CITY UNIVERSITY OF HONG KONG Course code & title : GE1359 Algebra Session : Summer Term , 20 2 2 Time allowed : T wo h ours This paper has FIVE pages (including this cover page). The Academic Honesty Pledge is shown on page 2. The questions start on page 3. The trigonometric identities are provided on page 5 Instructions to candidates: 1. This paper consists of SIX questions 2 Answer ALL questions 3 Start each question on a new page. 4 Show ALL steps clearly This is a closed - boo k examination. Candidates are allowed to use the following materials/aids: Approved calculator s Materials/aids other than those stated above are not permitted. Candidates will be subject to disciplinary action if any unauthorized ma terials or aids are found on them. Departmental hotline: 3442 8 40 4 - 2 - Academic Honesty Pledge “I pledge that the answers in this examination are my own and that I will not seek or obtain an unfair advantage in producing these answers. Specifically, ❖ I will not plagiarize (copy without citation) from any source; ❖ I will not communicate or atte mpt to communicate with any other person during the examination; neither will I give or attempt to give assistance to another student taking the examination; and ❖ I will use only approved devices (e.g., calculators) and/or approved device models. ❖ I understand that any act of academic dishonesty can lead to disciplinary action.” Please reaffirm the honesty pledge by writing “I pledge to follow the Rules on Academic Honesty and understand that violations may lead to severe penalties” onto the first examination answer sheet. - 3 - Question 1 For each of the following functions, find its largest possible domain and largest possible range. Give your answers in set/interval notations. ( a ) 𝑓 ( 𝑥 ) = 5 𝑥 + 3 2 𝑥 − 1 [ 6 marks ] ( b ) 𝑔 ( 𝑥 ) = 3 sin ( 5 𝑥 ) − 2 cos ( 5 𝑥 ) [ 7 marks ] Question 2 Consider the following system of linear equations { 𝑥 − 𝑦 + 2 𝑧 + 𝑤 = − 2 − 2 𝑥 + 2 𝑦 − 5 𝑧 + 𝑤 = 3 3 𝑥 − 3 𝑦 + 𝑎𝑧 + 12 𝑤 = 𝑏 , where 𝑎 and 𝑏 are real number s Using Gaussian elimination, determine all possible value s o f 𝑎 and 𝑏 such that the system has (a) no solution; (b) i nfinitely many solutions with one free variable; (c) infinitely many solutions with two free variables Then solve the system in part (c). [1 5 marks] Question 3 ( a ) If 𝛼 , 𝛽 , 𝛾 are the roots of the equation 𝑥 3 + 𝑝 𝑥 2 + 𝑞𝑥 + 𝑟 = 0 , where 𝑝 , 𝑞 and 𝑟 are real numbers , e xpress 𝛼 2 𝛽 2 + 𝛽 2 𝛾 2 + 𝛼 2 𝛾 2 and 𝛼 3 𝛽 3 + 𝛽 3 𝛾 3 + 𝛼 3 𝛾 3 in terms of 𝑝 , 𝑞 and 𝑟 [10 marks] ( b ) Let 𝑥 , 𝑦 > 0 Use the AM - GM inequality to find the minimum value of 𝑆 = 𝑥 𝑦 2 + 2 𝑦 𝑥 2 + 𝑥 and determine the values of 𝑥 and 𝑦 at which this minimum value of 𝑆 is attained. [ 10 marks] Question 4 (a) Prove by mathematical induction that 𝑛 3 + 5 𝑛 is divisible by 6 for all 𝑛 = 1 , 2 , 3 , ... [ 9 marks] (b) A sequence { 𝑎 𝑛 } is defined by 𝑎 1 = 5 , 𝑎 2 = 7 and 𝑎 𝑛 + 2 = 3 𝑎 𝑛 + 1 − 2 𝑎 𝑛 for all 𝑛 = 1 , 2 , 3 , ... Prove by mathematical induction that 𝑎 𝑛 = 3 + 2 𝑛 for all 𝑛 = 1 , 2 , 3 , ... [ 9 marks] - 4 - Question 5 (a) How many different arrangements can be obtained from the letters of the word STATISTICS ? [ 4 marks] (b) How many different arrangements can be obtained from the letters of the word SUMMER such that no two vowels are next to each other ? [5 marks] (c) How many different arrangements can be formed by taking 5 letters from the word ARRANGEMENT ? Assume that the letters are taken without replacement. [ 9 marks] Question 6 (a) Find the general solution (in radians) of the equation sin 𝑥 cos 𝑥 = sin 5 𝑥 cos 5 𝑥 [ 10 marks] (b) Without using calculator, find the exact value of 𝑃 = sin 20° sin 40° sin 80° [ 6 marks] (Hint: You may use the results of sine and cosine for some special angles as shown below.) 30° 45° 60° sin 1 2 √ 2 2 √ 3 2 cos √ 3 2 √ 2 2 1 2 - END - - 5 - Trigonometric identities - Formula sheet ➢ Compound Angle Formulae sin ( 𝐴 + 𝐵 ) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵 sin ( 𝐴 − 𝐵 ) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵 cos ( 𝐴 + 𝐵 ) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵 cos ( 𝐴 − 𝐵 ) = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 tan ( 𝐴 + 𝐵 ) = tan 𝐴 + tan 𝐵 1 − tan 𝐴 tan 𝐵 tan ( 𝐴 − 𝐵 ) = tan 𝐴 − tan 𝐵 1 + tan 𝐴 tan 𝐵 ➢ Double Angle Formulae sin 2 𝐴 = 2 sin 𝐴 cos 𝐴 cos 2 𝐴 = cos 2 𝐴 − sin 2 𝐴 ➢ Half Angle Formulae cos 2 𝐴 = 1 2 ( 1 + cos 2 𝐴 ) , sin 2 𝐴 = 1 2 ( 1 − cos 2 𝐴 ) ➢ Product - to - Sum Formulae sin 𝐴 cos 𝐵 = 1 2 [ sin ( 𝐴 + 𝐵 ) + sin ( 𝐴 − 𝐵 ) ] cos 𝐴 sin 𝐵 = 1 2 [ sin ( 𝐴 + 𝐵 ) − sin ( 𝐴 − 𝐵 ) ] cos 𝐴 cos 𝐵 = 1 2 [ cos ( 𝐴 + 𝐵 ) + cos ( 𝐴 − 𝐵 ) ] sin 𝐴 sin 𝐵 = − 1 2 [ cos ( 𝐴 + 𝐵 ) − cos ( 𝐴 − 𝐵 ) ] ➢ Sum - to - Product Formulae sin 𝑥 + sin 𝑦 = 2 sin ( 𝑥 + 𝑦 2 ) cos ( 𝑥 − 𝑦 2 ) sin 𝑥 − sin 𝑦 = 2 cos ( 𝑥 + 𝑦 2 ) sin ( 𝑥 − 𝑦 2 ) cos 𝑥 + cos 𝑦 = 2 cos ( 𝑥 + 𝑦 2 ) cos ( 𝑥 − 𝑦 2 ) cos 𝑥 − cos 𝑦 = − 2 sin ( 𝑥 + 𝑦 2 ) sin ( 𝑥 − 𝑦 2 )