Question 1: Expand / Simplify the following:

(vii)

Let

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)

Note: We will use the following identify:

(xvi)

(xvii)

(xviii)

(xix)

(xx)

(xx)

(xxi)

We will use the identity

(xxii)

We will use the identity

Therefore

(xxiii)

We will use the identity

(xxiv)

We will use the identity

(xxv)

We will use the identity

Question 2: Evaluate the following identities:

We will use the identity

(vii)

We will use the identity

(viii)

We will use the identity

(ix)

(x)

We will use the identity

(xi)

We will use the identity

(xii)

We will use the identity

(xiii)

We are going to use this identity

Question 3:

(i) If the number is more than number and the sum of the squares of and is , find the product of

Answer:

Therefore

(ii) If the number is less than the number and the sum if the square of and is , find

Answer:

(iii) If the sum of two numbers is and the sum of their cubes is , find the sum of their squares.

Answer:

Therefore

Question 4:

(i) If , find (a) (b)

Answer:

(a)

(b)

(ii) If , find (a) (b) (c)

Answer:

(a)

(b) Since

Therefore

(c)

Therefore

(iii) If , find (a) (b)

Answer:

(a)

(b)

, find the value of

Answer:

, find the value of

Answer:

, find the value of and

Answer:

Now

(vii) If , find the value of

Answer:

(viii) If , find the value of

Answer:

(ix) If , find the value of

Answer:

(x) If , find the value of

Answer:

Now

(xi) If , find the value of (a) (b) (c)

Answer:

(a)

(b)

(c)

(xii) If , find the values of (a) (b)

Answer:

(a)

(b)

(xiii) If find the value of

Answer:

Therefore

(xiv) If and , find the value of

Answer:

Given: and

We know:

(xv) If and , find the value of

Answer:

Given: and

We know:

(xvi) If and , find the value of

Answer:

Given: and

We know:

(xvii) If and , find the value of

Answer:

Given: and

We know:

(xviii) If , find the value of

Answer:

Given

If

Similarly, if

Then

(xix) If , find the value of

Answer:

Given

If

Similarly, if

Then

(xx) If , find the value of

Answer:

Given

Now

If

Similarly, if

Then

(xxi) If and , find the value of

Answer:

Given and

(xxii) If and , find the value of

Answer:

Given and

(xxiii) , find

Answer:

If

(xxiv) , find

Answer:

If

(xxv) , find

Answer:

Given

If

Similarly, if

Then

(xxvi) , find

Answer:

Given

Now

Similarly, if

Then

(xxvii) , find

Answer:

Given

Given

If

Similarly, if

Then

(xxviii) , find , and

Answer:

Now

(xxix) , find , and

Answer:

Given

Therefore

If

Similarly, if

Then

(xxx) , find

Answer:

(xxxi) , find , and

Answer:

Given

(xxxii) find

Answer:

If

Similarly, if

Then

(xxxiii) If prove that

Answer:

If

Hence proved.

(xxxiv) If and , find the value of and

Answer:

Given: and

also

(xxxv) If and , find the value of

Answer:

Given: and

(xxxvi) If and , find the value of

Answer:

Given: and

Question 5:

(i) If and , find the value of

Answer:

and , find the value of

Answer:

(iii) If , prove that

Answer:

Given

(iv) If and , find the value of

Answer:

(v) If and , find the value of

Answer:

(vi) If and , find the value of

Answer:

(vii) Prove that is always non negative for all values of .

Answer:

To prove that is always non-negative or To prove that is always non-negative

which is always not negative.

(viii) If and , find

Answer:

(ix) If and , find

Answer:

(x) If , prove that

Answer:

Given:

$

(xi) If and , find the value of

Answer:

Given: and

(xii) If and , find the value of

Answer:

Given: and

(xiii) If and , find the value of

Answer:

Given: and

(xiv) If , find the value of

Answer:

Given:

Therefore

(xv) If and , find the value of

Answer:

We know