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NDA 2 2024 Maths, question 78

Topic: Differential calculus

Q78

If f(x)=9x−8xf(x)=9x-8\sqrt{x} such that g(x)=f(x)−1g(x)=f(x)-1, then which one of the following is correct?

  1. (a) g(x)=0g(x)=0 has no real roots
  2. (b) g(x)=0g(x)=0 has only one real root which is an integerCorrect answer
  3. (c) g(x)=0g(x)=0 has two real roots which are integers
  4. (d) g(x)=0g(x)=0 has only one real root which is not an integer

Solution.

Answer: (b) g(x)=0g(x)=0 has only one real root which is an integer

  1. Put t=√x≥0.
  2. The equation g(x)=0 becomes 9t²−8t−1=(9t+1)(t−1)=0.
  3. Reject t=−1/9 because a square root is nonnegative.
  4. Therefore t=1 and x=1, the single real root, is an integer.