UNIVERSITY OF SOUTHAMPTON. This paper has two parts, part A and part B: you should answer all questions from part A and part B.

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1 UNIVERSITY OF SOUTHAMPTON MATH1017W1 SEMESTER 2 EXAMINATION 2011/12 Mathematics for Electronic and Electrical Engineering Duration: 120 min This paper has two parts, part A and part B: you should answer all questions from part A and part B. Part A consists of 20 multiple-choice questions each worth 2 marks. Enter your answers into the designated answer sheet (AS/MATH1017/Part A/2011). Part B consists of 5 questions each worth 12 marks. answers into the standard University answer booklet. Write your Using the provided tag, attach the completed answer sheet for part A to the answer booklet(s) for Part B. Formula sheet FS/ /11 will be provided. Printed foreign language dictionaries MAY be consulted. The University Approved Calculator MAY be used. Copyright 2012 c University of Southampton Number of Pages 10

2 2 MATH1017W1 SECTION A Exactly one of the 5 options (a), (b), (c), (d), (e) in each of the following 20 questions is correct. (Sometimes that is (e) none of the above!) Find that option and blob it on the designated answer sheet. Correct answers will attract 2 marks, no answer 0 marks and wrong answers 0.5 marks. dt +2x = sin(t) is t (a)t 2, (b)exp ( 2 3 t3), (c)t, (d)exp(2t 2 ), A1. (2 marks) An integrating factor for the differential equation dx A2. (2 marks) The equation is exact if (a)g(x) = tsinx, (b)g(x) = cosx, (c)g(x) = sinx, (d)g(x) = 6x 2 t 2, (3x 2 t 2 +tcosx) dx dt +2x3 t = G(x) A3. (2 marks) Which of the following is a particular integral to the equation? d 2 x dt 2 5dx dt +6x = e3t (a)e 3t, (b)te 3t, (c)e 2t, (d) 1 2 e3t, (e) none of the above?

3 3 MATH1017W1 A4. (2 marks) Which of the following ordinary differential equations is nonlinear? (a) d2 x dt 3dx +6x = 2 t5 dt (b) exp(t 3 ) d3 x x dt 3 5d2 dt (c) d2 x dt 2 5sin(t)dx dt (d) dx dt t x = 0, (e) none of the above? 2 +6x = 0, +x = cos(t), A5. (2 marks) The expression tanh (a)j, (b) 1 2 (1+j), (c)1, (d) ( j π ) can be simplified to 4 1 2, A6. (2 marks) The locus of the pointz that satisfies the equation z 1+j = 4 is a (a) a circle centred on( 1,1) with radius 2, (b) a circle centred on(1, 1) with radius 2, (c) a circle centred on( 1,1) with radius 4, (d) a circle centred on(1, 1) with radius 4, A7. (2 marks) The determinant of the matrixc = (a) 36, (b) 36, (c) 12, (d)12, is TURN OVER

4 4 MATH1017W1 ( ) 5 3 A8. (2 marks) The eigenvalues of the matrix C = are (a) λ = 5 + 3j and λ 3 = 5 3j, (b)λ 1 = 2 andλ 2 = 8, (c)λ 1 = λ 2 = 8, (d)λ 1 = 1 andλ 2 = 3, A9. (2 marks) The following matrices are given: A = ( ) 2 1, B = Which of the following statements is true? (a)ab andbc exist butac does not exist, (b)bc andac exist butab T does not exist, (c)ba exists but neitherc T B norab T exist, (d)bc andac T exist butac does not exist, and C = ( A10. (2 marks) Given the matrix A = 1 0 γ γ 1 1, which of the following conditions onγ ensures that the equationax = 0 has a unique solutionx? (a) γ = 4, (b) γ = 1, (c) γ = 4 and γ = 1 (d) γ < 0, (e) none of the above. ). A11. (2 marks) Which of the following is the Laplace transform of the function te 2t? 1 (a) (s+2) 2, (b) 1 (s 2) 2, (c) 1 (s+2), (d) 2 (s 2) 2,

5 5 MATH1017W1 A12. (2 marks) Given the functions f(t) = t 3 cost and g(t) = sin ( t 2), which of the following is true? (a)g is neither odd nor even andf is even, (c) both are odd, (d)f is odd andg is even, (b)g is odd andf is even, A13. (2 marks) The inverse Laplace transforml 1 { } 1 s 2 +3s+2 equals (a) e t e 2t, (b) e t sin2t, (c) 1 2 e t sin2t, (d) 1 2 et sin2t, A14. (2 marks) The inverse Laplace transform of (a) 1 2 H(t 2)sin(2(t 2)), (b) 1 2 H(t+2)sin(2(t+2)), (c) 1 2 H(t+2)cos(2(t+2)), (d)2h(t+2)cos(2(t+2)), e 2s s 2 +4 is A15. (2 marks) The value of the double integral 1 y 2 y=0 x=1 (x 2 y +1)dxdy is (a)5/9, (b) 19/24, (c)7/24, (d)7/9, TURN OVER

6 6 MATH1017W1 A16. (2 marks) The integral over the unit disk in the plane, is (use polar coordinates) x 2 +y 2 1 x 2 dxdy (a)4π/3, (b)4π, (c)π, (d)π/4, A17. (2 marks) If f = cosxcoshy, a function ofxandy, wherexandy both depend on t, which of the following is equal to df dt? (a) sinxcoshy dx dt +cosxsinhydy dt, (b) cosxsinhy dx dt +sinxcoshydy dt, (c) sinxcoshy dx dt cosxsinhydy dt, (d) sinxcoshy dx dt cosxsinhydy dt, A18. (2 marks) The equation of the plane through the pointsa = (1,3,5),B = (0,2,4) andc = (7,1,0) is (a) 3x 11y +8z = 10, (b)3x 11y +8z = 10, (c) 3x+11y +8z = 10, (d) 3x 11y +8z = 6, A19. (2 marks) Given that which of the following is 2 f y x? f(x,y) = e y/x, (a) y x 3ey/x, (b) 1 x 2ey/x, (c) e y/x, (d) ( 1 x + y ) e y/x 2 x 3,

7 7 MATH1017W1 A20. (2 marks) Which of the following functions f(t) has a Fourier series of the form f(t) = a n cos(nt)? n=1 (a)f(t) = 1+t 2 in π < t < π and f(t+2π) = f(t), (b)f(t) = t 2 in 1 < t < 1 and f(t+2) = f(t), (c)f(t) = t in π < t < π and f(t+2π) = f(t), (d)f(t) = tcost in 1 < t < 1 and f(t+2) = f(t), TURN OVER

8 8 MATH1017W1 SECTION B B1. (a) (3 marks) Evaluate the determinant of the matrix C = α and state the value ofαfor which the inverse ofcdoes not exist. (b) (5 marks) FindC 1 whenα = 4. (c) (4 marks) Hence, or otherwise, solve the set of linear equations x + 2y + 3z = 8, y + 4z = 6, 5x + 6y = 17, B2. (a) (6 marks) Show thatλ = 2 is an eigenvalue of the matrix A = and find the other eigenvalues. (b) (6 marks) Find three independent eigenvectors of A.

9 9 MATH1017W1 B3. (a) (4 marks) Make use of the table of Laplace transforms to find the following inverse transforms { 1 (i) L 1 s 1 { (ii) L 1 } } s+3 s 2 +6s+13 { (iii) L 1 1 s 2 +6s+13 }. (b) (5 marks) Use partial fractions and part (a) to obtain { } L 1 20 (s 1)(s 2. +6s+13) (c) (3 marks) Use Laplace transforms and part (b) to find the solution to the differential equation d 2 x dt + 6dx 2 dt + 13x = 20et which satifies the initial conditions x = 0 and dx dt = 0 when t = 0. B4. (a) (6 marks) Find the solution of differential equation t 2dx dt = x2 +3xt 3t 2, which satisfies the initial conditionx = 1 whent = 1. (b) (6 marks) Obtain the general solution of the second order differential equation d 2 y dx 2 +5dy dx +6y = e 2x. TURN OVER

10 10 MATH1017W1 B5 (a) (10 marks) R V(t) L C A circuit consists of an inductance, a capacitor and a resistor in series with a voltage source. The voltage applied is V(t) = V 0 cosωt, where L = 1H, C = 10 4 F, R = 200Ω, V 0 = 10V and ω = 50s 1. Write down the differential equation for the charge q(t) flowing through the circuit and hence the differential equation for the current I(t). Find the steady state current I(t) and its amplitude. (b) (2 marks) Check your result for the amplitude of the steady state current by calculating the impedance of the circuit. END OF PAPER

UNIVERSITY OF SOUTHAMPTON

UNIVERSITY OF SOUTHAMPTON MATH03W SEMESTER EXAMINATION 0/ MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING Duration: 0 min This paper has two parts, part A and part B. Answer all questions from part