MATH1025.03M

FW00

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.

(ii)

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:

(i) ;

(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

Augmented matrix:

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:

(a) B^T;

(b) tr(B); 2 + 0 = 2

(c) AB

7. (3 marks) If A is an m´n matrix and C is a p´q matrix and if the matrix product A^TBC 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.