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Questions › IIT JAM › IIT JAM 2021 › Q26
2 marks, −0.667Mathematics › Sets, relations and functions › Sets and set operationsModerateabout 2 min

IIT JAM 2021 Question 26: Sets, relations and functions

For S1,S2⊂RS_1, S_2 \subset \mathbb{R}S1​,S2​⊂R define S1−S2={x∈S1,x∉S2}S_1 - S_2 = \{x \in S_1, x \notin S_2\}S1​−S2​={x∈S1​,x∈/S2​}. Let P={x∈R:x2−2x−3≤0}P = \{x \in \mathbb{R} : x^2 - 2x - 3 \le 0\}P={x∈R:x2−2x−3≤0} and Q={x∈R:log⁡5(1+x2)≤1}Q = \{x \in \mathbb{R} : \log_5(1 + x^2) \le 1\}Q={x∈R:log5​(1+x2)≤1}. Then,

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Answer and worked solution

Answer: (D)

PPP solves (x−3)(x+1)≤0(x-3)(x+1) \le 0(x−3)(x+1)≤0, giving P=[−1,3]P = [-1, 3]P=[−1,3]. QQQ solves 1+x2≤51 + x^2 \le 51+x2≤5, i.e. x2≤4x^2 \le 4x2≤4, giving Q=[−2,2]Q = [-2, 2]Q=[−2,2].

Then Q−PQ - PQ−P removes from [−2,2][-2,2][−2,2] everything in [−1,3][-1,3][−1,3], leaving [−2,−1)[-2, -1)[−2,−1). The answer is (D).

The trap: (A) P−Q=[2,3]P - Q = [2, 3]P−Q=[2,3]

P−QP - QP−Q works out to (2,3](2, 3](2,3], not [2,3][2, 3][2,3]; the clean interval is Q−P=[−2,−1)Q - P = [-2, -1)Q−P=[−2,−1), so option (D) is the one that is exactly right.

Plain-text version

For S₁, S₂ ⊂ ℝ define S₁ - S₂ = {x ∈ S₁, x ∉ S₂}. Let P = {x ∈ ℝ: x² - 2x - 3 ≤ 0} and Q = {x ∈ ℝ: log₅(1 + x²) ≤ 1}. Then, (A) P - Q = [2, 3] (B) Q - P = (1, 2] (C) P - Q = [-3, -2) (D) Q - P = [-2, -1)

Also written as: For S_1, S_2 subset R define S_1 - S_2 = {x in S_1, x not in S_2}. Let P = {x in R: x^2 - 2x - 3 <= 0} and Q = {x in R: log_5(1 + x^2) <= 1}. Then,

Sets, relations and functions has come up 65 times in 6 exams since 2004.

What this question tests

This question is on Sets and set operations, part of Sets, relations and functions in Mathematics.

Unions, intersections, complements and differences of sets, and the identities that connect them.

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