JEE Main Sets, Relations, Functions & Trigonometry: Sets, Relations and Functions – Chapter Test

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Difficulty: 10 medium · 10 hard

Sets, Relations and Functions MCQ questions with answers

The first 3 of 20 questions. Unlock the series for the rest, each with a solution.

1. Q1. Assertion (A): If f : A → B and g : B → C are both one-one functions, then g ∘ f : A → C is one-one. Reason (R): The composition of two injective functions is injective. Choose the correct option.

   अभिकथन (A): यदि f : A → B तथा g : B → C दोनों एकैकी फलन हैं, तो g ∘ f : A → C एकैकी है। कारण (R): दो एकैकी (injective) फलनों का संयोजन एकैकी होता है। सही विकल्प चुनिए।

   • A. Both A and R are true and R is the correct explanation of A / A और R दोनों सत्य हैं तथा R, A की सही व्याख्या है
   • B. Both A and R are true but R is not the correct explanation of A / A और R दोनों सत्य हैं परंतु R, A की सही व्याख्या नहीं है
   • C. A is false but R is true / A असत्य है परंतु R सत्य है
   • D. A is true but R is false / A सत्य है परंतु R असत्य है
   Solution

   If (g∘f)(x₁) = (g∘f)(x₂) then g(f(x₁)) = g(f(x₂)); injectivity of g gives f(x₁) = f(x₂) and injectivity of f gives x₁ = x₂, so both statements are true and R explains A.

   यदि (g∘f)(x₁) = (g∘f)(x₂) तो g(f(x₁)) = g(f(x₂)); g की एकैकिता से f(x₁) = f(x₂) तथा f की एकैकिता से x₁ = x₂ प्राप्त होता है, अतः दोनों कथन सत्य हैं और R, A की व्याख्या करता है।

2. Q2. Among the following functions from R to R, three share a common structural property while one does not. Which is the odd one out?

   R से R में निम्नलिखित फलनों में से, तीन एक समान संरचनात्मक गुणधर्म साझा करते हैं जबकि एक नहीं। कौन-सा भिन्न (odd one out) है?

   • A. f(x) = x|x|
   • B. f(x) = sin x
   • C. f(x) = x²
   • D. f(x) = x³
   Solution

   sin x, x³ and x|x| are all odd functions, whereas x² is an even function, making it the odd one out.

   sin x, x³ और x|x| सभी विषम फलन हैं, जबकि x² एक सम फलन है, जो इसे भिन्न बनाता है।

3. Q3. The number of real values of x satisfying |x − 1| + |x − 2| = 1 is:

   |x − 1| + |x − 2| = 1 को संतुष्ट करने वाले x के वास्तविक मानों की संख्या है:

   • A. 0
   • B. exactly 2 / ठीक 2
   • C. infinitely many / अनंत रूप से अनेक
   • D. exactly 1 / ठीक 1
   Solution

   The sum of distances from 1 and 2 equals 1 (the gap between them) for every x in the closed interval [1, 2], so there are infinitely many solutions.

   1 और 2 से दूरियों का योग 1 (उनके बीच का अंतराल) बंद अंतराल [1, 2] में प्रत्येक x के लिए होता है, अतः अनंत रूप से अनेक हल हैं।

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