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MCQEasyJEE 2023Sets & Operations

JEE Mathematics 2023 Question with Solution

Let the sets AA and BB denote the domain and range respectively of the function f(x)=1∣x∣−xf(x) = \frac{1}{\sqrt{|x|}-x}, where [x][x] denotes the smallest integer greater than or equal to xx. Then among the statements:

  • A

    A∩B=(1,∞)−NA \cap B = (1, \infty) - N

  • B

    A∪B=(1,∞)A \cup B = (1, \infty)

  • C

    only (S1) is true

  • D

    both (S1) and (S2) are true

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The function is f(x)=1∣x∣−xf(x) = \frac{1}{\sqrt{|x|}-x}.

Find: Which statement among the given ones is true.

From the extracted solution, the function is defined for x>1x > 1 but excludes natural numbers N\mathbb{N} from the domain due to the square root denominator constraint.

So, according to the solution,

A∩B=(1,∞)−NA \cap B = (1, \infty) - \mathbb{N}

Therefore, (S1) is true.

For (S2), the extracted solution states that

A∪B≠(1,∞)A \cup B \neq (1, \infty)

so (S2) is false.

Therefore, only (S1) is true, so the correct option is A.

Common mistakes

  • Assuming that both set statements must be true together. This is wrong because the question asks to test each statement separately. Evaluate (S1) and (S2) independently before choosing the option.

  • Ignoring the domain restriction coming from the denominator. This is wrong because values making the denominator zero must be excluded. Always check where ∣x∣−x=0\sqrt{|x|}-x = 0 before writing the domain.

  • Confusing intersection with union. This is wrong because A∩BA \cap B means common elements, whereas A∪BA \cup B means all elements from either set. Use the correct set operation while matching statements.

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