Question 1: Assuming that is a positive real number and
are rational numbers, show:
Answer:
(xi) If , show that
Given
Question 2: Assuming are positive real numbers, simplify the following:
Answer:
Question 3: Assuming are positive real numbers, show that:
Answer:
(ii) If , prove that
, where
are different.
Question 4:
Answer:
RHS
LHS
Therefore RHS = LHS. Hence proved.
Question 5:
(i) If are distinct positive primes such that
, find
(ii) If are distinct positive primes such that
, find
Answer:
(i) If are distinct positive primes such that
, find
(ii) If are distinct positive primes such that
, find
Therefore
Hence
Question 6: If , find
. Then compute the value of
Answer:
Therefore
Hence =
Question 7:
(i) If , show that
(ii) Determine , if
(iii) If and
, find the value of
Answer:
(i) If , show that
We know:
Therefore
Therefore
Hence Proved.
(ii) Determine , if
Therefore
(iii) If and
, find the value of
Similarly,
Hence
Question 8:
(i) If and
, then show that
(ii) If , show that
(iii) If , show that
Answer:
(i) If and
, then show that
Let
Therefore ,
,
Now,
(ii) If , show that
Let
Therefore ,
,
(iii) If , show that
Let
Therefore ,
,
Question 9: Solve the following equations:
Answer:
Let
Therefore if
If
Let , Therefore
Therefore if
If
Let
Therefore if
If
Question 10: Given , find (i) the integral value of
(ii) the value of
Answer:
Therefore