City University of Hong Kong Course code & title: MA1201 Calculus and Basic Linear Algebra II Session: Semester B, 2021-2022 Time allowed: Three hours Academic honesty pledge for the online final assessment : I pledge that the answers in this exam 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 attempt to communicate with any other person during the exam; neither will I give or attempt to give assis- tance to another student taking the exam; • 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 disci- plinary action. Name: Student ID: Signature: or Please write “I pledge to follow the Rules on Academic Honesty and understand that violations may lead to severe penalties” onto the first examination answer sheet. Instructions : • Please show your work. Unsupported answers will receive NO cred- its. • Make sure you write down the correct lecture session you have regis- tered for, together with your full name and student ID on the front page of your answer script. • Exams submitted to wrong lecture sessions will NOT be graded and will receive 0 POINTS This paper has 4 pages (including this cover page). 1. This paper consists of 7 questions (100 points in total). 2. Attempt ALL questions. 3. Start each question on a new page. 4. Show your work clearly to receive full credits. This is a closed-book examination. Candidates are allowed to use the following materials/aids: • Non-programmable portable battery operated calculator. Materials/aids other than those stated above are not permitted. Candi- dates will be subject to disciplinary action if any unauthorized materials or aids are found on them. – 3 – 1. [15] Evaluate the following integrals. (a)[5] ∫ (sin x + 2 cos x ) 2 dx (b)[5] ∫ 1 √ x 3 √ xdx (c)[5] ∫ 1 0 x + 1 3 √ x + 2 dx 2. [20] Evaluate the following integrals. (a)[6] ∫ xe sin − 1 x √ 1 − x 2 dx (b)[7] ∫ xe x sin xdx (c)[7] ∫ 12 x 2 + 2 ( x + 2)(9 x 2 − 6 x + 2) dx 3. [15] (a)[7] Let R be the region bounded by the curve y = − x 2 − 4 x − 3 and the line y = − 1. Find the volume of the solid generated by revolving the region R about the line x = 1. (b)[8] Find the area of the surface generated by revolving the curve x = sin 3 t and y = cos 3 t , where 0 ≤ t ≤ π 2 , about the line y = − 1. 4. [15] (a)[8] Find the equation of the plane passing points A = (2 , 3 , 1) and B = (0 , 1 , − 1), and parallel to the line, which is the intersection between the plane x + y + z = 3 and the plane − y − z + 2 x = 2. (b)[7] Let A = (1 , − 2 , 1), B = (2 , − 3 , 2) be two points in R 3 . Let the line L 1 passing through A and B Let the parametric equations of line L 2 be x = 1 − 2 t, y = 2 + t, z = 3 t + 2 Here t ∈ R is the parameter. Calculate the distance between L 1 and L 2 5. [15] (a) [8] Compute ( − i + sin( π 4 ) + i cos( π 4 ) ) 4 5 and express your answer in Euler’s form. Here i = √− 1. (b) [7] Solve z 6 − 3 z 3 + 2 = 0. Here z is complex number. 6. [10] For any positive integer n , we define a n × n matrix D n = a 1 b 2 · · · b n c 2 a 2 . . . c n a n , where a 1 , · · · , a n are non-zero real numbers, and b 2 , · · · b n , c 2 , · · · , c n are real numbers. Notice that for any 1 ≤ i, j ≤ n , we have [ D n ] ij = a i if i = j, b j if i = 1 and j ≥ 2 , c i if i ≥ 2 and j = 1 , 0 otherwise Here [ D n ] ij is the ( i, j )-th component of the matrix D n Compute det( D n ). Your answer should depend on n , a i , b i and c i 7. [10] Given a system of linear equations as follows. x 1 + 2 x 2 + 3 x 3 − x 4 = 1 , 3 x 1 + 2 x 2 + x 3 − x 4 = 1 , 2 x 1 + 3 x 2 + x 3 + x 4 = 1 , 2 x 1 + 2 x 2 + 2 x 3 − x 4 = 1 , 5 x 1 + 5 x 2 + 2 x 3 = 2 Solve the above linear system by Gaussian elimination. End Useful Elementary Integrals Constant and powers 1. ∫ k dx = kx + C 2. ∫ x n dx = x n + 1 n + 1 + C , n 6 = − 1 ln | x | + C , n = − 1 Exponentials 3. ∫ e x dx = e x + C 4. ∫ a x dx = a x ln a + C , a 6 = 1 , a > 0. Trigonometric functions 5. ∫ sin x dx = − cos x + C 6. ∫ cos x dx = sin x + C 7. ∫ sec 2 x dx = tan x + C 8. ∫ csc 2 x dx = − cot x + C 9. ∫ sec x tan x dx = sec x + C 10. ∫ csc x cot x dx = − csc x + C 11. ∫ tan x dx = ln | sec x | + C 12. ∫ cot x dx = ln | sin x | + C 13. ∫ sec x dx = ln | sec x + tan x | + C 14. ∫ csc x dx = ln | csc x − cot x | + C 15. ∫ sec 3 x dx = 1 2 [ sec θ tan θ + ln | sec θ + tan θ | ] + C Algebraic functions 16. ∫ 1 1 + x 2 dx = tan − 1 x + C 17. ∫ 1 √ 1 − x 2 dx = sin − 1 x + C Hyperbolic functions 18. ∫ sinh x dx = cosh x + C 19. ∫ cosh x dx = sinh x + C Useful Trigonometric Identities Pythagorean identities 1. sin 2 θ + cos 2 θ = 1. 2. 1 + tan 2 θ = sec 2 θ 3. 1 + cot 2 θ = csc 2 θ Double-angle formulas 4. sin 2 θ = 2 sin θ cos θ 5. cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ Half-angle formulas 6. sin 2 θ = 1 2 ( 1 − cos 2 θ ) 7. cos 2 θ = 1 2 ( 1 + cos 2 θ ) Compound-angle formulas 8. sin ( A ± B ) = sin A cos B ± cos A sin B 9. cos ( A ± B ) = cos A cos B ∓ sin A sin B 10. tan ( A ± B ) = tan A ± tan B 1 ∓ tan A tan B Sum-to-product formulas 11. sin A + sin B = 2 sin A + B 2 cos A − B 2 12. sin A − sin B = 2 cos A + B 2 sin A − B 2 13. cos A + cos B = 2 cos A + B 2 cos A − B 2 14. cos A − cos B = − 2 sin A + B 2 sin A − B 2 Product-to-sum formulas 15. sin A cos B = 1 2 [ sin ( A + B ) + sin ( A − B ) ] 16. cos A sin B = 1 2 [ sin ( A + B ) − sin ( A − B ) ] 17. cos A cos B = 1 2 [ cos ( A + B ) + cos ( A − B ) ] 18. sin A sin B = − 1 2 [ cos ( A + B ) − cos ( A − B ) ] Euler’s formulas 19. e ± i θ = cos θ ± i sin θ 20. e i θ + e − i θ = 2 cos θ , cos θ = 1 2 ( e i θ + e − i θ ) 21. e i θ − e − i θ = 2i sin θ , sin θ = 1 2i ( e i θ − e − i θ ) Remark . Formulas of the form A ± B = C ± D contain two separate formulas A + B = C + D , and A − B = C − D Likewise, formulas of the form A ± B = C ∓ D contain two separate formulas A + B = C − D , and A − B = C + D