ANSWERS TO TERM TEST 1 - Test A
October 4, 2000
1. (13 marks)
(a) Convert the polar co-ordinatesinto cartesian form (x, y);
(b) Convert the cartesian co-ordinatesinto polar form ;
(b) Identify the 2-dimensional curve
Multiply by r to getor x2 + y2 + 6x = 0.
Completing the square yields x2 + 6x + 9 + y2 = 9 or (x+3)2 + y2 = 9 which is a circle,
center (-3, 0) and radius 3.
2. (17 marks)
(a) Convert the cartesian co-ordinates into the spherical polar form
(b) Convert the spherical polar co-ordinates into the cartesian form (x, y, z);
(c) Describe the following curves or surfaces
(i) 3r = 5z;
Convert to spherical polar coordinatesor which means that this surface is a cone.
The first equation is a sphere with centre at the origin and the second equation is a cone with its vertex at the origin so their intersection is a circle.
3. (15 marks)
(a) Use the polar form of complex numbers to evaluate:
(ii) (-3 -3i)7
(b) Using Euler's formula find in terms of trigonometric functions of q and f.
4. (20 marks)
(a) Use Gaussian elimination to solve the following system of equations:
2x - 6y - 4z = 8
y + 3z = 2
3x - 3y + 2z = 4
Note that marks will only be given for a correct use of Gaussian elimination even if the final answer is correct.
Augmented matrix: R1 -> 1/2 R1; R3 -> R3 -3 R1
R3 -> R3 + 6 R2 R3 -> 1/(-10) R3
By back substitution z = 2; y = 2 - 3z = -4; x = 4 + 3y + 2z = -4
(b) Use Gauss-Jordan elimination to solve the following system of equations:
3x + 6y - 9z = -12
2x + y - 3z = -8
x + 5y - 6z = -4
R1 -> 1/3 R1; R2 -> R2 - 2R1; R3 -> R3 - R1
Thus z = t (or some other parameter), y = t, x = t - 4
R2 -> 1/(-3) R2 R1 -> R1 - 2 R2
R3 -> R3 - 3 R2
5. (4 marks) Use Gaussian elimination to find the condition on k such that the following system of equations has an infinite number of solutions:
2x - 4y = 8
3x + ky = 12
Augmented matrix:R1 -> 1/2 R1; R2 -> R2 - 3 R1
For an infinite number of solutions the last row must be zero. Thus k + 6 = 0 or k = -6.
6. (8 marks) Given A = and B = evaluate:
(b) tr(B); 5 + 0 = 5
7. (3 marks) If A is an m´n matrix and C is a p´q matrix and if the matrix product ABCT is defined, what is the size of B?
Let B be of size r by s. Then the matrix productis defined. For this to be true we must have r = n and s = q, i.e. B is of size n by q.