, if x is even f(x) = 3x + 1, if x is odd Let fn(x) denote the n-fold composition of f(x). What is the smallest integer n such that: fn(13) = 1?

Find the Smallest Integer n Such That fⁿ(13) = 1

45. Let XYZ be an equilateral triangle and let P, Q, R be the midpoints of YZ, XZ, and XY, respectively. If: r = Area(△PQR) / Area(△XYZ) then find the value of r.

Find the Ratio of Area of Triangle PQR

44. Let ƒ(x) = (x — 1)(x — 2)(x — 3)(x — 4) and let α = ƒ(3/2), β = ƒ (5/2) and γ = ƒ(7/2). Which of the following is/are CORRECT? (A) α and β have the same sign (B) α and γ have the same sign (C) β and γ have the same sign (D) αβ and βγ have the same sign

Determine the Signs of α, β and γ

43. Let U = {1, 2, … , 15}. Let P ⊆ U consist of all prime numbers, Q ⊆ U consist of all even numbers and R ⊆ U consist of all multiples of 3. Let T = P — Q. Then, which of the following is/are CORRECT? (A) |T| = 5 and |T ∪ R| = 9 (B) |T| = 6 and |T ∪ R| = 9 (C) |T| = 5 and |T ∩ R| = 1 (D) |T| = 6 and |T ∩ R| = 1

Find |T|, |T ∪ R| and |T ∩ R| Using Set Difference

42. Simplify: sin A/(1 + cos A) + (1 + cos A)/sin A (A) 2 sec A (B) 2 cosec A (C) sec A (D) cosec A

Simplify sin A/(1 + cos A) + (1 + cos A)/sin A

41. In how many ways can one write the elements 1, 2, 3, 4 in a sequence x1, x2, x3, x4 with xi G i 6i? (A) 9 (B) 10 (C) 11 (D) 12

Number of Ways to Arrange 1, 2, 3, 4

40. Let a = (√5 + 1)/2 and b = (√5 − 1)/2. Then, evaluate: limn→∞ (an + bn)/(an − bn) (A) is 1 (B) is 1/2 (C) is 0 (D) does not exist

Evaluate the Limit Involving

39. Let U = {1, 2, 3, 4, 5}. A subset S is chosen uniformly at random from the non-empty subsets of U. What is the probability that S does NOT have two consecutive elements? (A) 9/31 (B) 10/31 (C) 11/31 (D) 12/31

Probability That a Random Subset

38. Which one of the points P = (3/2, 1/2), Q = (1/2, 3/2), R = (3/2, 11/2), and S = (11/2, 3/2) lies above the parabola y = 2x² and inside the circle x² + y² = 4? (A) P (B) Q (C) R (D) S

Find the Point Above the Parabola and Inside the Circle

37. Let a⃗ = 4î − 2ĵ + 6k̂ and b⃗ = 7î + ĵ − 12k̂. If a⃗ × b⃗ = αî + βĵ + γk̂, then the value of α + β + γ equals _______.

Find α + β + γ Using the Cross Product of Two Vectors

36. In a geometric progression, the 3rd term is 36 and the 5th term is 324. The 7th term of the same progression will be _______. (in integer)

Geometric Progression 3rd and 5th Term Problem

35. Let N be the set of natural numbers and let f : N → N be defined by: f(x) = x/2, if x is even and f(x) = 3x + 1, if x is odd. Let fn(x) denote the n-fold composition of f(x). What is the smallest integer n such that:

Find the Smallest n Such That fⁿ(13) = 1

34. Let XYZ be an equilateral triangle and let P, Q, and R be the midpoints of YZ, XZ, and XY, respectively. If r = Area(△PQR) / Area(△XYZ), then the value of r is ______.

Area Ratio of Midpoint Triangle PQR to Equilateral Triangle XYZ

33. Let ƒ(x) = (x — 1)(x — 2)(x — 3)(x — 4) and let α = ƒ(3/2), β = ƒ (5/2) and γ = ƒ(7/2). Which of the following is/are CORRECT? (A) α and β have the same sign (B) α and γ have the same sign (C) β and γ have the same sign (D) αβ and βγ have the same sign

Find the Signs of α, β and γ for f(x) = (x − 1)(x − 2)(x − 3)(x − 4)

32. Let U = {1, 2, … , 15}. Let P ⊆ U consist of all prime numbers, Q ⊆ U consist of all even numbers and R ⊆ U consist of all multiples of 3. Let T = P — Q. Then, which of the following is/are CORRECT? (A) |T| = 5 and |T ∪ R| = 9 (B) |T| = 6 and |T ∪ R| = 9 (C) |T| = 5 and |T ∩ R| = 1 (D) |T| = 6 and |T ∩ R| = 1

Set Difference, Union and Intersection Problem

31. Simplify the following trigonometric expression: sin A/(1 + cos A) + (1 + cos A)/sin A (A) 2 sec A (B) 2 cosec A (C) sec A (D) cosec A

Simplify sin A/(1 + cos A) + (1 + cos A)/sin A

30. In how many ways can one write the elements 1, 2, 3, 4 in a sequence x1, x2, x3, x4 with xi G i 6i? (A) 9 (B) 10 (C) 11 (D) 12

Number of Ways to Arrange 1, 2, 3, 4 Such That xᵢ ≠ i for Every i

29. Let a = (√5 + 1)/2 and b = (√5 − 1)/2. Then, limn→∞ (an + bn)/(an − bn) is ______. (A) 1 (B) 1/2 (C) 0 (D) Does not exist

Evaluate lim n→∞ (aⁿ + bⁿ)/(aⁿ − bⁿ) for Golden Ratio Values

28. Let U = {1, 2, 3, 4, 5}. A subset S is chosen uniformly at random from the non-empty subsets of U. What is the probability that S does NOT have two consecutive elements? (A) 9/31 (B) 10/31 (C) 11/31 (D) 12/31

Probability That a Random Subset

27. Which one of the points P = (3/2, 1/2), Q = (1/2, 3/2), R = (3/2, 11/2), and S = (11/2, 3/2) lies above the parabola y = 2x2 and inside the circle x2 + y2 = 4? (2019) (A) P (B) Q (C) R (D) S

Find the Point Above y = 2x² and Inside the Circle x² + y² = 4

26. The length of the edge of a variable cube is increasing at the rate of 25 cm s⁻¹. If the initial length of the edge of the cube is 10 cm, the rate of increase of the surface area of the cube is _________ cm² s⁻¹. (answer in integer)

Rate of Increase of Surface Area of a Cube

25. The number of 7-letter words (with or without meaning) starting with the letter B that can be formed using the letters of the word BIOLOGY is ______________. (answer in integer)

Number of 7-Letter Words Starting with B

24. The area bounded by the curve y = sin x and the x-axis between x = 0 and x = 3π/2 is ______ sq. units (answer in integer).

Area Bounded by y = sin x and the x-Axis from 0 to 3π/2

23. The limit of the function limx→2 [(2x2 + 2x − 12)/(x2 − 4)] is __________. (rounded off to 1 decimal)

Limit of a Rational Function as x Approaches 2

22. If P = ( cos α sin α −sin α cos α ) and P + PT = I, then the value of α, where 0 ≤ α ≤ π/2, is ______. (A) π/2 (B) π/3 (C) 3π/2 (D) 0

Find α If P + Pᵀ = I for a 2×2 Trigonometric Matrix

21. If a fair coin is tossed two times, the probability that the first or second toss Swill be heads is __________ (rounded off to two decimal places).

Probability That the First or Second Toss Will Be Heads

20. The value of limx→3 (x2 − 9)/(x2 − 4x + 3) is ______ (rounded off to the nearest integer).

Evaluate lim x→3 (x² − 9)/(x² − 4x + 3)

19. For a given square, if the area of its incircle is 100 cm², then the area of its circumcircle is cm² (rounded off to the nearest integer).

Area of Circumcircle of a Square When Incircle Area Is 100 cm²

18. If a variable Z shows a standard normal distribution, then the percent probability that 0 ≤ Z ≤ 1 is ______ (rounded off to the nearest integer). (A) 34 (B) 68 (C) 95 (D) 99

Percent Probability That 0 ≤ Z ≤ 1 in a Standard Normal Distribution

17. A random variable X and its probability distribution are given below. The value of P(X < 5) is ______ (rounded off to one decimal place). X 0 1 2 3 4 5 P(X) 0 k 2k 3k 6k 8k

Find P(X < 5) from a Discrete Probability Distribution

16. Given data consists of distinct values of xi occurring with frequencies fi. The mean value for the data is ______ (rounded off to one decimal place). (2023) The given data is: xi 5 6 8 10 fi 8 12 10 12

Calculate the Mean of Distinct Values with Frequencies

15. The value of limx→−3 (2x + 6)/(x + 3) is ______.

Evaluate lim x→−3 (2x + 6)/(x + 3)

14. The order of the differential equation d3y/dx3 + 2d2y/dx2 − 3dy/dx + 6x4y = 0 is ______.

Find the Order of the Differential Equation

13. A restriction endonuclease has a recognition site of 3 bases. Assuming random arrangement of nucleotides, the probability that this endonuclease will cut a piece of DNA is ______ (rounded off to three decimal places).

Probability of DNA Cutting by a Restriction Endonuclease

12. The number of possible unique combination(s) of linear tetrapeptides that can be made from four different amino acids using each amino acid only once in the chain is/are ______.

Number of Unique Linear Tetrapeptides

11. Given the following sets: A = {2, 4, 6, 8, 10, 12} B = {8, 10, 12, 14, 16, 18} C = {7, 8, 9, 10, 11, 12, 13} Find: (A ∩ B) ∪ (B ∩ C) (A) {8, 10, 12, 14} (B) {8, 10, 12} (C) {7, 8, 10, 11, 12, 13, 14} (D) {4, 6, 7, 8, 10, 11, 12, 13}

Solve (A ∩ B) ∪ (B ∩ C) for Given Sets

10. Given that A = (sin θ cos θ tan θ + sin θ cos θ cot θ) the value of A is ______.

Find the Value of A = sin θ cos θ tan θ + sin θ cos θ cot θ

9. The smallest positive (non-zero) integer “n” for which the expression ((1 + i)/(1 − i))n = 1 holds true is ______.

Smallest Positive Integer n for ((1 + i)/(1 – i))^n = 1

8. The average of all positive even integers less than or equal to 40 is ___.

Average of All Positive Even Integers Less Than or Equal to 40

7. A deck of ten cards is given to you as shown below in the figure. One card is drawn at random from this deck. The probability of selecting a number less than 9 is ____ (to one decimal place).

Probability of Selecting a Number Less Than 9 from a Deck

6. The equation sin(θ/2) [sin(θ/2) + cos(θ/2)] = β has a solution, where β is a natural number. Then β is ______.

Find β in the Trigonometric Equation

5. The distance between the parallel lines 2x + 5y = 7 and 2x + 5y = 15 is ______ (rounded off to 2 decimals).

Distance Between Parallel Lines 2x + 5y = 7 and 2x + 5y = 15

4. A function f : D → ℝ is defined as f(x) = x2 + 1 x2 + x + 1 where D ⊆ ℝ is the domain. The domain(s) on which the function f(x) is one-to-one is/are: (A) Natural numbers (B) Integers (C) Rational numbers (D) Irrational numbers

Domain on Which f(x) = (x² + 1)/(x² + x + 1) Is One-to-One

3. The value of the integral ∫04 (x − f(x)) dx where the function f(x) is defined as: f(x) = ⎧ 0, 0 ≤ x < 1 ⎪ 1, 1 ≤ x < 2 ⎪ 2, 2 ≤ x < 3 ⎪ 3, 3 ≤ x < 4 ⎩ 4, 4 ≤ x < 5 is: (A) 2 (B) 1 (C) −1 (D) −2

Evaluate ∫₀⁴(x − f(x)) dx for a Piecewise Function

2. If x + 1/x = 1, then the value of x6 + 1/x6 is: (A) −2 (B) −1 (C) 1 (D) 2

If x + 1/x = 1, Find x⁶ + 1/x⁶

1. Let A = ⎛ 2 1 ⎞ ⎝ 1 1 ⎠ and B = ⎛ 2 −5 ⎞ ⎝ 0 1 ⎠ If AX + 3B = 0, then the determinant of X is: (A) −18 (B) −6 (C) 6 (D) 18

Determinant of Matrix X When AX + 3B = 0

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