**MATH1025.03M**

**FW00**

**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 x^{2} + y^{2} + 6x = 0.

Completing the square yields x^{2} + 6x
+ 9 + y^{2} = 9 or (x+3)^{2} + y^{2} = 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.

(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 - 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

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:

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:

(a) A^{T};

(b) tr(B); 5 + 0 = 5

(c) AB

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