This set of Mathematics Objective Questions and Answers for Class 11 focuses on “Relationship Between A.M. and G.M.”.

1. If A.M. of two numbers is 15/2 and their G.M. is 6, then find the two numbers.

a) 6 and 8

b) 12 and 3

c) 24 and 6

d) 27 and 3

View Answer

Explanation: We know, A.M. of two numbers a and b is (a + b)/2

=> (a + b)/2 = 15/2 => a + b = 15.

Also, G.M. of two numbers a and b is \(\sqrt{ab}\)

=> \(\sqrt{ab}\) = 6 => ab = 36.

=> a(15-a) = 36 => a=3 or 12.

For a=3, b=12.

For a=12, b=3.

So, the two numbers are 3 and 12.

2. Which of the following is true if A means arithmetic mean and b means geometric mean of two numbers?

a) A>G

b) A≥G

c) G<A

d) G≤A

View Answer

Explanation: We know, A.M. of two numbers a and b is (a + b)/2

Also, G.M. of two numbers a and b is \(\sqrt{ab}\)

A-G = (a + b)/2 – \(\sqrt{ab} = ((a + b) – 2\sqrt{ab})/2 = (\sqrt{a} – \sqrt{b})^2\) / 2 ≥ 0

So, A≥G.

3. If the sum of two numbers is 4 times the geometric mean then find the ratio of numbers.

a) \(\frac{8±3\sqrt{5}}{1}\)

b) \(\frac{8±3\sqrt{7}}{1}\)

c) \(\frac{6±3\sqrt{5}}{1}\)

d) \(\frac{6±3\sqrt{7}}{1}\)

View Answer

Explanation: We know, G.M. of two numbers a and b is √ab.

So, a + b = 4 √ab

Squaring we get, a

^{2}+b

^{2}= 16ab

=>(a/b) + (b/a) = 16

Let x = a/b.

So, x + 1/x = 16 => x

^{2}– 16x + 1 = 0

=>x = \(\frac{16±\sqrt{256-4}}{2} = \frac{16±\sqrt{252}}{2} = \frac{16±6\sqrt{7}}{2} = \frac{8±3\sqrt{7}}{1}\).

4. The ratio of the A.M. and G.M. of two positive numbers a and b is 5: 3. Find the ratio of a to b.

a) 9:1

b) 3:5

c) 1:9

d) 3:1

View Answer

Explanation: (A.M.)/(G.M.) = 5/3

=> \(\frac{a+b}{2\sqrt{ab}} = \frac{5}{3}\)

Applying componendo and dividendo rule, we get

=> \(\frac{a+b+2\sqrt{ab}}{a+b-2\sqrt{ab}} = \frac{8}{2}\)

=> \((\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}})^2=4 \)

=> \((\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}})^1=2\)

=> \((\frac{\sqrt{a}}{\sqrt{b}})^1=3\)

Again applying componendo and dividendo rule, we get

a/b = (3/1)

^{2}= 9. So, a:b =9:1.

**Sanfoundry Global Education & Learning Series – Mathematics – Class 11**.

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