ma12001-aug12-exam
TRANSCRIPT
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