HE1004 Oct Quiz (2021) PDF
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2021
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This is a mathematics quiz from October 2021, containing multiple choice questions on topics such as irrational numbers, summations, functions, and calculus. The quiz covers various mathematical concepts and is likely part of an undergraduate mathematics course.
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(a) Each question is worth 1 point. Total Marks: 20 points (b) Key in your answers in MS Form. 1. Which of the following is an irrational number? a. β 0.58 b. 0.66666β¦ c. β16 d. e β100 ππ=1 ππ 2. β3ππ=1 β3 i...
(a) Each question is worth 1 point. Total Marks: 20 points (b) Key in your answers in MS Form. 1. Which of the following is an irrational number? a. β 0.58 b. 0.66666β¦ c. β16 d. e β100 ππ=1 ππ 2. β3ππ=1 β3 is equal to_____. ππ=0(ππβ1) a. 2525 b. 5050 c. 7575 d. 10100 3. Which of the following is correct? 2 3 a. βπ₯π₯1 π₯π₯ 2 = π₯π₯ 3 b. (π₯π₯ 5 π₯π₯ 3 )2 = π₯π₯ 30 c. (π₯π₯ ππ + π₯π₯ ππ )2 = π₯π₯ 2ππ + 2π₯π₯ ππ+ππ + π₯π₯ 2ππ 1 d. (ππ 4 )4 = 1 4. Suppose the sets πΉπΉ = {ππππππ ππππππππ ππππππππππππππ}, πΈπΈ = {ππππππ rational numbers}, and π΅π΅ = {ππππππ natural numbers}. Which of the following relationship descriptions is correct? a. πΈπΈ is a sufficient condition for π΅π΅. b. πΉπΉ is a necessary condition for π΅π΅. c. πΉπΉ is a sufficient condition for πΈπΈ. d. πΈπΈ is neither a sufficient nor necessary condition for πΉπΉ. 5. Suppose the demand function is ππ = 8 β ππ and the supply function is ππ = ππ β 4 What is the equilibrium quantity? a. 2 b. 4 c. 6 d. 8 1 6. The supply function for a good is given by: ππ = 2ππ β 10. To encourage production, the government subsidies the producers $2 per unit. What is the new supply function? a. ππ = 2ππ β 6 b. ππ = 2ππ β 5 c. ππ = 2ππ β 2 d. ππ = 2ππ β 1 7. If the production function is ππ = 8βπΏπΏ β πΏπΏ , where ππ denotes output and πΏπΏ denotes the size of the workforce, find out the marginal product of labor (πππππΏπΏ ) when πΏπΏ = 4. a. 5 b. 4 c. 3 d. 1 8. On 2 August 2021, Temasek launched an offering of 50-year Singapore bonds with a yield 2.8% per annum. If we invest $10,000 in this bond now, how much will we get in total by the end of year 50? a. 10, 000 Γ (1 + 0.028 Γ 50) b. 10,28050 c. (10,000 Γ 1.028)50 d. 10,000 Γ (1 + 0.028)50 9. If Country Aβs population is decreasing by 3.2% annually, then how many years will it take for the population to reach below 5 million from 5.7 million? a. About 4 years b. About 5 years c. About 6 years d. About 7 years 10. Suppose ππ(π₯π₯) = 3π₯π₯ 2 and ππ(π₯π₯) = ln (3π₯π₯ + 1). What is (ππ β ππ)(π₯π₯)? a. 3π₯π₯ 2 ln(3π₯π₯ + 1) b. 3[ln(3π₯π₯ + 1)]2 c. ln (9π₯π₯ 2 + 1) d. 9 ln(3π₯π₯ + 1) 2 11. Consider a closed economy with no government intervention. Households can then spend the money in one of the two ways. Income can be used for the consumption of goods or can be put into savings, i.e., πΆπΆ = πΆπΆ(ππ) and ππ = ππ(ππ). Suppose πΆπΆ = 1500 + 0.80ππ. What of the following statements is incorrect? a. The saving function is ππ = β1500 + 0.2ππ. b. The income multiplier is 5. c. The marginal propensity to save (MPS) is 0.2. d. The marginal propensity to consume (MPC) is 0.8. e. All the above are correct. 12. Suppose the profit function is ππ(ππ) = β4ππ 2 + 400ππ β 50. What is the optimal ππ, production level, such that the firm can maximize its profit? a. 0 b. 50 c. 100 d. β e. None of the above 13. Suppose ππ(π₯π₯) = ππ 3π₯π₯ + 2π₯π₯. What is the slope of the inverse function ππ β1 (π₯π₯) evaluated at π₯π₯ = 0? a. 0 b. 0.2 c. 1 d. 5 e. None of the above 2 14. Calculate the third derivative of ππ π₯π₯ evaluated at π₯π₯ = 1. a. 6e b. 12e c. 16e d. 20e e. None of the above 3 15. Which of the following is incorrect? a. lim (3π₯π₯ + 2) = 8 π₯π₯β2 b. lim π₯π₯ π₯π₯ π₯π₯ ββ¦ = 5 π₯π₯β5 c. lim π₯π₯ π₯π₯ π₯π₯ββ¦ = 0 π₯π₯β0 3π₯π₯ d. lim =0 π₯π₯ββ 4π₯π₯ e. All the above are correct. 16. Which of the following is/are incorrect? π₯π₯π₯π₯ a. ln π§π§ = ln(π₯π₯) + ln (π¦π¦) β ln (π§π§) b. ln π₯π₯ππ ln π¦π¦ = ln(π₯π₯) + ln (π¦π¦) c. ln (π₯π₯ 2 ) = 2π₯π₯ d ππ 2 ln(π₯π₯+π¦π¦) = π₯π₯ 2 + 2π₯π₯π₯π₯ + π¦π¦ 2 e. (c) and (d) π₯π₯ 2 β4 17. Calculate lim π₯π₯β2 π₯π₯β2 a. 0 b. 2 c. 4 d. β e. None of the above 18. Which of the following is incorrect? ππππ 2π₯π₯+2π¦π¦ a. π¦π¦ = π₯π₯ 2 + 2π₯π₯π₯π₯ implies ππππ = 1β2π₯π₯ ππππ 2π₯π₯ 2 b. π¦π¦ = π₯π₯ 2 ln(2π₯π₯ + 1) implies ππππ = 2π₯π₯ ln(2π₯π₯ + 1) + 2π₯π₯+1 ππππ π₯π₯π₯π₯π₯π₯π₯π₯(π¦π¦)βπ¦π¦2 c. π₯π₯ π¦π¦ = π¦π¦ π₯π₯ implies ππππ = π₯π₯π₯π₯π₯π₯π₯π₯(π₯π₯)βπ₯π₯2 ππππ 3 +2π₯π₯ d. ln(π¦π¦) = π₯π₯ 3 + 2π₯π₯ implies ππππ = (3π₯π₯ 2 + 2)ππ π₯π₯ e. All the above are correct. 4 1 1 19. Suppose π₯π₯Μ = ππ βππππ=1 π₯π₯ππ and π¦π¦ = ππ βππππ=1 π¦π¦ππ. Then, βππππ=1(π¦π¦ππ β π¦π¦ )(π₯π₯ππ β π₯π₯Μ ) is equal to_______. a. βππππ=1(π¦π¦ππ β π¦π¦ )π₯π₯ππ b. βππππ=1(π₯π₯ππ β π₯π₯Μ )π¦π¦ππ c. βππππ=1 π₯π₯ππ π¦π¦ππ β βππππ=1(π₯π₯Μ π¦π¦ ) d. (a) and (b) e. (a), (b), and (c) 20. Consider ππ(π₯π₯) = β3π₯π₯ 2 + 3π₯π₯ β 2 Which of the following is incorrect? a. ππ(π₯π₯) is a convex function. b. β3π₯π₯ 2 + 3π₯π₯ β 2 + ππ(ππ) + ππ β² (ππ)(π₯π₯ β ππ), where ππ is close to π₯π₯. c. ππ(2) = β8 d. ππ(π₯π₯) is strictly decreasing when π₯π₯ > 0.5. e. The range of the function ππ(π₯π₯) is (ββ, β1.25). Bonus Question 21. Which of the following is correct? a. If a function ππ(π₯π₯) is continuous at π₯π₯ = ππ, then ππ(π₯π₯) must also be differentiable at π₯π₯ = ππ. b. If the first and second derivatives of ππ(π₯π₯) are both positive, then ππ(π₯π₯) is a concave function. c. If lim ππ(π₯π₯) and lim ππ(π₯π₯) do not exist, then lim [ππ(π₯π₯) + ππ(π₯π₯)] does not exist. π₯π₯βππ π₯π₯βππ π₯π₯βππ 1 d. Suppose the function ππ(π₯π₯) = 2π₯π₯ 4 β 1. The first derivative of its inverse function is 8π₯π₯ 3. 6π₯π₯ 4 +8π₯π₯ 3 β4 ln π₯π₯ 2 +10π₯π₯β5 e. lim 5π₯π₯ 5 +4π₯π₯ 4 +2π₯π₯ 2 ββ2π₯π₯ = 0. π₯π₯ββ -- END OF PAPER-- GOOD LUCK! 5
