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

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

  1. If A = [[0, 3], [−3, 0]] is a square matrix, which of the following relations is true (A is skew-symmetric, i.e. Aᵀ = −A)? [1 marks]
  2. 3x + 3y = 6 and x + y = 2 are two simultaneous equations. (i) Write the system in matrix form AX = B where A, X and B have their usual meanings. [1] (ii) Find the determinant of A. [1] (iii) Does the inverse of A exist? [1] (iv) Does the system have a unique solution? [1] (v) On the basis of the Gauss elimination method, what type of solution exists? [1] [5 marks]
  3. (a) A rectangular garden requires 144 m of fencing wire for its complete fencing. What will be its shape if its area is to be maximized? Justify your reason. [4+1] (b) Find the area between the curve y = x³ and the x-axis when x = −2 and x = 2. [3] [8 marks]
  4. What is the value of the determinant |1 −1 −1; −1 0 1; 0 1 0|? (a) 0 (b) 1 (c) −1 (d) 3 [1 marks]
  5. If A = [[1, 2, −3], [2, −1, 4], [3, 1, 3]], find A⁻¹ and verify that A·A⁻¹ = I₃. [5] [5 marks]
  6. (a) The ages in years of three students are in G.P. such that the sum and the product of their ages are 21 and 64 respectively. Find their ages in years. [4] (b) A donor decides to grant a scholarship of Rs. 50,000 yearly for 5 years to a student of medicine. What sum of money should be deposited in the bank at a compound interest rate of 8% per year so that the sum is sufficient to grant the scholarship at the end of each year? [4] [8 marks]
  7. Which of the following entries will make the technology matrix satisfy the Hawkins–Simon conditions? (a) −0.2 (b) 0 (c) 0.5 (d) 0.01 [1 marks]
  8. (a) Write average cost (AC) in terms of total cost (C) and output (Q). [1] (b) What does ∫ₐᵇ f(x) dx represent geometrically? [1] (c) Write the degree of the differential equation dy/dx + 2y = 5. [1] (d) If P is the demand function and Q is output, write the corresponding revenue function. [1] (e) Write the formula for elasticity of demand (Ed). [1] [5 marks]
  9. (a) The regression equations in two variables X and Y are 3X + 2Y − 26 = 0 and 6X + Y − 31 = 0. (i) Find the mean values of X and Y. [2] (ii) Find the two regression coefficients. [2] (b) In a season, 70% of visitors went to Janakpur and 30% to Lumbini. It is estimated that 35% of Janakpur visitors and 65% of Lumbini visitors are Nepali. A visitor is selected at random and found to be Nepali. (i) What is the probability that the visitor is a Janakpur visitor? [2] (ii) What is the probability that the visitor is a Lumbini visitor? [2] [8 marks]
  10. On calculation of elasticity of demand, if Ed = −2.25 then (a) Demand is unit elastic (b) Demand is elastic (c) Demand is inelastic (d) Demand is viable [1 marks]
  11. The demand function is Pd = 196 − x² and the supply function is Ps = x² + 4x + 126. Find the consumer and producer surplus at market equilibrium. [5] [5 marks]
  12. What is the integrating factor of the differential equation dy/dx − y/x = eˣ? (a) −1/x (b) x (c) log x (d) 1/x [1 marks]
  13. The birth rate of the population in a certain country is given by dP/dt = 0.15(6000 − P), where t is the time in years (P = 0 when t = 0). (a) Solve the above differential equation. [3] (b) Calculate the population after 10 years. [2] [5 marks]
  14. If P is the present value, PT the present value of future return, T the time of cash flow and i the discount rate per annum, which of the following is applied to find the present value of future return? (a) P = PT/(1+i)ᵀ (b) P = PT/(1−i)ᵀ (c) PT = P/(1+i)ᵀ (d) PT = P/(1−i)ᵀ [1 marks]
  15. Using the simplex method, maximize z = 6x − 9y subject to the constraints x + y ≤ 20, 2x − 3y ≤ 6, x, y ≥ 0. [5] [5 marks]
  16. What is the geometric mean between 1 and 64? (a) 5 (b) 7 (c) 8 (d) 9 [1 marks]
  17. The height (ft) and weight (kg) of five students of a class are given: Height = 4, 4.5, 5, 5.5, 6; Weight = 30, 38, 42, 58, 70. Find the correlation coefficient between height and weight and interpret the result. [4+1] [5 marks]
  18. There are m simultaneous linear equations and n unknown variables (n > m). Which of the following is the number of basic solutions? (a) m + n (b) n − m (c) m − n (d) n [1 marks]
  19. A binomial distribution has 5 independent trials. If the probabilities of 1 and 3 successes are 1/4 and 144/625 respectively, find p and q. Also find the probability of 2 successes. [5] [5 marks]
  20. The original cost of a machine is Rs. 50,000. It depreciates at 20% per annum. What is the value of the machine after 2 years? (a) Rs. 10,000 (b) Rs. 20,000 (c) Rs. 25,000 (d) Rs. 32,000 [1 marks]
  21. In a frequency distribution, the difference between the mean and the mode is 1.4 and the coefficient of skewness SK(P) is 0.625. What is the value of the standard deviation? (a) 1.24 (b) 2.24 (c) 1.34 (d) 2.34 [1 marks]
  22. If P(A) and P(B) are the probabilities of events A and B respectively, which of the following conditions is true? (a) P(A/B) = P(A∩B)/P(A) (b) P(A/B) = P(A∩B)/P(B) (c) P(A/B) = P(A)/P(A∩B) (d) P(A/B) = P(B)/P(A∩B) [1 marks]