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: 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????? pages, including this cover page. Check that you have them all. 4. ANSWER ALL QUESTIONS. For Marker Only 1 2 3 4 5 6 7 8 9 total
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 2 QUESTION 1 [6 marks] (a) Let f = (1, 3, 5) and c = (2, 2, 1). Determine the following: (i) The unit vector ĉ in the direction of c; (1) ĉ = c (2, 2,1) c 2 = 1(2, 2, 1) = 2 +( 2) 2 +1 2 3 (2, 3 2, 1). 3 3 (ii) The projection of f on c. (2) (f ĉ)ĉ = [(1, 3, 5) ( 2 3, 2 3, 1 3 )](2 3, 2 3, 1 3 ) = ( 2 3 2 + 5 3 )(2 3, 2 3, 1 3 ) = ( 10 9, 10 9, 5 9 ). (b) Suppose u and v are unit vectors such that u v = 3. Determine the angle θ, (0 < 2 θ < π ), between u and v. (3) 2 Hence θ = π 6 or θ = 30. cosθ = u v u v = u v = 3 2.
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 3 QUESTION 2 [10 marks] Let l be the line with generic equation r(t) = (0, 2, 3) + t(1, 2, 2). (a) Prove that (1, 0, 1) is not on l. (1, 0, 1) is on l if there exist a unique t such that (1, 0, 1) = r(t) = (t, 2 + 2t, 3 + 2t). That amounts to the three equations 1 = t, 0 = 2 + 2t and 1 = 3 + 2t. Solving gives t = 1 and t = 1. t is not unique and then (1, 0, 1) is not on l. (b) Find two points on l that are at distance 3 from (1, 0, 1). (4) Any point on l has the form r(t) for some t. Now r(t) (1, 0, 1) 2 = (t 1, 2 + 2t, 2 + 2t) 2 = (t 1) 2 + 2 (2 + 2t) 2 = 9t 2 + 14t + 9. We want r(t) (1, 0, 1) 2 = 3 2 = 9 or 9t 2 + 14t + 9 = 9 or t(9t + 14) = 0. Solve to get t = 0 or t = 14 9. The two points are then r(0) = (0, 2, 3) and r( 14 9 ) = ( 14 9, 10 9, 1 9 ). (c) Find the foot Q of the perpendicular from (1, 0, 1) to l. (5) Q = (t, 2 + 2t, 3 + 2t) for some parameter t. Now the displacement from (1, 0, 1) to Q is perpendicular to the direction of the line l. Meaning (Q (1, 0, 1)) (1, 2, 2) = 0 or (t 1, 2 + 2t, 2 + 2t) (1, 2, 2) = 0 or 9t = 7 which gives t = 7 9. Finally Q = r( 7 9 ) = ( 7 9, 4 9, 13 9. (1)
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 4 QUESTION 3 [11 marks] Consider the points A = (1, 2, 1), B = (0, 1, 3) and C = ( 1, 2, 5). (a) Show that the points A, B and C are not collinear and so define a unique plane Π. (2) AB = B A = ( 1, 1, 2) and AC = C A = ( 2, 0, 4). AB is not parallel to AC. Hence A, B and C are not collinear and define a unique plane. (b) (i) Find a generic equation with parameter t and s for Π. (2) r(t, s) = OA +tab + sac = (1, 2, 1)+t( 1, 1, 2) +s( 2, 0, 4) = (1 t 2s, 2 t, 1 + 2t + 4s) (ii) Find a normal n to Π. (2) i j k n = AB AC = 1 1 2 = 4i 2k = ( 4, 0, 2). 2 0 4 (iii) Find the cartesian equation for Π. (2) 4x 2z + d = 0. Now substitute A to have 4 2 + d = 0 or d = 6. Finally the cartesian equation of the plane is 4x 2z + 6 = 0.
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 5 QUESTION 3 (continued) (b) Find a formula for the foot Q of the perpendicular from X = (x 1, x 2, x 3 ) to Π. (4) We can write Q = (x 1, x 2, x 3 )+tn = (x 1 4t, x 2, x 3 2t) for some t. This means that Q is on the line passing through X and along the normal n. Now, Q is also on the plane, 4(x 1 4t) 2(x 3 2t) + 6 = 0. Solve for t to find t = 3 + x 1 10 5 + x 3. Substitute in Q 10 to finally have Q = ( 6 5 + x 1 5 2x 3 5, x 2, 3 5 2x 1 5 + 4x 3 5 ) (c) Consider the transformation that maps X = (x 1, x 2, x 3 ) R 3 to Q of (b), all vectors seen as column matrices. Write Q in matrix form and prove that it is not a linear transformation. (5) Q = = 6 + x 1 5 5 2x 3 5 x 2 3 2x 1 + 4x 3 5 5 5 6 1 5 0 + 3 2 0 5 5 0 2 5 5 0 1 0 4 5 When X = 0, Q = ( 6, 0, 5 3 ). Hence it is not a linear transformation. 5 x 1 x 2 x 3
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 6 QUESTION 4 [9 marks] Let a = (1, 2, 1), b = (1, 3, 1) and c = (1, 1, 1) be vectors. (a) Work out the cross products a b and a c. (2) i j k a b 1 2 1 = 5i + 2j + k = ( 5, 2, 1). 1 3 1 i j k a c 1 2 1 = i + 2j 3k = ( 1, 2, 3). 1 1 1 (b) Work out the triple scalar product a (b c). (2) a (b c) = (a b) c = ( 5, 2, 1) (1, 1, 1) = 8. (c) Using your answer from (b), decide if the vector a, b and c are linearly independent. (3) a a (b c) = b = 8 0. Hence the rows a, b and c are linearly independent. c (d) What is the area of the triangle ABC with edges located at a, b and c. (2) AB = b a = (0, 1, 2) and AC = c a = (0, 3, 2) Now the area of the triangle is given by 1 2 AB AC = 1 2 (3, 0, 0) = 3 2.
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 7 QUESTION 5 [10 marks] (a) Consider the matrix 1 4 3 2 A = 3 4 1 2 2 9 7 5. 5 1 4 9 Solve the system of equations Ax = 0 for x R 4. (6) 1 4 3 2 3 4 1 2 R 2 + 3R 1 2 9 7 5 R 3 + 2R 1 5 1 4 9 R 4 5R 1 1 4 3 2 0 8 8 8 0 1 1 1 R2 R 0 19 19 19 3 1 4 3 2 0 1 1 1 0 8 8 8 0 19 19 19 R 3 + 8R 2 R 4 19R 2 1 4 3 2 0 1 1 1 0 0 0 0 0 0 0 0 Put x s = s and x 4 = t parameters. Then x 2 = s + t and x 1 = 7s 2t and finally 7s st x = s + t s. t
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 8 QUESTION 5 (continued) (b) What is the rank of the matrix A in (a)? (1) Number of non zero rowsin the row echelon form of A : 2. (c) Are the columns of A linearly dependent? Explain. (3) No. A has 4 columns and rank A = 2 < 4. or column 3 depends linearly on columns 1 and 2, etc.
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 9 QUESTION 6 [11 Marks] (a) Find the inverse D 1 of the matrix 1 3 1 D = 2 0 3 1 4 2 By the method of cofactors. 3( 7) = 1. Hence (b) Find the determinant of D. D = 2 ( 10) 3( 7) = 1. D 1 1 = 12; D 1 2 = 1; D 1 3 = 8 D 2 1 = 10; D 2 2 = 1; D 2 3 = 7 D 3 1 = 9; D 3 2 = 1; D 3 3 = 6 12 10 9 D 1 = 1 1 1 8 7 6 and D = 2 ( 10) (4) (1)
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 10 QUESTION 6 (Continued) (c) Use your inverse found in (a) to solve y 1 + 3y 2 y 3 = 3 2y 1 3y 3 = 1 y 1 + 4y 2 + 2y 3 = 2 The equation in matrix form reads Dy = [ 3 1 2 ] T. Hence 3 12 10 9 3 44 y = D 1 1 = 1 1 1 1 = 1 2 8 7 6 2 29. (2) (d) Use your inverse and the transpose of D to solve x 1 + 2x 2 x 3 = 4 3x 1 + 4x 3 = 6 x 1 3x 2 + 2x 3 = 7 The equation in matrix form reads D T x = [ 4 6 7 ] T. Hence x = (D T ) [ 1 4 6 7 ] = (D 1 ) [ T 4 6 7 ], and finally 12 1 8 x = 10 1 7 9 1 6. 4 6 = 7 98 83. 84 (4)
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 11 QUESTION 7 [15 marks] The light bar AB is supported by the cables BC and BD and a ball and sucket support at A. The force F acts along the vertical at distance of 1/2 of the length of the bar from A. The magnitude of F is 200 N. The point B has coordinates B = (4, 6, 2) m. 4m y 6m B C D 4m A F x 2m z (a) Determine the displacements AB, BD and BC, the vector F and the tensions in the cables as functions of the displacements. (4) (b) Draw the free-body diagram of the bar AB showing all the external forces acting on it. (3)
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 12 QUESTION 7 (Continued) (c) Evaluate the sum of moment vectors of all the external forces acting on the bar about point A. (4) (e) Write the equilibrium equations for the bar. (you are not asked to solve them). (4)
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 13 QUESTION 8 [14 marks] Consider the truss depicted in the figure below with a pin support at B and a roller support at I. C E G I H A 000000000 111111111 000000000 111111111 000000000 111111111 B D F 300N 30 (a) Find the reactions at the supports A and I. (4) (b) Using the method of joints, find the axial force in the member BD. (4)
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 14 QUESTION 8 (Continued) (c) Using the method of sections, find the axial forces in the members EG, EF and DF and decide if the members are in tension (T) or in compression (C). (6)
MATHEMATICS 132 - Applied Mathematics 1A (Engineering) Supp Exam 2011 PAGE 15 QUESTION 9 [8 Marks] Consider the figure below. D 2m A B 200N C 500N 300 N.m 2m E 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 (a) Draw the free body diagram of the structure showing all the external forces acting on it. (2) (b) Determine the reactions at the supports (6)