Note:
1. If there are Geometric Means inserted between
and
, then
2. Geometric mean between and
is
Question 1: Insert 6 geometric means between and
.
Answer:
Let be the
GMs inserted between
and
.
( As
geometric means are being inserted).
. Therefore,
Hence the 6 geometric means between and
are
Question 2: Insert 5 geometric means between and
.
Answer:
Let be the
GMs inserted between
and
.
( As
geometric means are being inserted).
. Therefore,
Hence the 5 geometric means between and
are
Question 3: Insert 5 geometric means between and
.
Answer:
Let be the
GMs inserted between
and
.
( As
geometric means are being inserted).
. Therefore,
Hence the 5 geometric means between and
are
Question 4: Find the geometric means of the following pairs of numbers:
i) ii)
iii)
Answer:
i)
Geometric mean between
ii)
Geometric mean between
iii)
Geometric mean between
Question 5: If is the G.M. of
and
, find
Answer:
Question 6: Find the two numbers whose A.M. is and GM is
.
Answer:
Given A.M.
… … … … … i)
Also GM … … … … … ii)
Solving i) and ii) we get
or
When
When
Hence the two numbers are and
Question 7: Construct a quadratic in such that A.M. of its roots is
and G.M. is
.
Answer:
Let the two roots of the quadratic equation be and
… … … … … i)
Also … … … … … ii)
We know the quadratic equation having roots and
is given by
Question 8: The sum of two numbers is times their geometric mean, show that the numbers are in the ratio
.
Answer:
Let the two numbers be and
Given
Applying componendo and dividendo
Applying componendo and dividendo once again
. Hence proved.
Question 9: If AM and GM of roots of a quadratic equation are and
respectively, then obtain the quadratic equation.
Answer:
Let the two roots of the quadratic equation be and
Given AM
… … … … … i)
Also GM
Therefore the quadratic equation will be
Question 10: If AM and GM of two positive numbers are
respectively, find the numbers.
Answer:
Given AM
… … … … … i)
Also GM … … … … … ii)
Solving i) and ii) we get
When
When
Hence the two numbers are and
.
Question 11: Prove that the product of geometric means between two quantities is equal to the
power of a geometric mean of those two quantities.
Answer:
Let be the
GMs inserted between
and
Let be the common ratio.
Let be the GM between
and
Now Given
. Hence proved.
Question 12: If the A.M. of two positive numbers and
is twice their geometric mean. Prove that:
Answer:
Let the two numbers be and
Given A.M. = 2 G.M.
Given
Applying componendo and dividendo
Applying componendo and dividendo once again
. Hence proved.
Question 13: If one A.M., and two geometric means
and
inserted between any two positive numbers, show that
Answer:
Let the two numbers be and
Let and
be two GM between
and
To prove :
LHS
RHS