8/12/2019 AppendixA1.Teaching Mathematics in Higher Education - The Basics and Beyond

http://slidepdf.com/reader/full/appendixa1teaching-mathematics-in-higher-education-the-basics-and-beyond 1/4

Appendix A1

A First Year Engineering MathematicsExamination Paper

ENGINEERING MATHEMATICS FIRST YEAR

Duration one hour 30 minutes

Answer ALL of Section A and 3 questions from Section BBoth Sections carry equal marks

Instructions to candidates1. Normal university regulations apply

2. The use of calculators is NOT permitted3. Show all working and explain your reasoning at all times

Materials provided

1. Script answer book

Section A

1. Factorise the polynomial x3 − 6x2 + 11x − 6.

2. Expressx + 1

(x − 1)(x + 2)

in partial fractions.

3. Expand (2 + 3x)5  by the Binomial Theorem.

4. Simplifyxe2 lnx

ln x3 + 2ln x

5. An arc of a circle of radius√ 

3 subtends an angle of  30◦ at the centre. A tangent is drawn to thecircle at the midpoint of the arc. Find the areas between the tangent, the arc and the extended radii bounding the arc. What is the length of the arc?

6. Express in the simplest form:

189

8/12/2019 AppendixA1.Teaching Mathematics in Higher Education - The Basics and Beyond

http://slidepdf.com/reader/full/appendixa1teaching-mathematics-in-higher-education-the-basics-and-beyond 2/4

190

(a)   sin2x cos x + sin x cos2x

(b)  1 − sin2 x + cos2 x

sin2x  .

7. Differentiate sin(3x − 2).

8. Differentiate x2 ln(x2 + 1).

9. Integrate    dx

x2 + 3x + 2

10. Integrate    1

0

xex dx

Section B

11. Factorise 1 − x + x2 − x3 and hence simplify the function

f (x) =  1 − x4

1 − x + x2 − x3

Expand [f (x)]6

Differentiate [f (x)]6

Integrate    1

0

[f (x)]6 dx

12. Write out the Binomial Theorem for a negative integer power.Expand the function

1

1 − 3x

 by the Binomial Theorem, up to the term in x3, simplifying the coefficients. Use the result to obtainan approximation to 2/0.97.

13. (a) Using the compound angle formulae for sine and cosine prove the following:

i.   cos2x = 2cos 2x − 1and give two alternative expressions for the right hand side.

ii.   tan(A − B) =  tan A − tan B

1 + tan A tan B.

iii.   sin A + sin B = 2 sin

A + B2

cos

A − B2

.

(b) A surveyor stands at the same horizontal level as the bottom of a vertical cliff, at a distance dfrom the cliff. The angle of elevation at a point P  on the cliff is θ  and the angle of elevationof the top of the cliff is φ  (in the same plane though the surveyor, perpendicular to the cliff).Show that the distance from the point P  to the top of the cliff is

d sin(θ − φ)

cos θ cos φ

8/12/2019 AppendixA1.Teaching Mathematics in Higher Education - The Basics and Beyond

http://slidepdf.com/reader/full/appendixa1teaching-mathematics-in-higher-education-the-basics-and-beyond 3/4

191

14. Using integration by parts twice, show that

   eax cos bxdx =  eax

a2 + b2[a cos bx + b sin bx]

where a and b are constants.

The root-mean-square value (RMS) of  f (x) over the range a < x < b is   1

b − a

   b

a

|f (x)|2 dx

Evaluate the RMS value of the function  ex cos x over the range 0  < x < π .

15. Integrate the following integrals:

I 1  =    x

−2

x2 − 4x + 7 dx

I 2  =

   x

x2 − 4x + 3dx

I 3  =

   e2x

e2x − 4ex + 3dx

END OF EXAMINATION PAPER

8/12/2019 AppendixA1.Teaching Mathematics in Higher Education - The Basics and Beyond

http://slidepdf.com/reader/full/appendixa1teaching-mathematics-in-higher-education-the-basics-and-beyond 4/4

192