MATHEMATICS 132 Applied Mathematics 1A (Engineering) EXAMINATION

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1 MATHEMATICS 132 Applied Mathematics 1A (Engineering) EXAMINATION DURATION: 3 HOURS 26TH MAY 2011 MAXIMUM MARKS: 100 LECTURERS: PROF J. VAN DEN BERG AND DR J. M. T. NGNOTCHOUYE EXTERNAL EXAMINER: DR K. ARUNAKIRINATHER INSTRUCTIONS 1. Fill in the following: Student Number Signature: 2. Write your answers on the question paper and in the space provided. Rough work can be done on the back of each page. 3. This paper comprises 20 pages, including this cover page. Check that you have them all. 4. For answers with decimal numbers, use two decimal places. You may use g = 9.81 N/kg for the value of the gravitational constant. 5. ANSWER ALL QUESTIONS. For Marker Only total

2 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 2 QUESTION 1 [6 marks] (a) Let f = (5, 3, 2) and c = ( 2, 2, 1). Determine the following: (i) The unit vector ĉ in the direction of c; (1) (ii) The projection of f on c. (1) (b) Suppose u and v are unit vectors such that u v = 1. Determine the angle θ between u and v. (2)

3 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 3 QUESTION 1 (continued) (c) Prove that if u = (u 1, u 2 ), v = (v 1, v 2 ) and w = (w 1, w 2 ) are any three vectors in R 2, then u (v + w) = u v + u w. (2) QUESTION 2 [9 marks] (a) Let l be the line with generic equation r(t) = (1, 2, 1)+t(1, 0, 1). Find the two points on l that are a distance 8 from (1, 2, 1). (2) (b) Let l be the line in Question 2(a). Give examples of the following. Provide a very brief explanation in each case. (Note that in each of (i), (ii) and (iii) below many examples exist satisfying the stated condition. You need only provide one example in each case.) (i) The generic equation of a line which is parallel to l but not equal to l. (1)

4 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 4 QUESTION 2(b) (continued) (b) (ii) The generic equation of a line that intersects with l and which is perpendicular to l. (1) (iii) The generic equation of a plane containing l. (2) (c) Let r 1 = ( 2, 1, 1) + t(1, 1, 1) and r 2 = (0, 3, 3) + t( 1, 0, 2) be the position of two particles at time t. (ii) Do the two particles collide? (1) (iii) As straight lines, do the paths of the two particle intersect? (2)

5 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 5 QUESTION 3 [10 marks] Consider the points A = (4, 1, 1), B = (1, 1, 3) and C = (5, 1, 2). (a) Show that the points A, B and C are not collinear and so define a unique plane Π. (1) (b) (i) Find a generic equation for Π. (2) (ii) Find a normal vector to Π. (2) (iii) Find the Cartesian equation for Π. (2)

6 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 6 QUESTION 3(b) (continued) (b) (iv) Find the foot of the perpendicular from point D = (5, 4, 2) onto Π. Hence find the shortest distance from D to Π. (3)

7 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 7 QUESTION 4 [13 marks] Consider the system of equations Ax = b where a A = and b = c d, e where a, c, d, e are parameters. (a) By row reductions, determine for which values of a, c, d and e does the equation Ax = b have a solution. (5)

8 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 8 QUESTION 4 (Continued) (b) Let a = 1, c = 2, d = 5, e = 5 so that b = [ ] T. What is the solution of the equation Ax = b (1) (c) Is [ ] T a linear combination of the A 1, A 2 and A 3? If your answer is yes, find the coefficients. (4)

9 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 9 QUESTION 4 (Continued) (d) What is the rank of A? (1) (e) Are the columns of A linearly independent? give reason. (1) (f) Find the matrix of the linear transformation that maps x R 3 1 to y = [ 3x 1 + 2x 2 5x 3 9x 1 6x x 3 x 1 11x x 3 ] T. (2)

10 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 10 QUESTION 5 [6 marks] (a) Let A be an m n matrix and let y = [ y 1 y 2... y m ] with at least one non zero entry such that ya = 0. What can you say about the rank of A. (2) (b) Let A = [a ij ] be a 3 3 matrix and A i j be the cofactor ij of the matrix A. (i) Define A i j and give a formula for the expansion of the determinant A of A by row 1. (1) (ii) Evaluate briefly (1) A 1 1 a 12 + A 2 1 a 22 + A 3 1 a 32 = (a) let a = (1, 3, 4) and b = (4, 2, 3). Using the geometrical meaning of the cross product, evaluate the sine of the angle α, (0 < α < π) between a and b, and hence the angle α. (2)

11 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 11 QUESTION 6 [16 marks] (a) B is a 3 3 matrix. (i) Find A if B 1 4B 3 + 2B 2 AB = 2B 1 + 5B 2 6B 3. 9B 1 + 2B 2 11B 3 (2) (ii) Find C if BC = [ B 1 ( 3) + B 2 + B 3 B 3 B 2 (4) + B 1 B 1 B 2 B 3 B 2 ]. (2) (b) Now let A = Find the inverse A 1 of A using two methods. (i) the method row reductions. (4)

12 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 12 QUESTION 6 (Continued) (ii) the method of cofactors. (2) (c) Using your answer of (b) above, solve the equation Ax = [ ]T. (2) (d) Evaluate the determinant (4)

13 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 13 QUESTION 7 [15 marks] The box located at A weigh 20 N and is supported by the smooth inclined surface and the strings AB and AC. The coordinates of A are given by A = (8, 3, 5) m. 2 y (0,8,2) (9,12,0) C D B 16m F A E 10m 3m x z (a) Determine the vector u = DF and v = DE and hence a normal n to the plane Π containing the points D, E and F. (You may use n = u v.) (2) (b) Draw the free-body diagram of the box showing all the external forces acting on it. (2)

14 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 14 QUESTION 7 (Continued) (c) Determine the tensions in the strings AB and AC and the normal reaction exerted by the inclined surface on the box. (4)

15 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 15 QUESTION 7 (Continued) (d) Bar EG is in the xy-plane and is parallel to the x-axis. The origin is at F. The tension in the cable GH is 3 6N. Determine the reactions at the fixed support F. (6) 6m y E 6m G F z 9m H 3m

16 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 16 QUESTION 8 [13 marks] (a) Consider the truss depicted in the figure below with a pin support at B and a roller support at A. 500N 30 E 4m C D 4m A B m 4m (i) Find the reactions at the supports A and B. (3)

17 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 17 QUESTION 8(a) (Continued) (ii) Using the method of joints, find the axial force in the member BC. (4)

18 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 18 QUESTION 8 (Continued) (b) The axial force in the member DF of the truss is 500 N and the member is in tension. Determine the weight W of the box and the axial force in the member EF. Indicate if the member EF is in compression or in tension. (6) F 3m D E W m 3m

19 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 19 QUESTION 9 [12 Marks] (a) The box of weight W in the picture below lies in equilibrium on the smooth surface. The normal force exerted by the surface on the box is 50 N. Determine the tension in the rope and the weight W of the box. (4)

20 MATHEMATICS Applied Mathematics 1A (Engineering) Exam 2011 PAGE 20 QUESTION 9 (Continued) (b) The support at the left end of the beam will fail if the moment about A of the 20 N force exceeds 30 N.m Based on this criterion, what is the largest allowable length l of the beam? (5) l 60 30

MATHEMATICS 132 Applied Mathematics 1A (Engineering) SUPPLEMENTARY EXAMINATION

MATHEMATICS 132 Applied Mathematics 1A (Engineering) SUPPLEMENTARY EXAMINATION DURATION: 3 HOURS 17TH JUNE 2011 MAXIMUM MARKS: 100 LECTURERS: PROF J. VAN DEN BERG AND DR J. M. T. NGNOTCHOUYE EXTERNAL EXAMINER: