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
View Answer

Answer: b
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.
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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

Answer: b
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

Answer: b
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
View Answer

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

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter