Cityu GE1359 2122S Problem Sets

Dr. Emily Chan Page 1 GE1359 Problem Set 1 Summer 2022 1. Determine which of the following five sets are equal: 𝐴 = {1, 2, 3} , 𝐵 = {2, 3, 1, 2} , 𝐶 = {3, 1, 3, 2} , 𝐷 = {2, 3, 2, 1} , 𝐸 = {1, 2, 3, 4} 2. Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓} , 𝐵 = {𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖} , 𝐶 = {𝑏, 𝑑, 𝑓, 𝑔} , 𝐷 = {𝑑, 𝑒} , 𝐸 = {𝑒, 𝑓} , 𝐹 = {𝑑, 𝑓} , and let 𝑌 be a set which satisfies the following conditions: 𝑌 ⊆ 𝐴 , 𝑌 ⊆ 𝐵 , 𝑌 ⊈ 𝐶 Determine which of the sets 𝐴 , 𝐵 , 𝐶 , 𝐷 , 𝐸 , 𝐹 can equal 𝑌 3. Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} , 𝐵 = {𝑏, 𝑑, 𝑓, ℎ} and 𝐶 = {𝑐, 𝑑, 𝑒, 𝑓} . Write down (a) 𝐴 ∩ 𝐵 , 𝐴 ∩ 𝐶 , 𝐵 ∩ 𝐶 (b) 𝐴 ∪ 𝐵 , 𝐴 ∪ 𝐶 , 𝐵 ∪ 𝐶 (c) 𝐴\𝐵 , 𝐵\𝐴 , 𝐵\𝐶 , 𝐶\𝐵 , 𝐴\𝐶 , 𝐶\𝐴 4. Let ℝ be the set of all real numbers, 𝐴 = [1,3] and 𝐵 = (2,4] be two closed intervals. Write down the following sets: (a) 𝐴 ∪ 𝐵 (b) 𝐴 ∩ 𝐵 (c) (ℝ\𝐴) ∩ 𝐵 (d) (ℝ\𝐵) ∩ 𝐴 (e) (ℝ\𝐴) ∩ (ℝ\𝐵) (f) (ℝ\𝐴) ∪ (ℝ\𝐵) (g) 𝐵 ∪ [𝐴 ∩ (ℝ\𝐵)] (h) [(ℝ\𝐴) ∩ 𝐵] ∪ [(ℝ\𝐵) ∩ 𝐴] 5. Let 𝑓(𝑥) = 𝑥 3 + 2 , 𝑔(𝑥) = 2 𝑥−1 and ℎ(𝑥) = √𝑥 Find the formulas for (a) (𝑓 + 𝑔)(𝑥) , (b) ( 𝑔 𝑓 ) (𝑥) , (c) (𝑔 ∘ 𝑓)(𝑥) , (d) (𝑓 ∘ 𝑔)(𝑥) (e) (𝑓 ∘ 𝑔 ∘ ℎ)(𝑥) and state their largest possible domains. 6. Let 𝐹(𝑥) and 𝐺(𝑥) be two functions defined by 𝐹(𝑥) = 3 2+𝑥 and 𝐺(𝑥) = 1 1+ 1 𝑥 (a) Find their largest possible domains. (b) Find (𝐹 ∘ 𝐺)(𝑥) and state its largest possible domain. Dr. Emily Chan Page 2 7. Find the largest possible domain and range for each of the following functions (a) 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3 (b) 𝑔(𝑥) = 𝑥−3 𝑥+2 (c) ℎ(𝑥) = √𝑥 2 + 4 8. Find the largest possible domain for each of the following functions (a) 𝑓(𝑥) = 5 − 4𝑥 − 𝑥 2 (b) 𝑔(𝑥) = 1 5−4𝑥−𝑥 2 (c) ℎ(𝑥) = √5 − 4𝑥 − 𝑥 2 9. (a) Show that if 𝑓(𝑥) is an odd function of 𝑥 and 𝑔(𝑥) is an even function of 𝑥 , then (𝑓𝑔)(𝑥) is an odd function of 𝑥 (b) Show that if 𝑓(𝑥) and 𝑔(𝑥) are both odd functions of 𝑥 , then (𝑓𝑔)(𝑥) is an even function of 𝑥 (c) Show that if 𝑓(𝑥) and 𝑔(𝑥) are both even functions of 𝑥 , then (𝑓𝑔)(𝑥) is an even function of 𝑥 10. For each of the following functions, determine whether it is an odd function, even function or neither of them. (a) 𝑓 1 (𝑥) = sin(𝑥 3 +𝑥) 𝑥 4 +3 (b) 𝑓 2 (𝑥) = |𝑥 5 + 1| (c) 𝑓 3 (𝑥) = cos 3 (2𝑥) 11. Let 𝐹(𝑥) be a function defined by 𝐹(𝑥) = 𝑥 2 + 2𝑥 − 3 for 𝑥 ≥ 0 (a) Determine whether the function 𝐹(𝑥) is one-to-one and give the range of 𝐹(𝑥) (b) Find the inverse function of 𝐹(𝑥) if it exists, then state the domain and range of 𝐹 −1 (𝑥) 12. For each of the following functions, show that it is one-to-one, then find its inverse function and state the largest possible domain of the inverse function. (a) 𝑝(𝑥) = 3𝑥 − 2 , 𝑥 ∈ ℝ (b) 𝑞(𝑥) = 𝑥 2 − 1 , 𝑥 ≥ 0 - End - Page 1 GE1359 Problem Set 2 Summer 2022 1. Convert the following angles from degrees to radians. (a) 48° (b) 120° (c) 315° 2. Convert the following angles from radians to degrees. (a) 𝜋 6 𝑟𝑎𝑑 (b) 123𝜋 180 𝑟𝑎𝑑 (c) − 2𝜋 5 𝑟𝑎𝑑 3. For each of the following functions, (i) plot its graph from 𝑥 = − 𝜋 2 to 𝑥 = 2𝜋 ; (ii) state it largest possible domain and largest possible range; (iii) determine whether the function is periodic or not. If it is periodic, find its period. (a) 𝑓(𝑥) = 5 sin 𝑥 + 1 (b) 𝑓(𝑥) = cos ( 𝑥 2 ) (c) 𝑓(𝑥) = tan (𝑥 + 𝜋 2 ) (d) 𝑓(𝑥) = −2|sin 𝑥| 4. Prove the following identities. (a) 1 cos 𝜃 −cos 𝜃 1 sin 𝜃 −sin 𝜃 = tan 3 𝜃 (b) csc 2 𝜃 1+tan 2 𝜃 = csc 2 𝜃 − 1 (c) cos(𝐴 + 𝐵) cos(𝐴 − 𝐵) = cos 2 𝐴 − sin 2 𝐵 (d) sin 2𝐴 cos 2𝐴+1 = tan 𝐴 (e) (sin 𝐴 − cos 𝐴) 2 = 1 − sin 2𝐴 (f) 4 cos 𝐴 cos ( 2𝜋 3 + 𝐴) cos ( 2𝜋 3 − 𝐴) = cos 3𝐴 5. Let  be an angle which lies between  180 and  270 , and 25 7 cos − =  . Without using a calculator, find the values of (a)   sin 25 tan 4 + (b) 2 cos  6. Simplify each of the following: (a) sin( 𝜋 2 +𝜃) cos( 3𝜋 2 −𝜃) sec(𝜃−𝜋) (b) tan(𝜃+ 3𝜋 2 ) cot(𝜋−𝜃) csc(𝜃− 𝜋 2 ) Page 2 7. Find the exact values of the following: (a) sin (sin −1 ( 2 5 )) (b) sin −1 (sin ( 𝜋 4 )) (c) sin −1 (sin (− 2𝜋 3 )) (d) sin −1 (tan ( 3𝜋 4 )) (e) cos (cos −1 ( 3 4 )) (f) cos −1 (cos ( 5𝜋 4 )) (g) cos −1 (sin (− 𝜋 6 )) (h) tan −1 (tan 𝜋) 8. Without using calculator, find the value of the following in surd form. (a) sin 35° cos 25° + sin 25° cos 35° (b) tan 165° (c) sin 75°−sin 15° cos 75°+cos 15° 9. If 5 3 ) cos( − = −   , 5 3 ) cos( = +   , where      −  2 and     2 2 3  +  , find  2 cos (Hint: ) ( ) ( 2      − − + = ) 10. Solve each of the following equations. (a) cos ( 𝑥 2 ) = √3 2 (b) 2 sin 2 𝑥 + sin 𝑥 − 1 = 0 11. Find all solutions of the equation   2 sin cos 2 2 = − for which   2 0   (Hint:    cos sin 2 2 sin = ) 12. (a) Starting from the formula B A B A B A sin cos cos sin ) sin( + = + , prove that x x x 3 sin 4 sin 3 3 sin − = (b) Solve the equation x x sin 2 3 sin = , giving all solutions for x such that  2 0   x 13. (a) Prove that x x x 2 cosec 2 cot tan = + (b) Find the general solution, in radians, of the equation x x cot 2 cosec 2 1 = + 14. Find the sum 𝑆 = sin 𝑥 + sin (𝑥 + 𝜋 4 ) + sin (𝑥 + 2𝜋 4 ) + ⋯ + sin (𝑥 + 99𝜋 4 ) - End - Page 1 GE1359 Problem Set 3 (Part I) Summer 2022 1. For each of the following functions, (i) express it in the standard form of quadratic function; (ii) find the vertex; (iii) sketch the graph; (iv) find its domain and range. (a) 𝑓(𝑥) = 3𝑥 2 + 12𝑥 − 36 (b) 𝑔(𝑥) = −2𝑥 2 + 12𝑥 + 14 (c) 𝑓(𝑥) = −𝑥 2 + 10𝑥 − 25 (d) 𝑔(𝑥) = 3𝑥 2 + 9𝑥 + 30 2. Find the quotient and the remainder for each of the following cases: (a) Dividend: 𝑝(𝑥) = 2𝑥 3 + 11𝑥 2 + 3𝑥 − 4 Divisor: 2𝑥 + 1 (b) Dividend: 𝑝(𝑥) = −3𝑥 3 + 13𝑥 2 − 10𝑥 + 29 Divisor: 3𝑥 − 1 (c) Dividend: 𝑝(𝑥) = −21𝑥 3 − 7𝑥 2 + 37𝑥 − 12 Divisor: 7𝑥 2 − 1 (d) Dividend: 𝑝(𝑥) = −6𝑥 4 + 8𝑥 2 − 2𝑥 + 21 Divisor: 3𝑥 2 + 𝑥 + 2 3. Use the remainder Theorem to find the remainder when (a) 𝑝(𝑥) = 2𝑥 3 + 16𝑥 2 − 3𝑥 − 9 is divided by 𝑥 − 2 ; (b) 𝑝(𝑥) = 6𝑥 3 − 11𝑥 2 + 𝑥 − 4 is divided by 2 𝑥 + 1 ; (c) 𝑝(𝑥) = −3𝑥 3 + 𝑥 2 + 𝑥 − 8 is divided by 3 𝑥 − 1 4. Factorize each of the following. (a) 𝑝(𝑥) = 𝑥 3 + 6𝑥 2 + 3𝑥 − 10 (b) 𝑝(𝑥) = 3𝑥 3 + 8𝑥 2 − 33𝑥 + 10 (c) 𝑝(𝑥) = 2𝑥 3 − 5𝑥 2 + 𝑥 + 2 (d) 𝑝(𝑥) = 𝑥 3 + 3𝑥 2 − 4 5. Find the largest possible domain of each of the following (real-valued) rational functions. (a) 𝑓(𝑥) = −𝑥 3 +5𝑥+11 𝑥 3 +9𝑥 2 +23𝑥+15 (b) 𝑔(𝑥) = −𝑥 4 +2𝑥−9 2𝑥 2 −3𝑥−2 (c) ℎ(𝑥) = 2𝑥 3 +𝑥−5 𝑥 3 −𝑥 2 +2𝑥−2 (d) 𝑓(𝑥) = (𝑥+3) 2 𝑥+3 Page 2 6. Resolve each of the following expressions into partial fractions. (a) 3𝑥 2 +18𝑥+18 𝑥 3 +7𝑥 2 +14𝑥+8 (b) 𝑥 2 +4𝑥+1 𝑥 3 +3𝑥 2 −𝑥−3 (c) −4𝑥 2 +9𝑥−23 𝑥 3 +5𝑥 2 +3𝑥+15 (d) 𝑥 2 +4𝑥+8 𝑥 3 +9𝑥 2 +27𝑥+27 (e) 4𝑥(𝑥+4) (𝑥 2 −4)(𝑥+2) (f) 𝑥 3 +9𝑥 2 +9𝑥+15 (𝑥 2 +1)(𝑥 2 +4𝑥+4) (g) 2𝑥 5 −𝑥 4 +2𝑥 3 +8𝑥 2 +5𝑥−1 𝑥 4 +𝑥 3 +𝑥+1 - End - Page 1 GE1359 Problem Set 3 (Part II) Summer 2022 1. Solve the following inequalities (a) |3𝑥 + 2| ≤ 4 (b) |2𝑥 − 7| < −5 (c) −3|𝑥 + 2| − 2 < −14 (d) |𝑥 + 3| ≥ 6𝑥 (e) |𝑥 − 2| ≤ 𝑥 2 (f) |2𝑥 − 1| < |𝑥 + 4| (g) |𝑥 − 3| + 4 < |3𝑥 + 1| 2. Let 𝑥, 𝑦 > 0 . Use the AM-GM inequality to find the minimum of 𝑆 = 50 𝑥 + 20 𝑦 + 𝑥𝑦 and the values of 𝑥 and 𝑦 at which this minimum value is attained. 3. Prove that for any 𝑎, 𝑏, 𝑐 > 0 , we have (𝑎 + 9𝑏)(𝑏 + 9𝑐)(𝑐 + 9𝑎) ≥ 216𝑎𝑏𝑐. 4. Let 𝑥 > 0 . Without using calculus, find the maximum value of 𝑓(𝑥) = (1 + 𝑥)(1 + 𝑥)(1 − 𝑥) and the value of 𝑥 at which this maximum value is attained. 5. Let 𝑎, 𝑏, 𝑐 > 0 . Without using calculus, find the minimum value of 𝑆 = 𝑎 𝑏 + √𝑏 𝑐 + √𝑐 𝑎 3 6. Using Gaussian elimination, solve the following systems (a) { 𝑥 − 𝑦 + 3𝑧 = 15 −3𝑥 + 2𝑦 + 𝑧 = 4 2𝑥 − 3𝑦 + 2𝑧 = 9 (b) { 2𝑥 + 𝑦 − 3𝑧 = 12 4𝑥 + 𝑧 = 5 3𝑥 − 𝑦 + 2𝑧 = 1 (c) { 𝑥 − 2𝑦 + 3𝑧 = 3 3𝑥 − 5𝑦 + 𝑧 = 4 (d) { 𝑥 + 𝑦 + 2𝑧 = 0 −3𝑥 + 4𝑦 + 𝑧 = 0 −2𝑥 + 5𝑦 + 3𝑧 = 0 (e) { 𝑥 + 𝑦 + 𝑧 = 9 2𝑥 + 5𝑦 + 11𝑧 = 52 2𝑥 + 𝑦 − 𝑧 = 0 (f) { 𝑥 + 3𝑦 − 2𝑧 − 𝑤 = 1 2𝑥 + 5𝑦 − 𝑧 + 3𝑤 = 2 −𝑥 − 𝑦 − 3𝑧 + 2𝑤 = −3 (g) { 𝑥 + 2𝑦 + 3𝑧 + 4𝑤 = −2 2𝑥 + 4𝑦 + 5𝑧 + 9𝑤 = 1 −3𝑥 − 6𝑦 + 𝑤 = 5 (h) { 𝑥 − 3𝑦 + 4𝑧 + 7𝑤 = 1 2𝑥 − 6𝑦 − 3𝑧 + 5𝑤 = 2 4𝑥 − 12𝑦 − 17𝑧 + 𝑤 = 4 Page 2 7. Consider the system { 𝑥 − 2𝑦 + 𝑧 = 1 𝑥 − 𝑦 + 2𝑧 = 2 𝑦 + 𝑐 2 𝑧 = 𝑐 Find all possible values of 𝑐 such that the system (a) has a unique solution. (b) has infinitely many solutions. (c) has no solution. 8. Consider the system { 2𝑥 + 𝑦 − 𝑏𝑧 = 3 𝑎𝑦 − 𝑧 = 2 −2𝑥 + 5𝑦 = 1 Find all possible values of 𝑎 and 𝑏 such that the system (a) has a unique solution. (b) has infinitely many solutions. (c) has no solution. 9. Solve the equation 𝑥 4 − 4𝑥 3 − 2𝑥 2 + 12𝑥 − 3 = 0 , given that 2 − √3 is one of the roots. 10. Find all possible (real and complex) roots the equation 2𝑥 3 − 7𝑥 2 + 6𝑥 + 5 = 0 , given that 2 + 𝑖 is one of the roots. 11. Let 𝛼 , 𝛽 be the roots of the equation 3𝑥 2 + 5𝑥 + 𝑐 = 0 . Find the following: (a) 𝛼 + 𝛽 (b) 𝛼 2 𝛽 + 𝛼𝛽 2 (c) 𝛼 2 + 𝛽 2 (d) 𝛼 3 + 𝛽 3 (e) 𝛼 4 + 𝛽 4 12. Let 𝛼 , 𝛽 be the roots of the equation 2𝑥 2 + 4𝑥 + 7 = 0 (a) Find a quadratic whose roots are 𝛼 2 − 1 and 𝛽 2 − 1 (b) Find a quadratic whose roots are 1 𝛼 2 +1 and 1 𝛽 2 +1 13. If 1, 1, 𝛼 are the roots of 𝑥 3 − 6𝑥 2 + 9𝑥 − 4 = 0 , find 𝛼 14. If − 3 2 , 1, 2 are the roots of 2𝑥 3 + 𝑎𝑥 2 + 𝑏𝑥 + 6 = 0 , find 𝑎 Page 3 15. If the product of the roots of 4𝑥 3 + 16𝑥 2 − 9𝑥 − 𝑐 = 0 is 9, find 𝑐 16. If 𝛼 , 𝛽 , 𝛾 are the roots of the equation 𝑥 3 + 𝑝𝑥 2 + 𝑞𝑥 + 𝑟 = 0 , find 1 𝛼 2 𝛽 2 + 1 𝛽 2 𝛾 2 + 1 𝛼 2 𝛾 2 17. Let 𝛼 , 𝛽 , 𝛾 ( ≠ 0 ) be the roots of 𝑥 3 + 𝑝𝑥 2 + 𝑞𝑥 + 𝑟 = 0 , where 𝑟 ≠ 0 Find a cubic equation (in terms of 𝑝 , 𝑞 and 𝑟 ) whose roots are 𝛽𝛾 𝛼 , 𝛼𝛾 𝛽 and 𝛼𝛽 𝛾 18. Solve the equation 3𝑥 4 + 16𝑥 3 + 24𝑥 2 − 16 = 0 , given that it has a multiple root. - End - Page 1 GE1359 Problem Set 4 Summer 2022 1. Prove by contradiction that (a) if 𝑥 is irrational, so is −𝑥 ; (b) there is no smallest positive rational number. 2. Prove by induction that for all positive integers 𝑛 , (a) 1 + 2 + 3 + 4 + ⋯ + 𝑛 = 𝑛(𝑛+1) 2 (b) 1 3 + 2 3 + 3 3 + 4 3 + ⋯ + 𝑛 3 = ( 𝑛(𝑛+1) 2 ) 2 (c) 1 + 1 2 2 + 1 3 2 + ⋯ + 1 𝑛 2 ≤ 2 − 1 𝑛 (d) 1 ∙ 𝑛 + 2 ∙ (𝑛 − 1) + 3 ∙ (𝑛 − 2) + ⋯ + (𝑛 − 1) ∙ 2 ∙ +𝑛 ∙ 1 = 𝑛(𝑛+1)(𝑛+2) 6 (e) (3𝑛 + 1) ∙ 7 𝑛 − 1 is divisible by 9 (f) 1 + 1 √2 + 1 √3 + ⋯ + 1 √𝑛 > 2(√𝑛 + 1 − 1) 3. Prove by induction that there exists 𝑁 0 ∈ ℕ such that 2 𝑛 ≥ 𝑛 3 for all 𝑛 ≥ 𝑁 0 4. Prove that for all positive integers 𝑛 > 3 , √3 3 > √𝑛 𝑛 5. Show that for every positive integer 𝑛 , (1 + √5 ) 𝑛 − (1 − √5 ) 𝑛 2 𝑛 ∙ √5 is a positive integer. 6. Let {𝑎 𝑛 } be a sequence of real numbers, where 𝑎 0 = 1 , 𝑎 1 = 6 , 𝑎 2 = 45 and 𝑎 𝑛 − 𝑎 𝑛+1 + 1 3 𝑎 𝑛+2 − 1 27 𝑎 𝑛+3 = 0 for 𝑛 = 0, 1, 2, ... Using mathematical induction to show that 𝑎 𝑛 = 3 𝑛 (𝑛 2 + 1) for 𝑛 = 0, 1, 2, ... - End - Page 1 GE1359 Problem Set 5 Summer 2022 1. If the letters in the word SMILEY are used to form a three-letter code word, find how many code words can be obtained. 2. How many combinations of a five-person subcommittee can be formed from a committee of twelve people? 3. Ten people form a committee. In how many different ways can a chairman, a vice-chairman and a secretary be selected. 4. (a) Find the number of distinguishable permutations obtained from the letters of the word TENNESSEE. (b) Find the number of distinguishable permutations obtained from the letters of the word TENNESSEE such that the words begin and end with the letter N. (c) Find the number of distinguishable permutations obtained from the letters of the word TENNESSEE such that the words end with the four E ’s 5. A committee of 5 is to be selected from 6 boys and 4 girls. How many selections are there if the committee must contain 2 or 3 boys. 6. How many three-digit numbers can be obtained from the digits 1, 2, 3 and 4 if it is assumed that no digit can be used more than once? 7. How many permutations of abcde are there in which the first character is a , b or c and the last character is c , d or e 8. Expand the following with the binomial theorem. (a) (2𝑥 − 3) 4 (b) (𝑧 − 1 𝑧 ) 5 (c) ( 2𝑥 𝑦 − 𝑦 4𝑥 2 ) 5 9. Find the coefficients of the terms specified in the expansions of the following. (a) ( 1 5 − 5𝑥) 9 , the term in 𝑥 6 (b) (2𝑦 − 3) 7 , the term in 𝑦 3 (c) (5𝑧 − 3 𝑧 3 ) 8 , the term in 1 𝑧 4 and the constant term Page 2 10. For any positive integer 𝑛 , let 𝐶 𝑟 𝑛 be the coefficient of 𝑥 𝑟 in the binomial expansion of (1 + 𝑥) 𝑛 , prove that (a) 𝐶 𝑟 𝑛 = 𝐶 𝑟 𝑛+1 − 𝐶 𝑟−1 𝑛 (b) 𝐶 𝑟 𝑛 + 2 𝐶 𝑟+1 𝑛 + 𝐶 𝑟+2 𝑛 = 𝐶 𝑟+2 𝑛+2 (c) (3𝑛 + 1)( 𝐶 𝑛 2𝑛 ) 2 = (𝑛 + 1) [( 𝐶 𝑛 2𝑛+1 ) 2 − ( 𝐶 𝑛−1 2𝑛 ) 2 ] where 𝑛 is an integer greater than 2. 11. Let 𝑚 , 𝑛 be positive integers. (a) Prove the addition formula: 𝐶 𝑟 𝑛 + 𝐶 𝑟−1 𝑛 = 𝐶 𝑟 𝑛+1 Hence show that 𝐶 0 𝑚 + 𝐶 1 𝑚+1 + 𝐶 2 𝑚+2 + ⋯ + 𝐶 𝑛 𝑚+𝑛 = 𝐶 𝑛 𝑚+𝑛+1 (b) Find the sum 𝑆 𝑛 = 𝑚! + (𝑚 + 1)! 1! + (𝑚 + 2)! 2! + ⋯ + (𝑚 + 𝑛)! 𝑛! - End -