National Examinations Board Grade 12 Mathematics — Model Question 2079 (2023)

National Examinations Board · Mathematics (Grade 12) · 2079 (2023) BS · Model Question — 22 questions — a free old question paper on Goodone.club’s exam-paper archive.

  1. What is an arrangement of the n natural numbers called? (A) Induction (B) Permutation (C) Combination (D) Expectation [1 marks]
  2. For any positive integer n, (a+x)ⁿ = c₀aⁿ + c₁aⁿ⁻¹x + c₂aⁿ⁻²x² + … + cₙxⁿ. (a) How many terms are there in the expansion? [1] (b) Write the binomial coefficients in the expansion. [1] (c) Write the general term of the expansion. [1] (d) Write the relation among C(n, r−1), C(n+1, r) and C(n, r). [1] (e) What is the value of C₀ + C₁ + C₂ + … + Cₙ? [1] [5 marks]
  3. (a) Using matrix methods, solve the system of linear equations: x + y + z = 4, 2x + y − 3z = −9, 2x − y + z = −1. [4] (b) Apply De Moivre's theorem to find the value of [2(cos15° + i sin15°)]⁶. [2] (c) Prove that (1 + 1/1! + 1/2! + 1/3! + …)(1 − 1/1! + 1/2! − 1/3! + …) = 1. [2] [8 marks]
  4. Let 1, ω, ω² be the cube roots of unity. Under which operation is the set A = {1, ω, ω²} closed? (A) Addition (B) Subtraction (C) Multiplication (D) Division [1 marks]
  5. (a) Using the principle of mathematical induction, show that 1 + 2 + 3 + … + n < (1/8)(2n+1)². [3] (b) Find the quadratic equation whose one root is 2 + √3. [2] [5 marks]
  6. (a) Find the direction cosines of the line joining the points (4, 4, −10) and (−2, 2, 4). [2] (b) Find the angle between the two diagonals of a cube. [4] (c) Find the vertices of the conic section 16(y−1)² − 9(x−5)² = 144. [2] [8 marks]
  7. What is the domain of sin⁻¹x? (A) x ≥ 1 or x ≤ −1 (B) (−∞, ∞) (C) −1 < x < 1 (D) −1 ≤ x ≤ 1 [1 marks]
  8. (a) Given y = sin⁻¹x and y > 0, express cos y and tan y in terms of x. [3] (b) If a, b and c are any three vectors such that a × b = a × c for a ≠ (0,0), show that b = c. [2] [5 marks]
  9. (a) If the limiting value of (f(x) − 5)/(x − 3) at x = 3 is 2 by using L'Hôpital's rule, find the appropriate value of f(x). [2] (b) Write any one homogeneous differential equation in (x, y) and solve it. [3] (c) The concept of anti-derivative is necessary for solving a differential equation. Justify this statement with an example. [3] [8 marks]
  10. ABCD is a parallelogram. Which one of the following represents the area of the parallelogram? (A) Magnitude of the vector product of two vectors along AB and BD (B) along AB and DC (C) along AC and BC (D) along AB and AD [1 marks]
  11. The price in Rupees (X) and demand in units (Y) over 6 days of a week are given as: X = 10, 12, 13, 12, 16, 15; Y = 40, 38, 43, 45, 37, 43. Calculate Pearson's coefficient of correlation and the regression coefficient of X on Y. [5] [5 marks]
  12. If a conic section has eccentricity e = √(a²−b²)/a, what is the equation of that conic section? (A) x²/a² − y²/b² = 1 (B) x²/a² + y²/b² = 1 (C) x²/a² + y²/a² = 1 (D) x²/b² + y²/b² = 1 [1 marks]
  13. (a) Define L'Hôpital's rule. [1] (b) Write the slope of the tangent and normal to the curve y = f(x) at (x₁, y₁). [1+1] (c) Write the integral of 1/(x² + a²) dx. [1] (d) What is the integral of sinh x dx? [1] [5 marks]
  14. If cos θ = −1/2 for integer n, what is the general value of θ? (A) 2nπ ± 2π/3 (B) nπ + (−1)ⁿ·π/3 (C) nπ − π/3 (D) nπ + π/3 [1 marks]
  15. (a) Solve: dy/dx = y/x. [2] (b) Verify Rolle's theorem for f(x) = x² + 3x − 4 in [−4, 1]. [3] [5 marks]
  16. 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)? (A) equal to P(B/A) (B) equal to P(A) (C) less than P(A ∩ B) (D) less than P(B/A) [1 marks]
  17. Using the simplex method, maximize P(x, y) = 15x + 10y subject to 2x + y ≤ 10, x + 3y ≤ 10, x, y ≥ 0. [5] [5 marks]
  18. The edge of a cube increases from 10 cm to 10.025 cm. What would be the approximate increment in volume? (A) 10³ cm³ (B) 10.025³ cm³ (C) 7.5187 cm³ (D) 7.5 cm³ [1 marks]
  19. A particle is projected with a velocity v and its greatest height is H. Prove that the horizontal range R is R = 4√{H(v²/2g − H)}. OR The cost function C(x) in thousands of rupees for producing x units of maths textbooks is C(x) = 30 + 20x − 0.5x², 0 ≤ x ≤ 15. (a) Find the marginal cost function. [3] (b) Find the marginal cost for producing 12,000 maths textbooks. [2] [5 marks]
  20. What is the integrating factor of the differential equation cos²x · dy/dx + y = 1? (A) tan x (B) e^{tan x} (C) sec²x (D) e^{sec x} [1 marks]
  21. What is the number of solutions of the system of linear equations x + y = 5 and x + y = 7? (A) One solution (B) No solution (C) Infinite solutions (D) More than one solution [1 marks]
  22. Forces P and Q are acting along the ceiling and floor of a rectangular room. What is the nature of the forces? (A) Like (B) Unlike (C) Collinear (D) Parallel OR If Δyₜ = yₜ₊₁ − yₜ, then Δ²yₜ is equal to (A) yₜ₊₂ − yₜ₊₁ (B) yₜ₊₁ − yₜ (C) yₜ₊₂ − yₜ₊₁ + yₜ (D) yₜ₊₂ − 2yₜ₊₁ + yₜ [1 marks]