Directions: Below question is followed by two statements labelled I and II. Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below: What is the value of X? I. If log 2, log (2x + 1) and log (2x + 1.5) are in AP II. If X is a non-negative integer

Directions: Below question is followed by two statements labelled I and II. Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below:

What is the value of X?

I. If log 2, log (2^x + 1) and log (2^x + 1.5) are in AP

II. If X is a non-negative integer

A Statement I alone is sufficient to answer the question

B Statement II alone is sufficient to answer the question.

C Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.

D Either Statement I or Statement I alone is sufficient to answer the question.

E Neither Statement I nor Statement II is sufficient to answer the question.

Solution

Correct Answer: Option A

Given, log 2, log (2^x + 1) and log (2^x + 1.5) are in AP.

⇒log (2^x + 1) = \(\frac{{\log 2 + \log ({2^x} + 3)}}{2} \Rightarrow 2\log \left( {{2^x} + 1} \right) = \log 2 + {\rm{log}}\left( {{2^x} + 1.5} \right)\)

⇒ log (2^x + 1)² = log (2 × (2^x + 1.5))

Equating the terms inside the logarithms (or, simply removing the logarithms),

(2^x + 1)²= (2 × (2^x + 1.5))

⇒ (2^x)² + (1)² + 2 × 1 × 2^x = 2(2^x + 1.5)

For simplicity, let us assume 2^x = a

⇒ a² + 1 + 2a = 2(a + 1.5)

⇒a² + 1 + 2a = 2a + 3

⇒a² + 1 = 3

⇒ a² = 2

⇒ a = √2 (We do not consider -√2 because we need to equate it with 2^x and not -2^x)

Substituting the original value of a,

2^x = √2 = 2^1/2

⇒ x = ½

Hence, statement I alone is sufficient to answer this question.

From statement II:

It is only given that x is a non-negative integer. In that case, x can take infinite values.

Hence, statement II alone is not sufficient to answer the question.

Solution

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