Sample Examination Questions for Engineering Mathematics
Chapter 6
1. Differentiate each of the following with respect to the variable in each case.
7 2 x6
2
3
(i) y  5 x  
(ii) y  2 x ln  
 9 x  26
x
3
 x
2 x
4e
(iii) y 
(iv) y  1  5x 2  4 x3
sin 5 x
Solutions
 2 
1. (i) 15 x 4  7 x 2  4 x5  9
(ii) 2 ln    1
  x 
(iii)
4e2 x  2sin  5 x   5cos  5 x  
x 5  6 x 
(iv)
sin 2  5 x 
1  5 x 2  4 x3
2. If y  a cosh  x / a   c , where a and c are constants, show that
d2 y 1
 dy 

1  
2
dx
a
 dx 
2
Solution
2. Use a hyperbolic identity cosh 2  t   sinh 2  t   1 .
3. A curve is given, parametrically, by
 t 
 t 
x  t  sin   , y  cos    t 2
 2
 2
Find the x- and y – coordinates of the point P which corresponds to the parameter t  1
dy
and find the value of
at P.
dx
Solution

3. P   0, 1 ,   2
2
Chapter 8
4. Find the integrals:
2
(i)
  4x
3
 /4
 x  1 dx
(ii)
1


 cos  2t  4  dt
0
(iv)   sin   d
(v)
3
  x  1 x  2
1
2
(ii)
1
2
(iv)  cos    sin    C
1
ln  x 2  1  C
2

x2
C
(v) ln
(vi)
x 1
6
(iii)
5. Evaluate each of the following definite integrals.
x
1/2
dx
Solutions
4. (i) 16
(iii)
(vi)

0
2
x
dx
1
1
1  x2
dx

4
3
(a)  8cos3  x  sin  x  dx
(b)
0

1
ln  x 
x
dx
Solution
15
5. (a)
(b) 4 ln  4   1
8
Chapter 7
6. Determine the dimensions of the rectangle of largest area with a fixed perimeter of
20 feet.
7. You are to construct an open (no top) rectangular box with a square base. Material
for the base costs 30 cents/in.2 and material for the sides costs 10 cents/in.2. If the
box’s volume is required to be 12 in.3, what dimensions will result in the least
expensive box?
Solutions
6. A square of edge 5 feet.
7. 2 in. by 3in.
Chapter 9
8. Find the area of the region bounded by the curve y  3x  x 2 and the x axis.
9. R is the region bounded by the curves y  3  sin  x  , y  sin  x  , x  0 and
x   . Find the area of R.
10. Using Simpson’s rule with 4 strips, find an approximate value of the integral
1
e
x2
dx .
0
4
11. Write down the Trapezoidal Rule approximation T3 for
 x cos  / x  dx . Leave
1
your answer expressed as a sum involving cosines.
[ T3 in this question means consider 3 intervals].
2
12. Approximate
x
 1 x
2
dx using the Trapezoidal Rule with n  4 subintervals.
0
13. Find the area under the graph of y  ln x between x  1 and x  3 .
Solutions
8. 9/2
9. 3  4
10. 1.464  3dp 
1
cos    2  2 cos  / 2   3cos  / 3   4 cos  / 4  
2
12. 0.78077  5dp 
11.
13. ln  27   2
Chapter 10
14. Given that z1  4  j 2 , z2  3  j find (i) 3z1  2 z2 (ii) z1 z2
15. Given z1  2  j , z2  3  j 4 , find (a) z1 z2
(b) Both values of
16. Find the four solutions to the equation
z 4  2  j 2 3
and display them on an Argand diagram.
17. Determine all complex 3rd roots of 8i .
18. Determine all complex numbers z such that z 5  32 .
Solutions
14. (i) 18  j 4
15. (a) 10  j5
z1
(ii) 14  j 2
(b) 51/ 4 13.28 and 51/ 4 193.28
16. z1  2  60  , z2  2 150 , z3  2  240  and z4  2  330 
17. z1  2  30 , z2  2 150 and z3  2  270
18. z1  2  36 , z2  2 108 , z3  2 180 , z4  2  252 and z5  2  324
Chapter 11
1 2
5 6
2
2
19. Let A  
 and B  
 . Determine (i)  A  B  A  B  (ii) A  B
3 4
7 8
20. Given the matrix
 3 2 6 
1

A   2 6 3 
7

 6 3 2 
(a) Compute A 2 and A3 .
(b) Based on these results, determine the matrices A 1 and A 2004 .
21. Let A, B and C be matrices defined by
1 1
1 0 1 


1
2
 1 1 1 0 



A  1 1 1  , B 
, C

 0 1 
 2 1 2 3 
1 2 3 




0 1
Which of the following are defined?
AT , AB, B  C, A  B, CB, BC T , A 2
Compute those matrices which are defined.
0 1 5
22. Find the determinant of the matrix A   3 6 9  .
 2 6 1 
23. The mesh equations of a circuit are given by:
i1  3i2  2i3  13
4i1  4i2
 3i3
5i1  i2  2i3
Determine the values of i1 , i2 and i3 .

3
 13
Solutions
 15
(ii) 4 
19
 3 2
1
3
2
20. (a) A  I and A   2 6
7
 6 3
19.
 4 5
(i) 16 

 4 5
17 

21
6 

3 
2 
(b) A1  A and A2004  I
 0 2 4 
 1 1 1
 0 2 




2
3  . AB, B  C , A  B and
21. A   0 1 2  , CB  
, A  3 3

1
1


 6 8 10 
 1 1 3 




t
BC is not valid.
22.
det  A   165
T
23.
i1  1 , i2  2 and i3  3