Note:
- If there are
Geometric Means inserted between
, then
- Geometric mean between
is
Answer:
( As
geometric means are being inserted).
Answer:
( As
geometric means are being inserted).
Answer:
( As
geometric means are being inserted).
Question 4: Find the geometric means of the following pairs of numbers:
Answer:
Geometric mean between
Geometric mean between
Geometric mean between
Answer:
.
Answer:
.. … … … … i)
.. … … … … ii)
Solving i) and ii) we get
or
.
Answer:
Let the two roots of the quadratic equation be
.. … … … … i)
.. … … … … ii)
We know the quadratic equation having roots is given by
Question 8: The sum of two numbers is times their geometric mean, show that the numbers are in the ratio
.
Answer:
Applying componendo and dividendo
Applying componendo and dividendo once again
Question 9: If AM and GM of roots of a quadratic equation are respectively, then obtain the quadratic equation.
Answer:
Let the two roots of the quadratic equation be
.. … … … … i)
Therefore the quadratic equation will be
Question 10: If AM and GM of two positive numbers respectively, find the numbers.
Answer:
.. … … … … i)
.. … … … … ii)
Solving i) and ii) we get
.
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:
be the
GMs inserted between
be the common ratio.
be the GM between
. Hence proved.
Question 12: If the A.M. of two positive numbers is twice their geometric mean. Prove that:
Answer:
Given A.M. = 2 G.M.
Applying componendo and dividendo
Applying componendo and dividendo once again
Question 13: If one A.M., and two geometric means
inserted between any two positive numbers, show that
Answer:
be two GM between