Class 12 Math Model Question 2082 – With Answers | NEB Education
Class 12 Math Model Questions 2082 – With Answers are the latest model questions prepared by the National Examination Board (NEB) for Class 12 students, designed to help them excel in the Math exam. Find model questions and their answers given below:
The candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks.
Time: 3 Hrs
Full Marks: 75Pass Marks: 27.5
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Group A [11 × 1 = 11]
Rewrite the correct answer in your answer sheet.
1. What is an arrangement of the n natural numbers called?
[1]
a. Induction
b. Permutation
c. Combination
d. Expectation
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2. Let 1, ω, ω² be the cube roots of unity. Under which operation is the set A = {1, ω, ω²} closed?
[1]
a. Addition
b. Subtraction
c. Multiplication
d. Division
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3. What is the domain of sin⁻¹x?
[1]
a. x ≥ 1 or x ≤ −1
b. (−∞, ∞)
c. −1 < x < 1
d. −1 ≤ x ≤ 1
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4. ABCD is a parallelogram. Which one of the following represents area of the parallelogram?
[1]
a. Magnitude of vector product of two vectors along AB and BD
b. Magnitude of vector product of two vectors along AB and DC
c. Magnitude of vector product of two vectors along AC and BC
d. Magnitude of vector product of two vectors along AB and AD
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5. If a conic section has eccentricity (e) = √(a² − b²)/a, what is the equation of that conic section?
[1]
a. x²/a² − y²/b² = 1
b. x²/a² + y²/b² = 1
c. x²/a² + y²/a² = 1
d. x²/b² + y²/b² = 1
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6. If cos θ = −1/2 for integer (n), what is the general value of θ?
[1]
a. 2nπ ± 2π/3
b. nπ + (−1)ⁿπ/3
c. nπ − π/3
d. nπ + π/3
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7. Let A and B be two dependent events. If P(A) = 1/2, P(B) = 3/4 and P(A∩B) = 2/5, what is the value of P(A/B)?
[1]
a. Equal to P(B/A)
b. Equal to P(A)
c. Less than P(A∩B)
d. Less than P(B/A)
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8. The edge of a cube increases from 10 cm to 10.025 cm. What would be the approximate increment in volume?
[1]
a. 10³ cm³
b. 10.025³ cm³
c. 7.5187 cm³
d. 7.5 cm³
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9. What is the integrating factor of the differential equation cos²x(dy/dx) + y = 1?
[1]
a. tan x
b. e^(tan x)
c. sec²x
d. e^(sec x)
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10. What is the number of solutions of the system of linear equations x + y = 5 and x + y = 7?
[1]
a. One solution
b. No solution
c. Infinite solutions
d. More than one solution
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11. Forces P and Q are acting along ceiling and floor of a rectangular room. What is the nature of the forces?
[1]
a. Like
b. Unlike
c. Collinear
d. Parallel
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Group B [8 × 5 = 40]
12. For any positive integer n, (a + x)ⁿ = c₀aⁿ + c₁aⁿ⁻¹x + c₂aⁿ⁻²x² + ⋯ + cₙxⁿ
[5]
(a) How many terms are there in the expansion? [1]
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(b) Write the binomial coefficients in the expansion. [1]
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(c) Write the general term of the expansion. [1]
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(d) Write the relation among C(n,r−1), C(n+1,r) and C(n,r). [1]
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(e) What is the value of C₀ + C₁ + C₂ + ⋯ + Cₙ? [1]
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13. [5]
(a) Using the principle of mathematical induction, show that: 1 + 2 + 3 + ⋯ + n < (1/8)(2n + 1)² [3]
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(b) Find the quadratic equation whose one of the roots is 2 + √3. [2]
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14. [5]
(a) Given y = sin⁻¹x and y > 0, express cos y and tan y in terms of x. [3]
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(b) If a⃗, b⃗ and c⃗ are any three vectors such that a⃗ × b⃗ = a⃗ × c⃗ for a⃗ ≠ (0,0), show that b⃗ = c⃗. [2]
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15. The price in Rupees (X) and demand in unit (Y) of 6 days of a week is given as:
[5]
X 10 12 13 12 16 15 Y 40 38 43 45 37 43
Calculate the Pearson's coefficient of correlation and the regression coefficients of X on Y.
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16. [5]
(a) What is the integral of ∫sinh x dx? [1]
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(b) Define L'Hospital's rule. [1]
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(c) Write the slope of the tangent and normal to the curve y = f(x) at (x₁, y₁). [1+1]
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(d) Write the integral of ∫1/(x² + a²) dx. [1]
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17. [5]
(a) Solve: dy/dx = y/x [2]
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(b) Verify Rolle's theorem for f(x) = x² + 3x − 4 in [−4, 1]. [3]
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18. Using simplex method, maximize P(x, y) = 15x + 10y subject to 2x + y ≤ 10, x + 3y ≤ 10, x, y ≥ 0
[5]
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19. A particle is projected with a velocity v and greatest height is H, prove the horizontal range R is: R = √[H(v²/2g − H)]
[5]
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Group C [3 × 8 = 24]
20. [8]
(a) Evaluate: lim(x→0) [(eˣ − e⁻ˣ − 2x) / (x − sin x)] [4]
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(b) Evaluate: ∫₀^(π/2) sin³x cos⁴x dx [4]
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21. [8]
(a) If α, β be the roots of the quadratic equation ax² + bx + c = 0, derive the relation between the roots and coefficients. [4]
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(b) The vertices of a triangle are A(2, 7), B(1, 1) and C(4, −1). Find the equations of its two altitudes and hence find the orthocenter. [4]
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22. [8]
(a) In usual notations, prove that: a/sin A = b/sin B = c/sin C [4]
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(b) In a triangle ABC, prove that: tan B/2 = (b − c)/(b + c) × cot A/2 [4]
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