By signing in you confirm you are 18 or older, or have your parent’s or guardian’s consent.

NDA 2 2024 Maths, question 84

Topic: Differential calculus

Passage · Q83 to Q84

Let f(x)=[x]2−[x2]f(x)=[x]^{2}-[x^{2}].

Q84

Consider the following statements :

I. f(x)f(x) is continuous at x=0x=0.

II. f(x)f(x) is continuous at x=1x=1.

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. Just left of 0, ⌊x⌋=−1 and ⌊x²⌋=0, giving f=1; at 0 and just right, f=0.
  2. So I is false.
  3. On both sides sufficiently close to 1, the two floor terms have equal squares/values and f=0, also f(1)=0.
  4. Hence II is true.