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NDA 2 2025 Maths, question 5

Topic: Algebra

Q5

If (10+log⁡10x)(10+\log _{10}x), (10+log⁡10y)(10+\log _{10}y) and (10+log⁡10z)(10+\log _{10}z) are in AP, then consider the following statements :

I. The GM of xx and zz is y2y^{2}.

II. The AM of log⁡10x\log _{10}x and log⁡10z\log _{10}z is log⁡10y\log _{10}y.

Which of the statements given above is/are correct?

  1. (a) I only
  2. (b) II onlyCorrect answer
  3. (c) Both I and II
  4. (d) Neither I nor II

Solution.

Answer: (b) II only

  1. The AP condition gives 2 log₁₀y = log₁₀x + log₁₀z.
  2. Thus y²=xz and, since logarithms require positive arguments, √(xz)=y, not y².
  3. Statement II is exactly the AP condition after removing the common constant 10.