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