Module MA12001 (Level 1)

Mathematics 1B

2 hours

August 2012

This paper contains 15 questions.

You should attempt ALL questions

MA12001

1. The position vectors of A, B and C relative to the origin O are a= [1,2,3], b= [�2,4,�2]and c = [1,0,1]

(a) Compute 3a and 3a�2b [2 marks]

(b) Find the unit vector in the direction of�!BC [2 marks]

(c) Show that�!OA and

�!OB are perpendicular. [2 marks]

(d) Find a⇥ c [3 marks]

2. Let 3 non-colinear points A, B and C have position vectors a,b and c respectively, rel-ative to an origin O. Let D be the mid-point of AB. Show that the centroid P of thetriangle ABC (i.e. the point having position vector 1

3(a+b+c)) lies on the line CD andcuts it in the ratio 2 : 1. [6 marks]

3. Let

A =

2

40 21 2

�2 �1

3

5 , B =

2

41 �2 11 3 1

�2 2 1

3

5 , x =

2

4123

3

5 , C =

1 5

�2 1

�.

Calculate BA and x

T BTx. Does C�1 exist? If so, find it. [7 marks]

4. Find the general solution of the following system of equations:

x1 +2x2 � x3 = 22x1 +5x2 +2x3 = �1

7x1 +17x2 +5x3 = �1

[8 marks]

5. By first converting the complex number to polar form, evaluate (�p

3+ i)4 using DeMoivre’s theorem. Write your answer in polar form. [6 marks]

6. Find the shortest distance between the two lines

r = [3,�2,1]+l [2,�3,0]

andr = [1,1,�1]+µ[3,1,0] [6 marks]

7. (a) Find the centre and radius of the sphere

x2 + y2 + z2 �2x�4y+2z+2 = 0. [4 marks]

(b) Hence find an equation for the tangent plane to the sphere at the point(2,3,

p2�1). [4 marks]

MA12001

8. Sketch and calculate the area of the finite region enclosed by the parabolas y = x2 andy = 2x� x2. [5 marks]

9. (a) Use the substitution u = sinx to evaluateZ cosx

sin2 xdx. [3 marks]

(b) Use the substitution u = 9� x2 to evaluate

Z p5

0

x3p

9� x2dx. [4 marks]

10. Write down the derivative of tanx.

Prove that1

1� sinx= sec2 x+

sinxcos2 x

.

Hence obtainZ 1

1� sinxdx. [5 marks]

11. Use the method of integration by parts to evaluate

(a)Z

x2 lnx dx, (b)Z

xex/2 dx. [8 marks]

12. (a) Expressx�9

x2 +3x�10in partial fractions. [3 marks]

(b) Hence evaluateZ 3

1

x�9x2 +3x�10

dx. [4 marks]

13. Use Simpson’s rule with 6 subintervals to obtain an approximate value ofZ 3

0e�

px dx.

Work to four decimal places and round your answer to two decimal places. [5 marks]

14. Find the solution y = y(x) to the differential equation

dydx

=xy

subject to the initial condition y = 2 at x = 0. [5 marks]

MA12001

15. Obtain the general solution y = y(x) to the differential equation

d2ydx2 �2

dydx

�3y = x+2. [8 marks]

END OF PAPER

MA12001