FACULTY OF SCIENCE DEPARTMENT OF APPLIED PHYSICS AND ENGINEERING MATHEMATICS B ENG TECH: ELECTRICAL, INDUSTRIAL, MECHANICAL, CIVIL, CHEMICAL, EXTRACTION/PHYSICAL METALLURGY AND CONSTRUCTION ENGINEERING CAMPUS: DFC ASSESSMENT: SEMESTER TEST 3 DATE: 28 OCTOBER 2017 TIME: 09:00-10:30 DURATION: 90 MINUTES MARKS: 50 MARKS AVAILABLE: 51 ASSESSORS: P DLAMINI, H KOTZE, I LETLHAGE, R DURANDT % MARKS OBTAINED 50 SURNAME AND INITIALS: ______________________________________________ STUDENT NUMBER: ___________________________________________________ CONTACT NUMBER: ___________________________________________________ LECTURER DR P DLAMINI MR I LETLHAGE MRS H KOTZE MRS R DURANDT INSTRUCTIONS: COURSE MARK WITH X CHEMICAL ENGINEERING EXTRACTION METALLURGY PHYSICAL METALLURGY MECHANICAL ENGINEERING INDUSTRIAL ENGINEERING CIVIL ENGINEERING CONSTRUCTION ENGINEERING ELECTRICAL ENGINEERING WRITE YOUR STUDENT NUMBER AND PARTICULARS IN THE SPACE PROVIDED. ANSWER ALL QUESTION IN THE SPACE PROVIDED. USE ONLY A BLACK OR BLUE PEN FOR WRITING AND DRAWING GRAPHS. ONE NON PROGRAMMABLE CALCULATOR PER STUDENT. THE NUMBER OF PAGES IN YOUR SCRIPT SHOULD BE 11, PLEASE VERIFY. -2MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 SECTION A [8 MARKS] INSTRUCTIONS: MARK THE LETTER (X) ON THE TABLE AT THE BOTTOM OF THE PAGE TO INDICATE THE CORRECT ANSWER. DO YOUR ROUGH WORK ON THE OPPOSITE BLANK PAGE. 1. The shaded region below is enclosed by the graphs of y sin x and y 1 between x 0 and x 2 . y 1 y sin x x π/2 The volume of the solid obtained by revolving the shaded region about the 𝑥-axis is NOT given by (2) 2 2 A V (1 sin x) 2 dx B V cos 2 x dx 0 0 1 C V 2 y sin 1 y dy D none of these 0 1 1 2 2. If one of the eigenvalues of matrix A 0 2 2 is 1, the eigenvector 1 1 3 associated with 1 is: (2) A 4 2 1 B 1 2 1 C 0 2 1 D none of these -3MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 Given two vectors a 2i 3j and b 3i 4 j. The resultant a 2b 3. (2) A 4i 5j B 8i 11j C 5i 7 j D none of these If the mean value of the function f x over the interval 0 x 4 is 4. 8 , then 3 4 f x dx (2) 0 A 4 3 B 32 3 C 2 3 D none of these 1. 2. 3. 4. A B C D A B C D A B C D A B C D SECTION B [10 MARKS] INSTRUCTIONS GIVE ONLY THE FINAL SIMPLIFIED ANSWER (CORRECT TO TWO DECIMAL PLACES WHERE APPLICABLE) IN THE SPACE PROVIDED. 5. Calculate the area of the shaded region in the figure below. y 0.5 y sin x cos2 x x π/2 π (2) -4MATE1B1 ASSESSMENT 3 1 0 1 6. Given C 2 4 and D 0 4 ; find C D. 2 1 2 2 28 OCTOBER 2017 (2) 7. For matrices C and D as defined in question 6 (above), is C D DC ? Motivate your answer. (2) 8. Two acceleration vectors a 10ms 2 at 60 to the horizontal and b 15ms 2 at 135 to the horizontal act at the same point. Sketch a diagram to illustrate a b. (2) -5MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 9. Express the vector illustrated in the diagram below in i, j, k notation. (2) a 6N 120° -6MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 SECTION C [33 MARKS] INSTRUCTIONS: SHOW ALL THE STEPS TAKEN AND GIVE YOUR FINAL ANSWER CORRECT TO TWO DECIMAL PLACES WHERE APPLICABLE. SIMPLIFY YOUR ANSWERS FULLY. 10. Calculate the area of the shaded region in the figure. y y (4) x x2 1 ↓ y x 10 ↓ x -7MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 11. Calculate the volume of the solid generated by rotating the area enclosed by the graphs of y x 2 8x 12 and y 2 x 4 about the line x 1 as shown in the figure below. (5) -8MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 12. Consider the following system of linear equations: 3x 4 y 10 7 x 9 y 20 12.1 Write the system in matrix form A X B . (1) 12.2 Find A1 (3) 12.3 Use the inverse matrix method to solve the system. (2) -9MATE1B1 13. ASSESSMENT 3 28 OCTOBER 2017 Three forces act as shown in the diagram below: 35° 15° 60° 13.1 Calculate the magnitude of the resultant force. (5) 13.2 Sketch the resultant. (1) 13.3 Calculate the direction of the resultant. (1) - 10 MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 12 kg/m , where x is x 1 measured in metres, from one end of the rod to the other. If the rod is 8m long, calculate the average (or mean) density of the rod. (4) 14. The linear density of a metal rod is given by x 15. Let the current i be given by i 30sin10 t k and suppose that the rms of i between t 0 and t 0.2 is known to be 23.45 A. Determine the value of k . (7) - 11 MATE1B1 ASSESSMENT 3 28 OCTOBER 2017 Use this space if you want to redo any question(s). Please indicate clearly at the relevant question(s) that the solution is on this page.