If (10 + _ 10 x) , (10 + _ 10 y) and (10 + _ 10 z) are in AP, then consider the following statements: I. The GM of x and z is y^2 . II. The AM of _ 10 x and _ 10 z is _ 10 y . Which of the statements given above is/are

Correct answer: B. II only Given (10 + _ 10 x) , (10 + _ 10 y) , and (10 + _ 10 z) are in AP. 2(10 + _ 10 y) = (10 + _ 10 x) + (10 + _ 10 z) 20 + 2 _ 10 y = 20 + _ 10 x + _ 10 z 2 _ 10 y = _ 10 x + _ 10 z _ 10 y = _ 10 x + _ 10 z 2 This shows that the AM of _ 10 x and _ 10 z is _ 10 y . Thus, Statement II is correct. From 2 _ 10 y = _ 10 x + _ 10 z , we get: _ 10 y^2 = _ 10 (xz) y^2 = xz The GM of x and z is xz = y^2 = y . Thus, Statement I is incorrect. Only Statement II is correct. Answer: II only

NDA 2025

Exam: NDA
Year: 2025
Subject: Mathematics
Chapter: Sequences and Series

NDA 2025 · Mathematics · Sequences and Series

If (10 + _ 10 x) , (10 + _ 10 y) and (10 + _ 10 z) are in AP, then...

(10 + \log_{10} x)

(10 + \log_{10} y)

and

(10 + \log_{10} z)

are in AP, then consider the following statements:
I. The GM of

x

and

z

y^2

.
II. The AM of

\log_{10} x

and

\log_{10} z

\log_{10} y

.
Which of the statements given above is/are correct?

Options

AI only
BII only
CBoth I and II
DNeither I nor II

Correct answer

B. II only

Step-by-step solution

Given

(10 + \log_{10} x)

(10 + \log_{10} y)

, and

(10 + \log_{10} z)

are in AP.

2(10 + \log_{10} y) = (10 + \log_{10} x) + (10 + \log_{10} z)

20 + 2\log_{10} y = 20 + \log_{10} x + \log_{10} z

2\log_{10} y = \log_{10} x + \log_{10} z

\log_{10} y = \dfrac{\log_{10} x + \log_{10} z}{2}

This shows that the AM of

\log_{10} x

and

\log_{10} z

\log_{10} y

. Thus, Statement II is correct.

From

2\log_{10} y = \log_{10} x + \log_{10} z

, we get:

\log_{10} y^2 = \log_{10} (xz)

y^2 = xz

The GM of

x

and

z

\sqrt{xz} = \sqrt{y^2} =

.<br><br>Thus, Statement I is incorrect.<br><br>Only Statement II is correct.<br><br>Answer: II only

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