250+ TOP MCQs on Arithmetic and Geometric Mean and Answers

Discrete Mathematics Multiple Choice Questions & Answers on “Arithmetic and Geometric Mean”.

1. Let A₁, A₂, be two AM’s and G₁, G₂ be two GM’s between a and b,then (A₁ + A₂) / G₁G₂ is equal to _______
a) (a+b) / 2ab
b) 2ab/(a+b)
c) (a+b)/(ab)
d) None of the mentioned

Answer: c
Clarification: A₁ + A₂ = a + b, G₁G₂ = ab.

2. The series a,(a+b)/2, b is in _______
a) AP
b) GP
c) HP
d) None of the mentioned

Answer: a
Clarification: (a+b)/2 is AM between a, b. Hence series is in AP.

3. The series a, (ab)^1/2, b is in _______
a) AP
b) GP
c) HP
d) None of the mentioned

Answer: b
Clarification: (ab)^1/2 is GM between a, b. Hence series is in GP.

4. If A and G be the A.M and G.M between two positive number then the numbers are A + (A² – G²)^1/2, A – (A² – G²)^1/2.
a) True
b) False

Answer: a
Clarification: The equation having its roots as given equation is
x² – 2Ax + G² = 0 which implies
x = A + (A² – G²)^1/2, A – (A² – G²)^1/2.

5. If one geometric mean G and two airthmetic mean A₁, A₂ are inserted between two numbers, then (2A₁ – A₂) (2A₂ – A₁) is equal to _______
a) 2G
b) G
c) G²
d) None of the mentioned

Answer: c
Clarification: Let a and b be two numbers then, G = (ab)^1/2, A₁ = (2a+b)/3, A₂ = (a+2b)/3, (2A₁ – A₂) = a, (2A₂ – A₁) = b, (2A₁ – A₂)(2A₂ – A₁) = G².

6. State whether the given statement is true or false.
AM ≤ GM.
a) True
b) False

Answer: b
Clarification: Airthmetic Mean is always greater or equal to the geometric mean.

7. If between two numbers which are root of given equation. x² – 18x + 16 = 0, a GM is inserted then the value of that GM is?
a) 4
b) 5
c) 6
d) 16

Answer: a
Clarification: x² – 2Ax + G² = 0, here G² = 16 and therefore G = 4.

8. If a₁, a₂, a₃ are in airthemetic as well as geometric progression then which of the following is/are correct?
a) 2a₂ = a₁ + a₃
b) a₂ = (a₁a₃)^1/2
c) a₂ – a₁ = a₃ -a₂
d) All of the mentioned are correct

Answer: d
Clarification: a₂ is AM, GM between a₁, a₃, also the series is in AP so common difference should be same.

9. If a₁, a₂, a₃ are in GP then 1/a₁, 1/a₂, 1/a₃ are in ___________
a) AP
b) GP
c) HP
d) None of the mentioned

Answer: b
Clarification: Let the terms be ar, a, a/r then reciprocals are 1/(ar), 1/a, r/a. Still the terms are in GP.

10. If a₁, a₂, a₃…….. are in AP then if a₇ = 15, then the value of common difference that would make a₂ a₇ a₁₂ greatest is?
a) 2
b) 0
c) 4
d) 9

Answer: b
Clarification: Let d be common difference of the AP. Then,
a₂ a₇ a₁₂ = (15 – 5d)(15)(15 + 5d) = 375(9 – d²)
For maximum value d=0.