sample examination questions for engineering mathematics

7/27/2019 Sample Examination Questions for Engineering Mathematics

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Sample Examination Questions for Engineering Mathematics

Chapter 6

1. Differentiate each of the following with respect to the variable in each case.

(i)

6

3 7 2

5 9 263

x

y x xx

= + + + (ii)2

2 lny x x

=

(iii)24

sin5

xey

x

= (iv) 2 31 5 4y x x= +

Solutions

1. (i) 4 2 515 7 4 9x x x + (ii)2

2 ln 1x

(iii)( ) ( )

( )

2

2

4 2sin 5 5cos 5

sin 5

xe x x

x

+ (iv)

[ ]2 3

5 6

1 5 4

x x

x x

+

2. If ( )cosh /y a x a c= + , where a and c are constants, show that22

2

d 1 d1

d d

y y

x a x

= +

Solution

2. Use a hyperbolic identity ( ) ( )2 2cosh sinh 1t t = .

3. A curve is given, parametrically, by

2sin , cos

2 2

t tx t y t

= =

Find thex- andy coordinates of the pointPwhich corresponds to the parameter 1t=

and find the value ofd

d

y

xatP.

Solution

3. ( )0, 1P= , 22

Chapter 8

4. Find the integrals:

(i) ( )2

3

1

4 1x x dx

+ (ii)/4

0

cos 24

t dt

(iii) 2 1

x dxx +

(iv) ( )sin d (v) ( ) ( )3

1 2dx

x x+ (vi)1/2

2

0

1

1dx

x

Solutions

4. (i)1

162

(ii)1

2(iii) ( )2

1ln 1

2x C+ +

(iv) ( ) ( )cos sin C + + (v)2

ln1

xC

x

+

+(vi)

6

5. Evaluate each of the following definite integrals.


2/4

(a) ( ) ( )3

3

0

8cos sinx x dx

(b)( )4

1

ln xdx

x

Solution

5. (a)

15

8 (b) ( )4 ln 4 1 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

boxs 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 curve23y x x= and thex axis.

9. R is the region bounded by the curves ( )3 siny x= , ( )siny x= , 0x = andx = . Find the area ofR.

10. Using Simpsons rule with 4 strips, find an approximate value of the integral

2

1

0

xe dx .

11. Write down the Trapezoidal Rule approximation 3T for ( )4

1

cos /x x dx . Leaveyour answer expressed as a sum involving cosines.

[ 3T in this question means consider 3 intervals].

12. Approximate

2

2

01

xdx

x+ using the Trapezoidal Rule with 4n = subintervals.

13. Find the area under the graph of lny x= between 1x = and 3x = .Solutions

8. 9/2

9. 3 4 10. ( )1.464 3dp

11. ( ) ( ) ( )( ) ( )1

cos 2 2cos / 2 3cos / 3 4cos / 42

+ + +

12. ( )0.78077 5dp

13. ( )ln 27 2


3/4

Chapter 10

14. Given that 1 4 2z j= + , 2 3z j= find (i) 1 23 2z z+ (ii) 1 2z z

15. Given 1 2z j= + , 2 3 4z j= , find (a) 1 2z z (b) Both values of 1z16. Find the four solutions to the equation

4

2 2 3z j= and display them on an Argand diagram.

17. Determine all complex 3rd roots of8i .

18. Determine all complex numberszsuch that 5 32z = .

Solutions

14. (i) 18 4j+ (ii) 14 2j+

15. (a) 10 5j (b) ( )1/ 45 13.28 and ( )1/ 45 193.2816. ( )1 2 60z = , ( ) ( ) ( )2 3 42 150 , 2 240 and 2 330z z z= = =

17. ( )1 2 30z = , ( )2 2 150z = and ( )3 2 270z = 18. ( )1 2 36z = , ( )2 2 108z = , ( )3 2 180z = , ( )4 2 252z = and ( )5 2 324z =

Chapter 11

19. Let1 2

3 4

=

A and

5 6

7 8

=

B . Determine (i) ( ) ( ) +A B A B (ii) 2 2A B

20. Given the matrix

3 2 61

2 6 37

6 3 2

=

A

(a) Compute 2A and 3A .

(b) Based on these results, determine the matrices 1A and 2004A .

21. Let A, B and C be matrices defined by

1 11 0 1

1 2 1 1 1 01 1 1 , ,

0 1 2 1 2 31 2 3

0 1

= = =

A B C

Which of the following are defined?2, , , , , ,T T+ A AB B C A B CB BC A

Compute those matrices which are defined.

22. Find the determinant of the matrix

0 1 5

3 6 9

2 6 1

=

A .

23. The mesh equations of a circuit are given by:

1 2 3

1 2 3

1 2 3

3 2 13

4 4 3 3

5 2 13

i i i

i i i

i i i

+ + =+ =+ + =

Determine the values of 1 2 3, andi i i .


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Solutions

19. (i)4 5

164 5

(ii)

15 174

19 21

20. (a) 2 =A I and3

3 2 612 6 3

76 3 2

=

A (b) 1 =A A and 2004 =A I

21.

1 1 1

0 1 2

1 1 3

T

=

A ,0 2

1 1

=

CB ,2

0 2 4

3 3 3

6 8 10

=

A . AB, +B C , A B and

tBC is not valid.

22. ( )det 165=A

23. 1 1i = , 2 2i = and 3 3i =

sample examination questions for engineering mathematics

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