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NDA 1 2024 Maths, question 28

Topic: Algebra

Q28

For x≥y>1x\ge y>1, let log⁡x(xy)+log⁡y(yx)=k\log _{x}\left( \frac{x}{y} \right)+\log _{y}\left( \frac{y}{x} \right)=k, then the value of kk can never be equal to

  1. (a) −1-1
  2. (b) −12-\frac{1}{2}
  3. (c) 00
  4. (d) 11Correct answer

Solution.

Answer: (d) 11

  1. Put t=ln x/ln y≥1.
  2. The expression becomes (1−1/t)+(1−t)=2−t−1/t≤0.
  3. It ranges continuously from zero down to negative infinity, so −1, −1/2 and 0 are possible, but 1 is not.