# question 2

1. Given:

1.1. Find Z

1.2. Convert Z into polar form. Show all steps. θmust be positive

1.3. Represent Z and ALL calculated values in 1.2 on the Argand diagram

2. Solve for x if:

3. Solve for x if:

4. Given:

Z = (2.4 |86.3° )

Use de movire’s theorem to calculate for Z. Leave the answer in a+jb form.

5. Given:

Z= -3 -j3

5.1. Find Z

5.2. Convert Z in polar form. θ must be positive. Show all steps

6. Solve for x if:

7. Make y the subject of the formula if:

8. Determine the length and breadth of a rectangle if the length of a rectangle is 3m more than its breadth and the area of a rectangle is 54m².

9. Given:

A + 4b + 8c =0

2a - 5b = -2 + 6c

B - 4c = -3

Solve for c by the use of cramer’s rule

10. Make P the subject of the formlua:

11. The hypotenuse of a right angles triangle is 10mm longer than the longest of the two sides. Calculate the lengths of the sides if the shortest side is 50mm long.

12. Solve for x if:

13. Given:

| 1        2         3|

| 4        -1         -2|

| -3       -4         -5|

13.1. Calculate the value of the detrminant

13.2. What is the minor of -5

13.3. What is the co-factor of -2

14. Solve for x by using the determinant if:

x + y = 10

X - y = 2

15. Given:

2x -- 2y - Z =  3

4x + 5y - 2Z = -3

3x + 4y - 3Z = -7

15.1. Solve for the value of Z by using Cramer’s Rule.

15.2. Determine the value of the cofactor of -3.

16. Given:

Make y the subject of the formula.

17. Solve for x:

18. The sum of two numbers is 10. The difference of their squares is 50. Calculate the two numbers.

19. Solve for x and  if:

20. Given:

Convert z into polar form. Show ALL steps. θ may only be positive.

21. Given:

Z = 15.9 |49,7°

Express z in a+jb form. Show all steps.

22. Given:

2x + 3y - z = 4

3x + y + 2x = 13

X + 2y - 5x = -11

Solve for the value of x with the aid of Cramer’s Rule.

23. Given:

Determine the value of x and y.

24. Given:

and

24.1. Determine the polar for of  and

24.2. Evaluate:

25. Simplify:

5)

Step 1:

The rectangular form is

Find

Step 2:

Convert Z in polar form.

Conversion from rectangular to polar equation :

and

Solution:

The value of

Polar form is

4)

Step 1:

The polar form is

Conversion from Polar to rectangular form :

and .

Substitute in the rectangular form.

The rectangular form is

Step 2:

De Moivres Theorem:

If is a complex number, then

where is a positive integer.

De Moivres theorem: .

Then,

Solution:

8)

Step 1:

The length of a rectangle is 3 m more than its breadth.

Consider the rectangle breadth is x.

Then, the equation is

The area of a rectangle is

Find the length and breadth of a rectangle.

The are of the rectangle is

Substitute

The length of the rectangle is always positive then consider the x value is positive.

Substitute in

Solution:

The length and breadth of a rectangle is

9)

Step 1:

The system of equations are

Rewrite the equations are

Convert the equations into matrices form ,

where is coefficient matrix, is variable matrix and is constant matrix.

Solve the equations by using Cramer's Rule.

Since Cramer's Rule is applicable.

Step 2:

Solution:

Question numbers

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11)

Step 1:

The hypotenuse of a right angles triangle is longer than the longest of the two sides.

The shortest side is long.

Find the lengths of the sides.

Consider the right angle triangle,

Let longest side is and hypotenuse is

Pythagorean theorem

Solution:

13)

Step 1:

The matrix is

Find the determinant.

Determinant value of the matrix is

Step 2:

The co factor of

The co factor of the

Solution:

Determinant value of the matrix is

The co factor of the

14)

Step 1:

The system of equations are

Convert the equations into matrices form ,

where is coefficient matrix, is variable matrix and is constant matrix.

Solve the equations by using Cramer's Rule.

Since Cramer's Rule is applicable.

Step 2:

Solution:

The value of

15)

Step 1:

The system of equations are

Convert the equations into matrices form ,

where is coefficient matrix, is variable matrix and is constant matrix.

Solve the equations by using Cramer's Rule.

Since Cramer's Rule is applicable.

Step 2:

The co factor of

The co factor of the

Solution:

The value of

The co factor of the

18)

Step 1:

Find the two numbers.

Consider the two numbers are

The sum of two numbers is

The difference of their squares is

Rewrite the equation is

Solve the two equations

The two numbers are

Solution: