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Question

Three non-zero real numbers form A.P. and the squares of these numbers taken in the same order form a G.P. Then the number of all possible common ratios of the G.P. is


A

1

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B

2

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C

3

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D

None of these

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Solution

The correct option is B

2


Explanation for the correct answer:

Let the three numbers in arithmetic progression be

a-d,a,a+d

The squares of these numbers respectively are

a-d2,a2,a+d2

According to given condition the squares of the numbers form a geometric progression

a-d2,a2,a+d2 are in a geometric progression

a22=a-d2a+d2 [y2=xz,whenx,y,zareinG.P]

a4=a2-d22 [x+yx-y=x2-y2]

a2=a2-d2 or a2=d2-a2

d2=0 or d2=2a2

d=0 or d=±2a

As the given terms are distinct d cannot be zero d=±2a

The common ratio r is the ratio of the consecutive terms

r=a2a-d2

r=a2a±2a2

r=a2a+2a2 or r=a2a-2a2

r=11+22 or r=11-22

Hence there are 2 possible values of the common ratio for the squares of the numbers to be in a geometric progression.

Hence, option B is the correct answer.


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