MATE1B1 TEST 3(1)

```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,
-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
(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&deg;
-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
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 A1
(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&deg;
15&deg;
60&deg;
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