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 A and part B for full marks. Part A consists of 0 multiple-choice questions each worth marks. Enter your answers into the designated answer sheet (AS/MATH03/Part A/0. Part B consists of 5 questions each worth 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/00-7/ will be provided. Printed foreign language dictionaries MAY be consulted. The University Approved Calculator MAY be used.

MATH03W PART A Exactly one of the 5 options (a, (b, (c, (d, (e in each of the following 0 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 marks, no answer 0 marks and wrong answers 0.5 marks. A. ( marks If f(x = ln( x, then f (x = (a x, (b x, (c x, (d x, A. ( marks Given the function f(x, y = x sin (y, which of the following is true? (a f x = 4x sin(y cos(y (b f x = x sin (y (c f x (d f x = x cos(y = sin(y cos(x f and y = x cos(y, and f y = x sin(y, f and y = x sin(y cos(y, f and y = sin(x cos(y, A3. ( marks If y xy = 6, then dy dx = x (a x y, (b y x y, (c y y x, (d y y x,

3 MATH03W A4. ( marks The region R in the first quadrant is enclosed by the lines y = and y = sin x from x = 0 to x = π. The volume of the solid obtained by revolving R about the x-axis is given by (a π (c π π 0 π 0 x sin x dx, (b π ( sin x dx, (d π π 0 π 0 x cos x dx, sin x dx, A5. ( marks If z = 3 + j, then z + z = (a 3 + 3j, (b 3 + 6j, (c + 3j, (d 3j, A6. ( marks The principal value of the argument of 4 + 3j is (a tan ( 3 4 + π, (b tan ( 4 3 π, (c tan ( 4 3 π, (d tan ( 3 4 π, A7. ( marks The general solution of the differential equation d y 9y = 0 is dx (a y = (Ax + Be 9x, (b y = Ae 3x + Be 3x, (c y = Axe 3x, (d y = Bxe 3x, A8. ( marks The general solution of the differential equation dy dx = y sec x is (a y = tan x + C, (b y = tan x + Ce x, (c y = e tan x + C, (d y = Ce tan x

4 MATH03W A9. ( marks The derivative of the function f(x = tan (x + x is x + (a cos (x + x, (b x + + (x + x, (c + (x + x, (d cos (x + x, A0. ( marks If 4 (a 6, (b 3, (c 0, (d 6, f(xdx = 6, what is the value of 4 f(5 xdx? A. ( marks For which value of k will the function f(x = x + k x at x =? (a 4, (b, (c, (d 4, have a local maximum ( π A. ( marks The derivative of the function f(x = ln cos is x π ( π (a x cos ( π, (b tan, (c x π ( π x tan, (d π ( π x x x tan x, A3. ( marks x e x3 dx = (a 3 ln ( e x3 + C, (b ex3 3 + C, (c 3e x3 + C, (d x 3 3e x3 + C,

5 MATH03W A4. ( marks The area of the region in the first quadrant, enclosed by the graph of the curve y = x( x and the x-axis is (a 3, (b 3, (c 5, (d, 6 A5. ( marks (a 4 ln x 4, 4 x 4 dx = (b tan ( x, (c ln x ln x +, (d ln x ln x +, A6. ( marks The improper integral (x 3 dx is (a not defined, (b equal to, (c equal to 3, (d equal to 3, A7. ( marks If a = i j k then (a a = 3 and â = (i j k, 9 (b a = 9 and â = (i j k, 3 (c a = 3 and â = i j k, (d a = 9 and â = (i j k, 9

6 MATH03W A8. ( marks If a = i + j + k, b = i j + k, and c = i j k, then a c b = (a 5, (b 6, (c 5i 6j 4k, (d 9i + 5j k, A9. ( marks Given that A = (a (AB T = ( 4 3 9 ( 4 3 (c (AB T does not exist, (d (AB T = and B = 3, (b (AB T = 6 9 5 3 0 8 5 7 ( 4, 3 9, then A0. ( marks Let A denote the cofactor of the matrix A = 0 3 3 4 in the second row and second column, and let det(a denote the determinant of A. Then (a A = and det(a = 5, (b A = 4 and det(a = 5, (c A = and det(a = 5, (d A = 4 and det(a = 5,

7 MATH03W PART B B. A function is defined as f(x = x x +. (a (4 marks Find all stationary points of the function and determine their nature. (b (3 marks Find all points of inflection of the function. (c ( marks Calculate lim f(x and lim f(x. x x (d (3 marks Sketch the graph of f(x showing clearly all your results from parts (a-(c. B. (a (6 marks Given that A = 3 3 determine (i A A T, (ii adj(a. (b (6 marks Given that a = i 3j + 4k and b = i j 3k find (i a b, (ii the angle θ between a and b, (iii the component of b in the direction of a.

8 MATH03W B3. (a (3 marks If y = sinh x use the exponential definition of sinh(y to show that y satisfies the equation e y xe y = 0. (b (5 marks Using u = e y rewrite the above equation as a quadratic and hence deduce that sinh x = ln(x + x +. (c (4 marks By differentiating the result in (b verify that d dx (sinh x = x +. B4. (a (6 marks Plot the number z = 3j on an Argand diagram and express z in exponential form. Hence, or otherwise, write z 5 and z both in exponential form (giving the principal value of the argument in each case and in the form x + jy. (b (6 marks Find the solution of the differential equation dy dx which satisfies y = when x = π. = xy sin x

9 MATH03W B5. (a (5 marks Use de Moivre s theorem to show that (i cos 4θ = cos 4 θ 6 cos θ sin θ + sin 4 θ (ii sin 4θ = 4 sin θ cos 3 θ 4 sin 3 θ cos θ. (b (3 marks Evaluate the integral x x 3 dx. (c (4 marks Use the substitution x = sinh u to evaluate the integral x dx + x. END OF PAPER