Question 1: If , prove that:
i)
ii)
iii)
Answer:
i)
Given
By componendo and dividendo
ii)
Given
By componendo and dividendo
or
iii)
Given
By componendo
Cross multiplying
or
or
Question 2: If , prove that:
.
Answer:
Given
By Componendo
or
or
Again
By Dividendo
or
or
Hence
Question 3: Given,
, prove that:
[2000]
Answer:
Given
By componendo and dividendo
By Alternendo
Question 4: If
; Then prove that
.
Answer:
By componendo and dividendo
.
Question 5: If ; prove that
Answer:
Given
By componendo and dividendo
. Hence Proved.
Question 6:
i) If
, find the value of:
ii) If
, find the value of:
Answer:
i) Given
By componendo and dividendo
… … … … … … i)
Similarly
By componendo and dividendo
… … … … … … ii)
Adding i) and ii)
ii) Given
By componendo and dividendo
… … … … … … i)
Similarly
By componendo and dividendo
… … … … … … ii)
Adding i) and ii) we get
Question 7: If ; prove that
Answer:
Given
By componendo and dividendo
Applying again componendo and dividendo
Simplifying
Hence Proved.
Question 8: If
, show that
.
Answer:
Given
By componendo and dividendo
Simplifying
Applying again componendo and dividendo
Simplifying
Question 9: If ; prove that:
.
Answer:
Given
Therefore
Hence Proved.
Question 10: If are in continued proportion, prove that:
i)
ii)
Answer:
Given
or
i)
LHS
RHS
Hence LHS = RHS. Hence Proved.
ii)
LHS
RHS
Hence LHS = RHS. Hence Proved.
Question 11: Using Properties of proportion, solve for :
i)
ii)
iii)
Answer:
i)
Applying componendo and dividendo
Squaring both sides
ii)
Applying componendo and dividendo
Squaring both sides
Applying componendo and dividendo
or
iii)
Applying componendo and dividendo
Squaring both sides
Question 12: If
, prove that:
. [2007]
Answer:
Given
Applying componendo and dividendo
Squaring both sides
Applying componendo and dividendo once again
simplifying
Hence proved.
Question 13: Using the properties of proportion, solve for Given:
. [2013]
Answer:
Given
Applying componendo and dividendo
Taking the square root of both sides
Question 14: If
, express
in terms of
.
Answer:
Given
Applying componendo and dividendo
Simplifying
Squaring both sides
Applying componendo and dividendo again
Simplifying
or
Question 15: If
, show that:
.
Answer:
Given
Applying componendo and dividendo
or
Applying componendo and dividendo again
Simplifying
or