HE1004 Oct Quiz (2021).pdf

(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. − 0.58 b. 0.66666… c. √16 d. e ∑100 𝑖𝑖=1 𝑖𝑖 2. ∑3𝑗𝑗=1 ∑3 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. − 0.58 b. 0.66666… c. √16 d. e ∑100 𝑖𝑖=1 𝑖𝑖 2. ∑3𝑗𝑗=1 ∑3 is equal to_____. 𝑖𝑖=0(𝑖𝑖−1) a. 2525 b. 5050 c. 7575 d. 10100 3. Which of the following is correct? 2 3 a. √𝑥𝑥1 𝑥𝑥 2 = 𝑥𝑥 3 b. (𝑥𝑥 5 𝑥𝑥 3 )2 = 𝑥𝑥 30 c. (𝑥𝑥 𝑎𝑎 + 𝑥𝑥 𝑏𝑏 )2 = 𝑥𝑥 2𝑎𝑎 + 2𝑥𝑥 𝑎𝑎+𝑏𝑏 + 𝑥𝑥 2𝑏𝑏 1 d. (𝑒𝑒 4 )4 = 1 4. Suppose the sets 𝑹𝑹 = {𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛}, 𝑸𝑸 = {𝑎𝑎𝑎𝑎𝑎𝑎 rational numbers}, and 𝑵𝑵 = {𝑎𝑎𝑎𝑎𝑎𝑎 natural numbers}. Which of the following relationship descriptions is correct? a. 𝑸𝑸 is a sufficient condition for 𝑵𝑵. b. 𝑹𝑹 is a necessary condition for 𝑵𝑵. c. 𝑹𝑹 is a sufficient condition for 𝑸𝑸. d. 𝑸𝑸 is neither a sufficient nor necessary condition for 𝑹𝑹. 5. Suppose the demand function is 𝑄𝑄 = 8 − 𝑃𝑃 and the supply function is 𝑄𝑄 = 𝑃𝑃 − 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: 𝑄𝑄 = 2𝑃𝑃 − 10. To encourage production, the government subsidies the producers $2 per unit. What is the new supply function? a. 𝑄𝑄 = 2𝑃𝑃 − 6 b. 𝑄𝑄 = 2𝑃𝑃 − 5 c. 𝑄𝑄 = 2𝑃𝑃 − 2 d. 𝑄𝑄 = 2𝑃𝑃 − 1 7. If the production function is 𝑄𝑄 = 8√𝐿𝐿 − 𝐿𝐿 , where 𝑄𝑄 denotes output and 𝐿𝐿 denotes the size of the workforce, find out the marginal product of labor (𝑀𝑀𝑀𝑀𝐿𝐿 ) when 𝐿𝐿 = 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 × (1 + 0.028 × 50) b. 10,28050 c. (10,000 × 1.028)50 d. 10,000 × (1 + 0.028)50 9. If Country A’s 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 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 2 and 𝑔𝑔(𝑥𝑥) = ln (3𝑥𝑥 + 1). What is (𝑓𝑓 ∘ 𝑔𝑔)(𝑥𝑥)? a. 3𝑥𝑥 2 ln(3𝑥𝑥 + 1) b. 3[ln(3𝑥𝑥 + 1)]2 c. ln (9𝑥𝑥 2 + 1) d. 9 ln(3𝑥𝑥 + 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., 𝐶𝐶 = 𝐶𝐶(𝑌𝑌) and 𝑆𝑆 = 𝑆𝑆(𝑌𝑌). Suppose 𝐶𝐶 = 1500 + 0.80𝑌𝑌. What of the following statements is incorrect? a. The saving function is 𝑆𝑆 = −1500 + 0.2𝑌𝑌. 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(𝑄𝑄) = −4𝑄𝑄 2 + 400𝑄𝑄 − 50. What is the optimal 𝑄𝑄, production level, such that the firm can maximize its profit? a. 0 b. 50 c. 100 d. ∞ e. None of the above 13. Suppose 𝑓𝑓(𝑥𝑥) = 𝑒𝑒 3𝑥𝑥 + 2𝑥𝑥. What is the slope of the inverse function 𝑓𝑓 −1 (𝑥𝑥) evaluated at 𝑥𝑥 = 0? a. 0 b. 0.2 c. 1 d. 5 e. None of the above 2 14. Calculate the third derivative of 𝑒𝑒 𝑥𝑥 evaluated at 𝑥𝑥 = 1. a. 6e b. 12e c. 16e d. 20e e. None of the above 3 15. Which of the following is incorrect? a. lim (3𝑥𝑥 + 2) = 8 𝑥𝑥→2 b. lim 𝑥𝑥 𝑥𝑥 𝑥𝑥 √… = 5 𝑥𝑥→5 c. lim 𝑥𝑥 𝑥𝑥 𝑥𝑥√… = 0 𝑥𝑥→0 3𝑥𝑥 d. lim =0 𝑥𝑥→∞ 4𝑥𝑥 e. All the above are correct. 16. Which of the following is/are incorrect? 𝑥𝑥𝑥𝑥 a. ln 𝑧𝑧 = ln(𝑥𝑥) + ln (𝑦𝑦) − ln (𝑧𝑧) b. ln 𝑥𝑥𝑒𝑒 ln 𝑦𝑦 = ln(𝑥𝑥) + ln (𝑦𝑦) c. ln (𝑥𝑥 2 ) = 2𝑥𝑥 d 𝑒𝑒 2 ln(𝑥𝑥+𝑦𝑦) = 𝑥𝑥 2 + 2𝑥𝑥𝑥𝑥 + 𝑦𝑦 2 e. (c) and (d) 𝑥𝑥 2 −4 17. Calculate lim 𝑥𝑥→2 𝑥𝑥−2 a. 0 b. 2 c. 4 d. ∞ e. None of the above 18. Which of the following is incorrect? 𝑑𝑑𝑑𝑑 2𝑥𝑥+2𝑦𝑦 a. 𝑦𝑦 = 𝑥𝑥 2 + 2𝑥𝑥𝑥𝑥 implies 𝑑𝑑𝑑𝑑 = 1−2𝑥𝑥 𝑑𝑑𝑑𝑑 2𝑥𝑥 2 b. 𝑦𝑦 = 𝑥𝑥 2 ln(2𝑥𝑥 + 1) implies 𝑑𝑑𝑑𝑑 = 2𝑥𝑥 ln(2𝑥𝑥 + 1) + 2𝑥𝑥+1 𝑑𝑑𝑑𝑑 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥(𝑦𝑦)−𝑦𝑦2 c. 𝑥𝑥 𝑦𝑦 = 𝑦𝑦 𝑥𝑥 implies 𝑑𝑑𝑑𝑑 = 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥(𝑥𝑥)−𝑥𝑥2 𝑑𝑑𝑑𝑑 3 +2𝑥𝑥 d. ln(𝑦𝑦) = 𝑥𝑥 3 + 2𝑥𝑥 implies 𝑑𝑑𝑑𝑑 = (3𝑥𝑥 2 + 2)𝑒𝑒 𝑥𝑥 e. All the above are correct. 4  1 1 19. Suppose 𝑥𝑥̅ = 𝑛𝑛 ∑𝑛𝑛𝑖𝑖=1 𝑥𝑥𝑖𝑖 and 𝑦𝑦 = 𝑛𝑛 ∑𝑛𝑛𝑖𝑖=1 𝑦𝑦𝑖𝑖. Then, ∑𝑛𝑛𝑖𝑖=1(𝑦𝑦𝑖𝑖 − 𝑦𝑦 )(𝑥𝑥𝑖𝑖 − 𝑥𝑥̅ ) is equal to_______. a. ∑𝑛𝑛𝑖𝑖=1(𝑦𝑦𝑖𝑖 − 𝑦𝑦 )𝑥𝑥𝑖𝑖 b. ∑𝑛𝑛𝑖𝑖=1(𝑥𝑥𝑖𝑖 − 𝑥𝑥̅ )𝑦𝑦𝑖𝑖 c. ∑𝑛𝑛𝑖𝑖=1 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 − ∑𝑛𝑛𝑖𝑖=1(𝑥𝑥̅ 𝑦𝑦 ) d. (a) and (b) e. (a), (b), and (c) 20. Consider 𝑓𝑓(𝑥𝑥) = −3𝑥𝑥 2 + 3𝑥𝑥 − 2 Which of the following is incorrect? a. 𝑓𝑓(𝑥𝑥) is a convex function. b. −3𝑥𝑥 2 + 3𝑥𝑥 ≈ 2 + 𝑓𝑓(𝑎𝑎) + 𝑓𝑓 ′ (𝑎𝑎)(𝑥𝑥 − 𝑎𝑎), where 𝑎𝑎 is close to 𝑥𝑥. c. 𝑓𝑓(2) = −8 d. 𝑓𝑓(𝑥𝑥) is strictly decreasing when 𝑥𝑥 > 0.5. e. The range of the function 𝑓𝑓(𝑥𝑥) is (−∞, −1.25). Bonus Question 21. Which of the following is correct? a. If a function 𝑓𝑓(𝑥𝑥) is continuous at 𝑥𝑥 = 𝑎𝑎, then 𝑓𝑓(𝑥𝑥) must also be differentiable at 𝑥𝑥 = 𝑎𝑎. b. If the first and second derivatives of 𝑓𝑓(𝑥𝑥) are both positive, then 𝑓𝑓(𝑥𝑥) is a concave function. c. If lim 𝑓𝑓(𝑥𝑥) and lim 𝑔𝑔(𝑥𝑥) do not exist, then lim [𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥)] does not exist. 𝑥𝑥→𝑎𝑎 𝑥𝑥→𝑎𝑎 𝑥𝑥→𝑎𝑎 1 d. Suppose the function 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 4 − 1. The first derivative of its inverse function is 8𝑥𝑥 3. 6𝑥𝑥 4 +8𝑥𝑥 3 −4 ln 𝑥𝑥 2 +10𝑥𝑥−5 e. lim 5𝑥𝑥 5 +4𝑥𝑥 4 +2𝑥𝑥 2 −√2𝑥𝑥 = 0. 𝑥𝑥→∞ -- END OF PAPER-- GOOD LUCK! 5