ANSWERS TO TERM TEST 1 - Test B
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 + 4y = 0.
Completing the square yields x2 + y2 + 4y + 4 = 4 or x2 + (y+2)2 = 4 which is a circle,
center (0, -2) and radius 2.
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) 2r = 3z;
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 + 4y - 4z = -4
y - 2z = 3
3x + 9y - 7z = -7
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 - 3 R2 R3 -> 1/5 R3
By back substitution z = -2; y = 3 + 2z = -1; x = -2 - 2y + 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:
3x + 6y = -3
2x - ky = 12
Augmented matrix:R1 -> 1/3 R1; R2 -> R2 - 2 R1
For no solutions the last row must be zero except for the last element. Thus -k - 4 = 0 or k = -4.
6. (8 marks) Given A = and B = evaluate:
(b) tr(B); 2 + 0 = 2
7. (3 marks) If A is an m´n matrix and C is a p´q matrix and if the matrix product ATBC 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 = m and s = p, i.e. B is of size m by p.