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