National Examinations Board Grade 12 Mathematics (Optional) — Grade Increment (Supplementary) 2080 (2023) (Set A)

National Examinations Board · Mathematics (Grade 12) · 2080 (2023) BS · Grade Increment (Supplementary) Examination — 22 questions — a free old question paper on Goodone.club’s exam-paper archive.

  1. In how many ways can 'r' letters be posted in 'n' letter boxes (n ≥ r)? (A) n×r (B) nʳ (C) n!/r! (D) n!/(n−r)! [1 marks]
  2. (a) What is the sum of the coefficients of even terms in the expansion of (1+x)ⁿ? [1] (b) How many terms are there in the expansion (x + 1/x)⁶? [1] (c) Express e^{−x} in an infinite series. [1] (d) Write the condition that ax² + bx + c = 0 (a ≠ 0) has equal roots. [1] (e) What does θ represent in z = r(cosθ + i sinθ)? [1] [5 marks]
  3. (a) Using the row-equivalent matrix method, solve the system of linear equations: 2x + y − z = 9, 3x − y + 2z = −1, 4x + y − 3z = 17. [4] (b) The sum of the roots of a quadratic equation is 5 and the sum of their squares is 13. Find the equation. [2] (c) Show that 1/2! + (1+2)/3! + (1+2+3)/4! + … = e/2. [2] [8 marks]
  4. Let (G, o) be a group such that a o b = 2a + 3b for all integers a, b. What is 2 o 3? (A) 10 (B) 12 (C) 13 (D) 15 [1 marks]
  5. (a) Evaluate (1 − √3 i)⁶ using De Moivre's theorem. [2] (b) Using the principle of mathematical induction, show that n(n+1)(2n+1) is divisible by 6 for all n ∈ N. [3] [5 marks]
  6. (a) Find the intercepts of the plane 3x + 4y + 6z − 24 = 0 on the axes. [2] (b) Find the equation of the ellipse in standard form with the length of its major axis 12 and eccentricity 2/3. [2] (c) Find the direction cosines of two lines which satisfy the relations 4l + 3m − 2n = 0 and lm − mn + nl = 0. Also find the angle between the lines. [4] [8 marks]
  7. Which one of the following is the general solution for x satisfying the equation tan²x = tan²θ? (n represents an integer) (A) nπ − θ (B) nπ + θ (C) nπ ± θ (D) 2nπ ± θ [1 marks]
  8. (a) Find the value of sin(cos⁻¹(1/2) + sin⁻¹(3/5)). [3] (b) Find the area of the parallelogram determined by the vectors −i − j + k and 3i − j + 2k. [2] [5 marks]
  9. (a) Integrate ∫ x²/[(x+2)²(x+3)] dx. What concept is used to integrate the above integral? [2] (b) A function f(x) is continuous in [a, b] and differentiable in (a, b). If f(a) ≠ f(b), does Rolle's theorem exist in [a, b]? Give reason. [3] (c) Write an example of a homogeneous differential equation of first order and solve it. [3] [8 marks]
  10. What is the range of y = sin⁻¹x? (A) −π/2 < y < π/2 (B) −π/2 ≤ y < π/2 (C) −π/2 < y ≤ π/2 (D) −π/2 ≤ y ≤ π/2 [1 marks]
  11. (a) Calculate the coefficient of rank correlation between age (in yrs.) and weight (in kg) of the following observations: Age (X) = 12, 14, 16, 18, 20; Weight (Y) = 25, 32, 40, 50, 56. [3] (b) If 20% of the bulbs produced by a machine are defective, determine the probability that out of 4 bulbs chosen at random two are defective. [2] [5 marks]
  12. What is the value of i·(j×k) + j·(k×i) − k·(i×j)? Where i, j, k are unit vectors along the x-, y- and z-axes respectively. (A) −3 (B) −1 (C) 1 (D) 3 [1 marks]
  13. (a) What does dy/dx represent geometrically at a particular point of a curve? [1] (b) Write the integral of ∫ 1/(a² + x²) dx. [1] (c) What is the difference between Δy and dy? [1] (d) What is the derivative of sinh x? [1] (e) Write a differential equation in linear form. [1] [5 marks]
  14. From a well-shuffled pack of 52 cards, two cards are drawn at random successively without replacement. What is the probability of getting both ace cards? (A) 1/221 (B) 1/169 (C) 2/221 (D) 2/13 [1 marks]
  15. (a) Solve: dy/dx = (1 + y²)/(1 + x²). [2] (b) Verify Lagrange's mean value theorem for f(x) = (x+1)(x−2) in [−1, 2]. [3] [5 marks]
  16. For what value of c are the planes 2x+3y+5z+11 = 0 and 4x+6y+cz+d = 0 parallel? (A) 2/5 (B) 5/2 (C) 5 (D) 10 [1 marks]
  17. Using the simplex method, maximize Z(x, y) = 10x + 15y subject to x + 3y ≤ 21, 2x + 3y ≤ 24, x, y ≥ 0. [5] [5 marks]
  18. Using L'Hôpital's rule, what is the value of lim(x→0) [2(e^{2x} − 1) / log(1+x)]? (A) 1 (B) 2 (C) 3 (D) 4 [1 marks]
  19. Two like parallel forces of magnitudes A and B are acting at the end points M and N of a rod MN of length l. If two opposite forces each of magnitude T are added to A and B, prove that the line of action of the new resultant will be displaced through a distance Tl/(A+B). [5] OR P and Q are the input-output coefficient matrix and the demand vector respectively: P = [[0.1, 0.4], [0.2, 0.2]], Q = [3200; 1800]. Find the total output and comment on your results. [5] [5 marks]
  20. What is the integral of ∫ dx/(x² − 36)? (A) (1/12)ln((x−6)/(x+6)) + C (B) (1/2)ln((x+6)/(x−6)) + C (C) (1/6)ln((x−6)/(x+6)) + C (D) (1/6)ln((x+6)/(x−6)) + C [1 marks]
  21. The Gaussian forward elimination step ends with a system of equations in matrix form [a₁₁ a₁₂ : c₁ ; 0 a₂₂ : c₂] such that a₂₂ ≠ 0. The system has (A) No solution (B) Infinitely many solutions (C) Finite solutions (D) Unique solution [1 marks]
  22. A body of mass 250 g, initially at rest, is subjected to a force of 3 N for 1 second. The velocity acquired during the second is (A) 3/250 ms⁻¹ (B) 4/3 ms⁻¹ (C) 12 ms⁻¹ (D) 250/3 ms⁻¹ OR The demand equation of a certain commodity is P = 60 − 2Q − Q², where P is price and Q is quantity. If the demand is 6, what is the consumer surplus? (A) 120 (B) 140 (C) 180 (D) 252 [1 marks]