BPSC TRE 3.0 Mathematics Question Paper (Class 11–12, 22 July 2024) with Official Answers

BPSC TRE 3.0 Mathematics question paper (Class 11–12, 22 July 2024): all 80 Mathematics questions from Part III with BPSC’s official revised final answers and a short explanation for each.

BPSC TRE 4.0 is now open for 33,320 teacher posts, 16,540 of them for Class 11–12. Your subject part is 80 of the 150 marks, so solving the last paper is the best way to see the level BPSC sets for Mathematics.

Table of Contents

• Exam overview
• Mathematics questions with answers (Q71–150)
• How to prepare
• FAQs

BPSC TRE 3.0 Mathematics (Class 11–12): Exam Overview

Exam	BPSC School Teacher (Re-)Examination, TRE 3.0 (Advt. No. 22/2024)
Subject	Mathematics, Class 11–12
Exam date	22 July 2024, 1st sitting
Paper	150 questions, 150 marks, 2 hours 30 minutes
Part I	Language (qualifying): Q1–30
Part II	General Studies: Q31–70 (solve here)
Part III	Mathematics: Q71–150 (on this page)
Answers	Revised Final Answer Key, notice dated 6 November 2024 (Set-E column). Changes vs Final Answer Key of 17 October 2024 for Q71–150: Q84 A → E. Q83 deleted in both., booklet Series E
Deleted by BPSC	Q83

BPSC TRE 3.0 Mathematics Questions with Answers (Q71–150)

Question numbers are as in the original booklet. Answers follow BPSC’s revised final answer key; explanations are by the ExamYaari team.

Q71. If A is any square matrix, then A + A′ is always

(A) Symmetric
(B) Anti Symmetric
(C) Anti Hermitian
(D) More than one of the above
(E) None of the above

Answer: (A) Symmetric
Explanation: (A + A′)′ = A′ + A = A + A′, so A + A′ equals its own transpose and is always symmetric.

Q72. If Aₘₓₙ and Bₚₓq are any two matrices and their product is defined as Aₘₓₙ · Bₚₓq = Cₘₓq, then which one is true?

(A) n < p
(B) n > p
(C) n = p
(D) More than one of the above
(E) None of the above

Answer: (C) n = p
Explanation: A product AB is defined only when the number of columns of A equals the number of rows of B, so n = p.

Q73. If A = [[1, 2], [−1, 1]], then the value of A² − 2A + 2I is equal to

(A) I
(B) O
(C) −I
(D) More than one of the above
(E) None of the above

Answer: (C) −I
Explanation: A² = [[−1, 4], [−2, −1]], −2A = [[−2, −4], [2, −2]] and 2I = [[2, 0], [0, 2]]. Adding gives [[−1, 0], [0, −1]] = −I.

Q74. The vectors (1, 0, 0), (0, 1, −1) and (0, 1, 2) are

(A) Linearly Dependent
(B) Linearly Independent
(C) Not confirm
(D) More than one of the above
(E) None of the above

Answer: (B) Linearly Independent
Explanation: The determinant of these three vectors is 1 × (1·2 − (−1)·1) = 3 ≠ 0, so they are linearly independent.

Q75. The value of determinants Δ₁ = |2 0 0; 3 4 0; 5 6 8| and Δ₂ = |4 4 8; 0 8 2; 0 0 2|, then which option is correct?

(A) Δ₁ > Δ₂
(B) Δ₁ < Δ₂
(C) Δ₁ = Δ₂
(D) More than one of the above
(E) None of the above

Answer: (C) Δ₁ = Δ₂
Explanation: Both are triangular determinants, so each equals the product of its diagonal: Δ₁ = 2 × 4 × 8 = 64 and Δ₂ = 4 × 8 × 2 = 64. So Δ₁ = Δ₂.

Q76. Set of square matrices of same order follows following laws

(A) Associative law
(B) Identity law
(C) Commutative law
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: Multiplication of square matrices of the same order is associative and has an identity (I), but is not commutative in general. Two options hold, so the answer is ‘More than one of the above’.

Q77. If A is any set of n elements, the number of all subsets of set A are

(A) n²
(B) 2ⁿ
(C) 2n
(D) More than one of the above
(E) None of the above

Answer: (B) 2ⁿ
Explanation: Each of the n elements is either in or out of a subset, giving 2 × 2 × … × 2 = 2ⁿ subsets.

Q78. If α + β = π, then value of (cos α + i sin α)(cos β + i sin β) is

(A) 0
(B) 1
(C) −1
(D) More than one of the above
(E) None of the above

Answer: (C) −1
Explanation: By De Moivre, the product equals cos(α + β) + i sin(α + β) = cos π + i sin π = −1.

Q79. If R is the set of real numbers and a mapping f : R → R is defined as f(x) = 2x + 1, ∀x ∈ R, then f is

(A) Many one and onto
(B) One-one and onto
(C) One-one and into
(D) More than one of the above
(E) None of the above

Answer: (B) One-one and onto
Explanation: 2x₁ + 1 = 2x₂ + 1 gives x₁ = x₂, so f is one-one; and any real y comes from x = (y − 1)/2, so f is onto.

Q80. If R is the set of real numbers and a mapping f : R → R is defined as f(x) = x − 2, ∀x ∈ R, then which elements satisfy the above relation?

(A) (4, 2)
(B) (8, 6)
(C) (5, 3)
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: Each pair (x, f(x)) must have second entry = first − 2: 4 − 2 = 2, 8 − 2 = 6 and 5 − 2 = 3. All three fit, so the answer is ‘More than one of the above’.

Q81. The sum of two roots of a quadratic equation is 5 and product is 6, then the equation will be

(A) x² − 5x + 6 = 0
(B) x² + 5x − 6 = 0
(C) x² − 6x + 5 = 0
(D) More than one of the above
(E) None of the above

Answer: (A) x² − 5x + 6 = 0
Explanation: A quadratic is x² − (sum)x + (product) = 0, so it is x² − 5x + 6 = 0.

Q82. If a + ib is one root of a polynomial f(x), then the value of f(a − ib) will be

(A) a − ib
(B) a
(C) b
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: For a polynomial with real coefficients, complex roots come in conjugate pairs, so a − ib is also a root and f(a − ib) = 0. Zero is not among the options.

Q83. The quadratic equation whose roots are inverse of roots of equation x² − 2x + 3 = 0, will be

(A) 2x² − 3x + 1 = 0
(B) 6x² + 4x + 2 = 0
(C) 3x² + 2x + 1 = 0
(D) More than one of the above
(E) None of the above

Answer: Question deleted (BPSC removed it in the final answer key)
Explanation: BPSC deleted this question in the final answer key. Replacing x by 1/x gives 3x² − 2x + 1 = 0, which matches option (C) in the Hindi version, but the English version prints it as 3x² + 2x + 1 = 0.

Q84. Number of positive roots of polynomial x³ − 3x² + x − 3 = 0, is

(A) At most 3 positive roots
(B) At most 2 positive roots
(C) All complex roots
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: x³ − 3x² + x − 3 = (x − 3)(x² + 1), so the roots are 3, i and −i: exactly one positive root. As per BPSC’s revised final key, the answer is (E); the earlier final key had given (A).

Q85. Solution of congruence relation 2x ≡ 1 (mod 7) is

(A) 4
(B) 2
(C) 1
(D) More than one of the above
(E) None of the above

Answer: (A) 4
Explanation: Try x = 4: 2 × 4 = 8 = 7 + 1, so 8 ≡ 1 (mod 7). Hence x = 4.

Q86. If [·] represents greatest integer function then the value of [5.7] + [3.2] is

(A) 9
(B) 10
(C) 8
(D) More than one of the above
(E) None of the above

Answer: (C) 8
Explanation: [5.7] = 5 and [3.2] = 3, so the sum is 8.

Q87. The greatest common divisor of f(x) = x³ − 12x² + 47x − 60 and g(x) = x² − 5x + 6, will be

(A) x − 1
(B) x − 2
(C) x − 3
(D) More than one of the above
(E) None of the above

Answer: (C) x − 3
Explanation: g(x) = (x − 2)(x − 3). f(2) = 8 − 48 + 94 − 60 = −6 ≠ 0 but f(3) = 27 − 108 + 141 − 60 = 0, so the common factor is x − 3.

Q88. If “a” is divided by “p” then remainder is 5. If a² is divided by “p” then remainder will be

(A) 5
(B) 25
(C) 125
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: a² leaves the same remainder as 25 when divided by p, and that depends on p (e.g. p = 7 gives 4, p = 6 gives 1). No single option is always right, so BPSC’s key gives (E).

Q89. Two numbers are in the ratio 2:3, if the product of LCM and HCF is 294, then the numbers will be

(A) 2, 3
(B) 14, 21
(C) 10, 15
(D) More than one of the above
(E) None of the above

Answer: (B) 14, 21
Explanation: LCM × HCF = product of the numbers, so 2k × 3k = 6k² = 294, giving k² = 49 and k = 7. The numbers are 14 and 21.

Q90. Which of the following is a set

(A) Set of Vowels in alphabets
(B) Collection of months in a year
(C) Set of natural numbers less than 10
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: All three are well-defined collections (anyone can say exactly what belongs), so all are sets and the answer is ‘More than one of the above’.

Q91. If A and B are two non empty sets then which option is correct

(A) A − B = A ∩ (Bᶜ)
(B) A − B = B ∩ (Aᶜ)
(C) A − B = (A ∩ B)ᶜ
(D) More than one of the above
(E) None of the above

Answer: (A) A − B = A ∩ (Bᶜ)
Explanation: A − B is the set of elements in A but not in B, which is exactly A ∩ Bᶜ.

Q92. If A = {1, 2, 3} and B = {4, 5, 6} the maximum number of possible functions from A to B are

(A) 3
(B) 6
(C) 27
(D) More than one of the above
(E) None of the above

Answer: (C) 27
Explanation: Each of the 3 elements of A can go to any of 3 elements of B, so there are 3³ = 27 functions.

Q93. The value of lim x→0 (e^2x − 1)/(5x), is

(A) 0
(B) 1/5
(C) 2/5
(D) More than one of the above
(E) None of the above

Answer: (C) 2/5
Explanation: Write it as (2/5) × (e^2x − 1)/(2x). Since (e^t − 1)/t → 1 as t → 0, the limit is 2/5.

Q94. The value of lim x→0 (log sin 2x)/(log sin x), is

(A) 1
(B) 2
(C) 4
(D) More than one of the above
(E) None of the above

Answer: (A) 1
Explanation: It is of the form ∞/∞, so by L’Hôpital’s rule the limit = lim (2 cot 2x)/(cot x) = lim (2 tan x)/(tan 2x) = 2 × (1/2) = 1.

Q95. Which one is not differentiable at point x = 1,

(A) f(x) = |x − 1|, x ∈ R
(B) f(x) = [x], x ∈ R
(C) f(x) = 1 + (1 − x)^1/3, x ∈ R
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: |x − 1| has a sharp corner at x = 1, [x] jumps (is not even continuous) at x = 1, and (1 − x)^1/3 has a vertical tangent there (derivative → ∞). None is differentiable at x = 1, so the answer is ‘More than one of the above’.

Q96. Differentiation of xˣ with respect to x, is

(A) xˣ(1 + logₑ x)
(B) xˣ(1 − logₑ x)
(C) x⁻ˣ
(D) More than one of the above
(E) None of the above

Answer: (A) xˣ(1 + logₑ x)
Explanation: Let y = xˣ, so log y = x log x. Differentiating, (1/y) dy/dx = log x + 1, hence dy/dx = xˣ(1 + logₑ x).

Q97. Value of integral ∫₋₁¹ sin⁵x · cos⁴x dx, is

(A) π/2
(B) π/4
(C) 0
(D) More than one of the above
(E) None of the above

Answer: (C) 0
Explanation: sin⁵x · cos⁴x is an odd function (changing x to −x changes its sign), and the limits −1 to 1 are symmetric, so the integral is 0.

Q98. The value of ∫₀^π/2 √(sin x)/(√(sin x) + √(cos x)) dx, is

(A) π/4
(B) π/2
(C) π
(D) More than one of the above
(E) None of the above

Answer: (A) π/4
Explanation: Replacing x by π/2 − x gives I = ∫₀^π/2 √(cos x)/(√(cos x) + √(sin x)) dx. Adding the two forms, 2I = ∫₀^π/2 1 dx = π/2, so I = π/4.

Q99. The value of ∫ √(1 + sin x) dx, is

(A) sin(x/2) + cos(x/2)
(B) 2(cos(x/2) − sin(x/2))
(C) 2(sin(x/2) − cos(x/2))
(D) More than one of the above
(E) None of the above

Answer: (C) 2(sin(x/2) − cos(x/2))
Explanation: 1 + sin x = (sin(x/2) + cos(x/2))², so the integrand is sin(x/2) + cos(x/2). Integrating gives −2cos(x/2) + 2sin(x/2) = 2(sin(x/2) − cos(x/2)) + C.

Q100. The area enclosed by the circle x² + y² = 16, in the first quadrant is

(A) π
(B) 2π
(C) 3π
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: Radius = 4, so the full area is 16π and the first-quadrant part is 16π/4 = 4π, which is not among the options.

Q101. If x and y are any two real numbers, x > 0 there exists a positive integer n, then Archimedean property is

(A) nx < y
(B) nx > y
(C) nx = y
(D) More than one of the above
(E) None of the above

Answer: (B) nx > y
Explanation: The Archimedean property says that for any real x > 0 and any real y, some positive integer n makes nx > y.

Q102. Consider a function f(x) = { x³ + 3 ; x ≠ 0 and 1 ; x = 0 }, then which is true?

(A) f(x) is continuous at x = 0
(B) f(x) is differentiable at x = 0
(C) Could not say about continuity
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: As x → 0, f(x) → 0 + 3 = 3, but f(0) = 1. So f is not continuous at 0, and hence not differentiable there; we can clearly say it is discontinuous. None of A–C is true.

Q103. Which statement is not correct?

(A) Every continuous function is differentiable
(B) Every differentiable function is continuous
(C) Every polynomial is continuous function for real number
(D) More than one of the above
(E) None of the above

Answer: (A) Every continuous function is differentiable
Explanation: |x| is continuous at 0 but not differentiable there, so statement (A) is false. (B) and (C) are true.

Q104. Which option is not correct?

(A) Every bounded above set exists its supremum
(B) Every bounded below set exists its infimum
(C) Every bounded set exist its suprimum and infimum
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: By the completeness property of real numbers, every (non-empty) set bounded above has a supremum and every set bounded below has an infimum, so a bounded set has both. All three statements are correct, so the answer is (E).

Q105. Order and degree of the equation d²y/dx² = (dy/dx)^1/3 + y, is

(A) (2, 1)
(B) (3, 2)
(C) (2, 3)
(D) More than one of the above
(E) None of the above

Answer: (C) (2, 3)
Explanation: Rewrite as d²y/dx² − y = (dy/dx)^1/3 and cube both sides: (d²y/dx² − y)³ = dy/dx. The highest derivative is second order and its power is 3, so (order, degree) = (2, 3).

Q106. What is the equation of a curve that passes through (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is 2x/y²?

(A) y³ = 3x² + 15
(B) y² = 3x³ + 5
(C) x² + y² = 25
(D) More than one of the above
(E) None of the above

Answer: (A) y³ = 3x² + 15
Explanation: dy/dx = 2x/y² gives y² dy = 2x dx, so y³/3 = x² + c. At (−2, 3): 9 = 4 + c, so c = 5 and y³ = 3x² + 15.

Q107. Solution of the differential equation dy/dx − y/x = 2x², is

(A) y = x² + c
(B) y = x + c
(C) y = x³ + cx
(D) More than one of the above
(E) None of the above

Answer: (C) y = x³ + cx
Explanation: This is linear with integrating factor e^−∫dx/x = 1/x. Then d(y/x)/dx = 2x, so y/x = x² + c, i.e. y = x³ + cx.

Q108. General solution of equation y = px + p², where p = dy/dx; is

(A) y = cx + 2c
(B) y = cx + c²
(C) y = x² + c
(D) More than one of the above
(E) None of the above

Answer: (B) y = cx + c²
Explanation: This is Clairaut’s equation y = px + f(p); its general solution is obtained by putting p = c, giving y = cx + c².

Q109. The particular solution of dy/dx = −4xy², given that y(0) = 1; is

(A) (2x² + 1)y = 1
(B) (2y² + 1)x = 1
(C) x² + 2y² = 1
(D) More than one of the above
(E) None of the above

Answer: (A) (2x² + 1)y = 1
Explanation: Separate: dy/y² = −4x dx, so −1/y = −2x² + c. y(0) = 1 gives c = −1, so 1/y = 2x² + 1, i.e. (2x² + 1)y = 1.

Q110. How many arbitrary constants are there in the general solution of the differential equation of order three (3)?

(A) 1
(B) 2
(C) 3
(D) More than one of the above
(E) None of the above

Answer: (C) 3
Explanation: The general solution of an n-th order differential equation has n arbitrary constants, so order 3 gives 3 constants.

Q111. The locus of solution of differential equation x dx − y dy = 0, represents a family of

(A) Ellipse
(B) Hyperbola
(C) Parabola
(D) More than one of the above
(E) None of the above

Answer: (B) Hyperbola
Explanation: Integrating gives x²/2 − y²/2 = c, i.e. x² − y² = k, which is a family of (rectangular) hyperbolas.

Q112. Solution of differential equation x dy + y dx = 0 is a curve, which is symmetric with respect to

(A) x = 0
(B) y = 0
(C) y = x
(D) More than one of the above
(E) None of the above

Answer: (C) y = x
Explanation: x dy + y dx = d(xy) = 0, so xy = c. Swapping x and y leaves xy = c unchanged, so the curve is symmetric about y = x (but not about either axis).

Q113. The area included between the parabola y² = 4x and x² = 4y is

(A) 16/3
(B) 15/4
(C) 8/3
(D) More than one of the above
(E) None of the above

Answer: (A) 16/3
Explanation: The curves meet at (0, 0) and (4, 4). Area = ∫₀⁴ (2√x − x²/4) dx = (4/3)·8 − 64/12 = 32/3 − 16/3 = 16/3.

Q114. The nature of function f(x) = sin 3x, in [0, π/6] is

(A) Decreasing
(B) Increasing
(C) Neither decreasing nor increasing
(D) More than one of the above
(E) None of the above

Answer: (B) Increasing
Explanation: For x in [0, π/6], 3x lies in [0, π/2], where sine rises from 0 to 1; also f′(x) = 3cos 3x ≥ 0. So f is increasing.

Q115. Minimum value of f(x) = |x − 3| attains at the point

(A) x = 1
(B) x = 2
(C) x = 3
(D) More than one of the above
(E) None of the above

Answer: (C) x = 3
Explanation: |x − 3| ≥ 0 and equals 0 only when x = 3, so the minimum is at x = 3.

Q116. Maximum value of f(x) = 3x⁴ + 4x³ − 12x² + 12, is

(A) 19
(B) 12
(C) 09
(D) More than one of the above
(E) None of the above

Answer: (B) 12
Explanation: f′(x) = 12x³ + 12x² − 24x = 12x(x + 2)(x − 1), so the critical points are 0, −2 and 1. f″(0) = −24 < 0, so x = 0 gives the (local) maximum value f(0) = 12.

Q117. The value of ∫ log x dx, is

(A) xˣ + c
(B) x(log x + 1)x + c
(C) x(log x − 1) + c
(D) More than one of the above
(E) None of the above

Answer: (C) x(log x − 1) + c
Explanation: By parts with u = log x, dv = dx: ∫ log x dx = x log x − ∫ x · (1/x) dx = x log x − x + c = x(log x − 1) + c.

Q118. Value of (1 + i)⁴ is

(A) −4
(B) 4e^iπ
(C) −4e^−iπ/2
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: (1 + i)² = 2i, so (1 + i)⁴ = (2i)² = −4. Also 4e^iπ = 4(−1) = −4, while −4e^−iπ/2 = −4(−i) = 4i. Options A and B are both right, so the answer is ‘More than one of the above’.

Q119. Argument of i is

(A) π/4
(B) π/2
(C) π
(D) More than one of the above
(E) None of the above

Answer: (B) π/2
Explanation: i = 0 + 1·i lies on the positive imaginary axis, so its argument is π/2.

Q120. Modulus of a + 3i is 5, then the value of a will be

(A) 4
(B) −4
(C) 2
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: √(a² + 9) = 5 gives a² = 16, so a = 4 or −4. Options A and B both work, so the answer is ‘More than one of the above’.

Q121. Value of 1³ + 2³ + 3³ + ……… + n³, is

(A) n(n + 1)/2
(B) n(n + 1)(n + 2)/6
(C) n(n + 1)(2n + 1)/6
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: The sum of the first n cubes is [n(n + 1)/2]², which is not among the options. (Option C is the sum of squares.)

Q122. The principal value of iⁱ, is

(A) e^−π/2
(B) e^π/2
(C) 1
(D) More than one of the above
(E) None of the above

Answer: (A) e^−π/2
Explanation: Principal value: i = e^iπ/2, so iⁱ = e^i·iπ/2 = e^−π/2, a real number.

Q123. If A′ is set of all limit points of a set A and Ā is closure of A, then which option is correct

(A) Ā = A ∪ A′
(B) Ā = A ∩ A′
(C) Ā = A′ ∪ A
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: The closure of A is A together with its limit points: Ā = A ∪ A′. Since union is commutative, A′ ∪ A is the same set, so A and C are both correct.

Q124. If x ∈ (a, b], then which option is correct

(A) a ≤ x < b
(B) a < x ≤ b
(C) a ≤ x ≤ b
(D) More than one of the above
(E) None of the above

Answer: (B) a < x ≤ b
Explanation: In (a, b] the round bracket excludes a and the square bracket includes b, so a < x ≤ b.

Q125. Which one is null set

(A) { }
(B) {φ}
(C) {a}
(D) More than one of the above
(E) None of the above

Answer: (A) { }
Explanation: { } has no elements, so it is the null set. {φ} contains one element (the empty set) and {a} contains a, so neither is empty.

Q126. The set [−1, 1] − {0} is equal to

(A) [−1, 0] ∪ [0, 1]
(B) (−1, 0) ∪ (0, 1)
(C) [−1, 0) ∪ (0, 1]
(D) More than one of the above
(E) None of the above

Answer: (C) [−1, 0) ∪ (0, 1]
Explanation: Removing only 0 keeps both endpoints −1 and 1, so the set is [−1, 0) ∪ (0, 1].

Q127. Which of the following is unit vector along 3j + 4k

(A) (1/5)(3j + 4k)
(B) (1/7)(3j + 4k)
(C) i + j + k
(D) More than one of the above
(E) None of the above

Answer: (A) (1/5)(3j + 4k)
Explanation: |3j + 4k| = √(9 + 16) = 5, so the unit vector is (1/5)(3j + 4k).

Q128. For what value of “a” are vectors 2i − j + k, i − 3j − 5k and 3i − 4j + ak coplanar

(A) 4
(B) −4
(C) 5
(D) More than one of the above
(E) None of the above

Answer: (B) −4
Explanation: Coplanar means |2 −1 1; 1 −3 −5; 3 −4 a| = 0: 2(−3a − 20) + (a + 15) + (−4 + 9) = −5a − 20 = 0, so a = −4.

Q129. If position vectors of two points A and B are i + 2j − k and −i + j + k respectively. Point Q divides line AB externally in ratio 2 : 1, then position vector of the point Q is

(A) −3i + 3k
(B) −3j + 3k
(C) 3i − 3j − 3k
(D) More than one of the above
(E) None of the above

Answer: (A) −3i + 3k
Explanation: External division 2 : 1 gives Q = (2B − A)/(2 − 1) = 2(−i + j + k) − (i + 2j − k) = −3i + 0j + 3k.

Q130. If a⃗ + b⃗ + c⃗ = 0⃗ and |a⃗| = 3, |b⃗| = 5 and |c⃗| = 7, then the angle between a⃗ and b⃗ is

(A) π
(B) π/2
(C) π/3
(D) More than one of the above
(E) None of the above

Answer: (C) π/3
Explanation: c⃗ = −(a⃗ + b⃗), so 49 = 9 + 25 + 2·3·5 cos θ, giving cos θ = 15/30 = 1/2 and θ = π/3.

Q131. If a⃗ = 5i − j − 3k and b⃗ = i + 3j − 5k, then a⃗ + b⃗ and a⃗ − b⃗ are

(A) Parallel
(B) Perpendicular
(C) Inclined with 45°
(D) More than one of the above
(E) None of the above

Answer: (B) Perpendicular
Explanation: (a⃗ + b⃗)·(a⃗ − b⃗) = |a⃗|² − |b⃗|² = 35 − 35 = 0, so the two vectors are perpendicular.

Q132. If angular points of a triangle are A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1), then area of triangle in square unit is

(A) √21
(B) √11/2
(C) √21/2
(D) More than one of the above
(E) None of the above

Answer: (C) √21/2
Explanation: AB⃗ = j + 2k and AC⃗ = i + 2j, so AB⃗ × AC⃗ = −4i + 2j − k with magnitude √21. Area = (1/2)√21 = √21/2.

Q133. The equation of the line which passes through the point (1, 2, 3) and parallel to vector 3i + 2j − 2k, is

(A) r⃗ = i + 2j + 3k + λ(3i + 2j − 2k), λ ∈ R
(B) r⃗ = 3i + 2j − 2k + λ(i + 2j + 3k), λ ∈ R
(C) r⃗ = 6i + λ(3i + 2j − 2k), λ ∈ R
(D) More than one of the above
(E) None of the above

Answer: (A) r⃗ = i + 2j + 3k + λ(3i + 2j − 2k), λ ∈ R
Explanation: A line through point a⃗ parallel to b⃗ is r⃗ = a⃗ + λb⃗, so r⃗ = i + 2j + 3k + λ(3i + 2j − 2k). Option C does not pass through (1, 2, 3).

Q134. For what value of “a” are the lines (x − 5)/7 = (y + 2)/a = z/1 and x/1 = y/2 = z/3 perpendicular

(A) −5
(B) 5
(C) 4
(D) More than one of the above
(E) None of the above

Answer: (A) −5
Explanation: For perpendicular lines the dot product of direction ratios is zero: 7·1 + a·2 + 1·3 = 0, so 2a = −10 and a = −5.

Q135. The region formed by constraints 5x + y ≤ 100, x + y ≤ 60 and x ≥ 0, y ≥ 0 is

(A) infeasible
(B) feasible
(C) not confirm
(D) More than one of the above
(E) None of the above

Answer: (B) feasible
Explanation: Points such as (0, 0) or (10, 40) satisfy all four constraints, so the region is non-empty, i.e. feasible.

Q136. The value of Maximize Z = 4x + y subjected to conditions x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0, is

(A) 40
(B) 80
(C) 120
(D) More than one of the above
(E) None of the above

Answer: (C) 120
Explanation: Corner points are (0, 0), (30, 0), (20, 30) and (0, 50), giving Z = 0, 120, 110 and 50. The maximum is 120 at (30, 0).

Q137. The optimal value of the objective function is attained at the point

(A) In-side the feasible region
(B) Centre of the feasible region
(C) Out side of the feasible region
(D) More than one of the above
(E) None of the above

Answer: (E) None of the above
Explanation: By the corner-point theorem, the optimal value of a linear objective function is attained at a corner (vertex) of the feasible region, which none of A–C says.

Q138. Solution by simplex method is given by

(A) G.B. Dantzig
(B) L. Katorovich
(C) T.C. Koopmans
(D) More than one of the above
(E) None of the above

Answer: (A) G.B. Dantzig
Explanation: The simplex method for solving linear programming problems was developed by George B. Dantzig in 1947.

Q139. The objective function of linear programming problem is

(A) A function to be maximize
(B) A function to be minimize
(C) A function to be neither maximize nor minimize
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: The objective function is the linear function we optimise, which may be maximised (e.g. profit) or minimised (e.g. cost). Both A and B fit, so the answer is ‘More than one of the above’.

Q140. Which of the following set is convex

(A) {(x, y) : x² + y² ≥ 1}
(B) {(x, y) : y² ≥ x}
(C) {(x, y) : y ≥ 2, y ≤ 4}
(D) More than one of the above
(E) None of the above

Answer: (C) {(x, y) : y ≥ 2, y ≤ 4}
Explanation: The strip 2 ≤ y ≤ 4 contains the whole segment between any two of its points, so it is convex. In (A), the segment from (−1, 0) to (1, 0) passes through (0, 0), which is outside the set; in (B), (1, 1) and (1, −1) are in the set but (1, 0) is not.

Q141. If A and B are two events such that P(A) = .6, P(B) = .3, P(A ∩ B) = .2, then P(A/B) is

(A) 1/2
(B) 2/3
(C) 1/3
(D) More than one of the above
(E) None of the above

Answer: (B) 2/3
Explanation: P(A/B) = P(A ∩ B)/P(B) = 0.2/0.3 = 2/3.

Q142. A pair of dice is thrown then the probability of getting 7 as the sum, is

(A) 1/6
(B) 5/6
(C) 1
(D) More than one of the above
(E) None of the above

Answer: (A) 1/6
Explanation: Sum 7 comes from (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): 6 out of 36 outcomes, so the probability is 1/6.

Q143. The probability that a leap year will have 53 Fridays, is

(A) 1/7
(B) 2/7
(C) 3/7
(D) More than one of the above
(E) None of the above

Answer: (B) 2/7
Explanation: A leap year has 52 weeks and 2 extra days. Of the 7 possible pairs of extra days, 2 contain a Friday (Thu–Fri, Fri–Sat), so the probability is 2/7.

Q144. The number of permutations of the letters of the word ALLAHABAD, is

(A) 5400
(B) 4500
(C) 7560
(D) More than one of the above
(E) None of the above

Answer: (C) 7560
Explanation: ALLAHABAD has 9 letters with A four times and L twice, so the count is 9!/(4! × 2!) = 362880/48 = 7560.

Q145. If A and B are events such that P(A/B) = P(B/A), then

(A) A ⊂ B
(B) P(A) = P(B)
(C) P(A) < P(B)
(D) More than one of the above
(E) None of the above

Answer: (B) P(A) = P(B)
Explanation: P(A ∩ B)/P(B) = P(A ∩ B)/P(A), and cancelling P(A ∩ B) (taken non-zero) gives P(A) = P(B).

Q146. A group consists of 8 girls and 7 boys. In how many ways can a team of 4 members be selected if team has no girls.

(A) 35
(B) 25
(C) 15
(D) More than one of the above
(E) None of the above

Answer: (A) 35
Explanation: All 4 members must be boys, chosen from 7: ⁷C₄ = (7 × 6 × 5 × 4)/(4 × 3 × 2 × 1) = 35.

Q147. There are 6 boys and 3 girls and they are sitting on a circular table, in how many ways they can sit with no restrictions,

(A) 9!
(B) 8!
(C) 7!
(D) More than one of the above
(E) None of the above

Answer: (B) 8!
Explanation: There are 9 people in all, and n people can sit around a circular table in (n − 1)! ways, so the answer is (9 − 1)! = 8!.

Q148. If average of 10 entries is 50, if we reduce each entry by 5, then new mean will be

(A) 55
(B) 50
(C) 45
(D) More than one of the above
(E) None of the above

Answer: (C) 45
Explanation: Subtracting the same number from every entry lowers the mean by that number: 50 − 5 = 45.

Q149. Which of the following option is correct?

(A) Mean − Mode = 3 (Mean − Median)
(B) 3 Median = Mode + 2 Mean
(C) Mean − Mode = Median
(D) More than one of the above
(E) None of the above

Answer: (D) More than one of the above
Explanation: The empirical relation is Mode = 3 Median − 2 Mean, i.e. 3 Median = Mode + 2 Mean (B). Rearranging, Mean − Mode = 3Mean − 3Median = 3(Mean − Median) (A). Both A and B are correct.

Q150. The coefficient of x⁶ in expansion of (2x − 3)⁶ is

(A) 8
(B) 16
(C) 64
(D) More than one of the above
(E) None of the above

Answer: (C) 64
Explanation: The x⁶ term is (2x)⁶ = 64x⁶, so the coefficient is 64.

How to Prepare for BPSC TRE Mathematics

• Solve this paper in one sitting with a timer: about 1 minute per question.
• For every wrong answer, read the explanation and revise that chapter from your NCERT and graduation books.
• Watch for (D) More than one and (E) None of the above: check each option, don’t stop at the first one that looks right.
• Practise the General Studies part too: BPSC TRE 3.0 GS paper (Class 11–12).
• Take timed mock tests on the ExamYaari app.

More BPSC TRE Resources

• यही प्रश्न पत्र हिंदी में देखें
• All BPSC TRE General Studies question papers
• BPSC TRE 3.0 General Studies (Class 11–12)

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