5)

Step 1:

The rectangular form is

Find

Step 2:

Convert Z in polar form.

Convert rectangular form into Polar form is

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 equation.

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 3m 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

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

Right angle property is

Solution:

13)

Step 1:

The matrix is

i)

Find the detrminant.

Determinant value of the matrix is

iii)

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 Cramers Rule.

Since Cramers 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 Cramers Rule.

Since Cramers 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:

21)

Step 1:

The polar form is

Conversion from Polar to rectangular form :

and .

Substitute in the rectangular equation.

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:

22)

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 Cramers Rule.

Since Cramers Rule is applicable. \ \

\ \

Solution:

The value of