Mathematics Questions and Answers – Relationship Between A.M. and G.M.

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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

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

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) $$\frac{6±3\sqrt{7}}{1}$$

Explanation: We know, G.M. of two numbers a and b is √ab.
So, a + b = 4 √ab
Squaring we get, a2+b2 = 16ab
=>(a/b) + (b/a) = 16
Let x = a/b.
So, x + 1/x = 16 => x2 – 16x + 1 = 0
=>x = $$\frac{16±\sqrt{256-4}}{2} = \frac{16±\sqrt{252}}{2} = \frac{16±6\sqrt{7}}{2} = \frac{8±3\sqrt{7}}{1}$$.
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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

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.

To practice Mathematics Objective Questions and Answers for Class 11, here is complete set of 1000+ Multiple Choice Questions and Answers.