Simple Learning HubCLASS 9–12 · JEE · NEET

Ch 2Relations and Functionsसंबंध एवं फलनRelations aur Functions

Ordered pairs and Cartesian product, relations (domain, codomain, range), functions, graphs of special functions and algebra of real functions — every NCERT exercise solved.क्रमित युग्म और कार्तीय गुणन, संबंध (प्रांत, सहप्रांत, परिसर), फलन, विशेष फलनों के आलेख और वास्तविक फलनों का बीजगणित — NCERT के सभी प्रश्नों के हल।Ordered pairs aur Cartesian product, relations (domain, codomain, range), functions, special functions ke graphs aur real functions ka algebra — NCERT ke saare questions ke hal.

Didn’t understand a paragraph? Tap it (or its अ button) → हिंदी / Hinglish / Simple English.कोई पैराग्राफ़ समझ नहीं आया? उस पर (या अ बटन पर) टैप करें → English / Hinglish।Koi paragraph samajh nahi aaya? Us par (ya अ button par) tap kijiye → हिंदी / English.

👆 Words with a open a diagram or graph — tap any label to read what it means.👆 वाले शब्द पर टैप करें — चित्र या ग्राफ़ खुलेगा (हर लेबल पर टैप = उसका अर्थ)।👆 wale word par tap kijiye — diagram ya graph khulega (har label par tap = uska matlab).
📘 NCERT chapter (English PDF)📗 NCERT हिंदी PDF∑ Formulas✎ NCERT solutions🎯 Quiz

Diagrams in this chapterइस अध्याय के चित्रIs chapter ke diagrams

Arrow diagram of a relation (Ex 2.2 Q4)संबंध का तीर आरेख (प्रश्न 2.2.4)Relation ka arrow diagram (Ex 2.2 Q4)Fig 2.7Graphs of special functions: identity, modulus, signum, greatest integerविशेष फलनों के आलेख: तत्समक, मापांक, चिह्न, महत्तम पूर्णांकSpecial functions ke graphs: identity, modulus, signum, greatest integerFig 2.8, 2.13, 2.14, 2.15Polynomial y = x² and rational function y = 1/xबहुपद y = x² और परिमेय फलन y = 1/xPolynomial y = x² aur rational function y = 1/xFig 2.10, 2.12
In this chapter (NCERT sections)इस अध्याय में (NCERT खंड)Is chapter mein (NCERT sections)
  1. 2.2Cartesian Products of Setsसमुच्चयों का कार्तीय गुणनSets ka Cartesian Product
  2. 2.3RelationsसंबंधRelations
  3. 2.4FunctionsफलनFunctions
  4. ★Yaad rakho — key formulasयाद रखो — मुख्य सूत्रYaad rakho — key formulas
  5. ⚠Common mistakesआम गलतियाँCommon mistakes
  6. ✎NCERT exercises — solutionsNCERT प्रश्नावली — हलNCERT exercises — solutions
  7. 🎯Quizक्विज़Quiz

2.2Cartesian Products of Setsसमुच्चयों का कार्तीय गुणनSets ka Cartesian Product

An ordered pair (a, b) has a first element and a second element; order matters, so (1, 2) \ne (2, 1). Two ordered pairs are equal only when the corresponding elements are equal: (a, b) = (x, y) \iff a = x and b = y.क्रमित युग्म (a, b) में पहला और दूसरा अवयव होता है; क्रम मायने रखता है, इसलिए (1, 2) \ne (2, 1)। दो क्रमित युग्म तभी समान हैं जब संगत अवयव समान हों: (a, b) = (x, y) \iff a = x और b = y।Ordered pair (a, b) mein pehla aur doosra element hota hai; order matter karta hai, isliye (1, 2) \ne (2, 1). Do ordered pairs tabhi equal hain jab corresponding elements equal hon: (a, b) = (x, y) \iff a = x aur b = y.

A \times B = \{(a, b) : a \in A
Cartesian product: all ordered pairs with first element from A and second from B. If A or B is empty, A \times B = \phi.कार्तीय गुणन: वे सभी क्रमित युग्म जिनका पहला अवयव A से और दूसरा B से हो। यदि A या B रिक्त हो तो A \times B = \phi।Cartesian product: wo saare ordered pairs jinka pehla element A se aur doosra B se ho. Agar A ya B empty ho to A \times B = \phi.
b \in B\}
n(A \times B) = n(A) \cdot n(B)
If A has p elements and B has q, then A \times B has pq pairs.यदि A में p और B में q अवयव हों तो A \times B में pq युग्म।Agar A mein p aur B mein q elements hon to A \times B mein pq pairs.
Example 1If (x + 1, y - 2) = (3, 1), find x and y.उदाहरण 1यदि (x + 1, y - 2) = (3, 1), तो x और y ज्ञात कीजिए।Example 1Agar (x + 1, y - 2) = (3, 1), to x aur y nikaliye.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. x + 1 = 3,\quad y - 2 = 1

✔ x = 2,\ y = 3

2.3RelationsसंबंधRelations

A relation R from a non-empty set A to a non-empty set B is a subset of A \times B, chosen by some rule linking the first and second elements. If (x, y) \in R we write x\,R\,y; y is the image of x. The set of first elements is the domain, the set of second elements is the range, and the whole set B is the codomain (range ⊂ codomain). See the arrow diagram.अरिक्त समुच्चय A से अरिक्त समुच्चय B में संबंध R, A \times B का एक उपसमुच्चय है, जो प्रथम और द्वितीय अवयवों को जोड़ने वाले किसी नियम से चुना जाता है। यदि (x, y) \in R तो x\,R\,y लिखते हैं; y, x का प्रतिबिंब है। प्रथम अवयवों का समुच्चय प्रांत, द्वितीय अवयवों का समुच्चय परिसर, और पूरा B सहप्रांत है (परिसर ⊂ सहप्रांत)। तीर आरेख देखिए।Non-empty set A se non-empty set B mein relation R, A \times B ka ek subset hai, jo pehle aur doosre elements ko jodne wale kisi rule se chuna jaata hai. Agar (x, y) \in R to x\,R\,y likhte hain; y, x ka image hai. Pehle elements ka set domain, doosre elements ka set range, aur poora B codomain hai (range ⊂ codomain). Arrow diagram dekhiye.

Arrow diagram of a relation (Ex 2.2 Q4)संबंध का तीर आरेख (प्रश्न 2.2.4)Relation ka arrow diagram (Ex 2.2 Q4)Fig 2.7 · 3 tappable partsचित्र 2.7 · 3 टैप करने लायक हिस्सेFig 2.7 · 3 tappable partsOpen
Number of relationsIf n(A) = p and n(B) = q, then n(A \times B) = pq and the number of relations from A to B is 2^{pq} (every subset of A \times B is a relation).संबंधों की संख्यायदि n(A) = p और n(B) = q, तो n(A \times B) = pq और A से B में संबंधों की संख्या 2^{pq} है (A \times B का हर उपसमुच्चय एक संबंध है)।Relations ki gintiAgar n(A) = p aur n(B) = q, to n(A \times B) = pq aur A se B mein relations ki ginti 2^{pq} hai (A \times B ka har subset ek relation hai).

2.4FunctionsफलनFunctions

FunctionA relation f from A to B is a function if every element of A has one and only one image in B. We write f : A \to B and f(x) = y. Domain of f = A, codomain = B, range = set of images. A function whose range is a subset of \mathbf{R} is a real-valued function; if its domain is also in \mathbf{R} it is a real function.फलनA से B में संबंध f एक फलन है यदि A के प्रत्येक अवयव का B में एक और केवल एक प्रतिबिंब हो। इसे f : A \to B और f(x) = y लिखते हैं। f का प्रांत = A, सहप्रांत = B, परिसर = प्रतिबिंबों का समुच्चय। जिस फलन का परिसर \mathbf{R} का उपसमुच्चय हो वह वास्तविक मान फलन है; यदि प्रांत भी \mathbf{R} में हो तो वास्तविक फलन।FunctionA se B mein relation f ek function hai agar A ke har element ka B mein ek aur sirf ek image ho. Ise f : A \to B aur f(x) = y likhte hain. f ka domain = A, codomain = B, range = images ka set. Jis function ki range \mathbf{R} ka subset ho wo real-valued function hai; agar domain bhi \mathbf{R} mein ho to real function.

Quick test: in the list of pairs, no first element may repeat with a different second element. \{(1, 3), (1, 5), (2, 5)\} is not a function (1 has two images), but \{(2, 1), (5, 1), (8, 1)\} is (many x may share one image).आसान जाँच: युग्मों की सूची में कोई प्रथम अवयव अलग द्वितीय अवयव के साथ दोहराया नहीं जाना चाहिए। \{(1, 3), (1, 5), (2, 5)\} फलन नहीं (1 के दो प्रतिबिंब), पर \{(2, 1), (5, 1), (8, 1)\} फलन है (कई x का एक ही प्रतिबिंब हो सकता है)।Aasaan check: pairs ki list mein koi pehla element alag doosre element ke saath repeat nahi hona chahiye. \{(1, 3), (1, 5), (2, 5)\} function nahi (1 ke do images), par \{(2, 1), (5, 1), (8, 1)\} function hai (kai x ka ek hi image ho sakta hai).

2.4.1Some functions and their graphsकुछ फलन और उनके आलेखKuch functions aur unke graphs

Graphs of special functions: identity, modulus, signum, greatest integerविशेष फलनों के आलेख: तत्समक, मापांक, चिह्न, महत्तम पूर्णांकSpecial functions ke graphs: identity, modulus, signum, greatest integerFig 2.8, 2.13, 2.14, 2.15 · 4 tappable partsचित्र 2.8, 2.13, 2.14, 2.15 · 4 टैप करने लायक हिस्सेFig 2.8, 2.13, 2.14, 2.15 · 4 tappable partsOpenPolynomial y = x² and rational function y = 1/xबहुपद y = x² और परिमेय फलन y = 1/xPolynomial y = x² aur rational function y = 1/xFig 2.10, 2.12 · 2 tappable partsचित्र 2.10, 2.12 · 2 टैप करने लायक हिस्सेFig 2.10, 2.12 · 2 tappable partsOpen

2.4.2Algebra of real functionsवास्तविक फलनों का बीजगणितReal functions ka algebra

For real functions f, g : X \to \mathbf{R} (X ⊂ R) we define them point by point:वास्तविक फलनों f, g : X \to \mathbf{R} (X ⊂ R) के लिए बिंदुवार परिभाषा:Real functions f, g : X \to \mathbf{R} (X ⊂ R) ke liye point by point definition:

(f \pm g)(x) = f(x) \pm g(x)
Sum, difference and productयोग, अंतर और गुणनफलSum, difference aur product
(fg)(x) = f(x)\,g(x)
(\alpha f)(x) = \alpha f(x)
Multiplying by a scalar; quotient (only where g(x) ≠ 0)अदिश से गुणा; भागफल (केवल जहाँ g(x) ≠ 0)Scalar se guna; quotient (sirf jahan g(x) ≠ 0)
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
g(x) \ne 0
Example 16Let f(x) = x^2 and g(x) = 2x + 1. Find (f + g)(x), (f - g)(x), (fg)(x) and \left(\frac{f}{g}\right)(x).उदाहरण 16मान लीजिए f(x) = x^2 और g(x) = 2x + 1। (f + g)(x), (f - g)(x), (fg)(x) और \left(\frac{f}{g}\right)(x) ज्ञात कीजिए।Example 16Maan lijiye f(x) = x^2 aur g(x) = 2x + 1. (f + g)(x), (f - g)(x), (fg)(x) aur \left(\frac{f}{g}\right)(x) nikaliye.
Solution — step by step (3)हल — चरण दर चरण (3)Hal — step by step (3)
  1. (f + g)(x) = x^2 + 2x + 1,\quad (f - g)(x) = x^2 - 2x - 1
  2. (fg)(x) = x^2(2x + 1) = 2x^3 + x^2
  3. \left(\frac{f}{g}\right)(x) = \frac{x^2}{2x + 1},\quad x \ne -\frac{1}{2}
Example 20f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\} is a linear function from Z to Z. Find f(x).उदाहरण 20f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}, Z से Z में रैखिक फलन है। f(x) ज्ञात कीजिए।Example 20f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}, Z se Z mein linear function hai. f(x) nikaliye.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. Linear means f(x) = mx + c.रैखिक अर्थात् f(x) = mx + c।Linear yaani f(x) = mx + c.
    f(0) = -1 \Rightarrow c = -1;\quad f(1) = 1 \Rightarrow m - 1 = 1 \Rightarrow m = 2

✔ f(x) = 2x - 1

★ Yaad rakho — key formulas★ याद रखो — मुख्य सूत्र★ Yaad rakho — key formulas

  1. (a, b) = (x, y) \iff a = x,\ b = y; n(A \times B) = n(A)\,n(B).(a, b) = (x, y) \iff a = x,\ b = y; n(A \times B) = n(A)\,n(B)।(a, b) = (x, y) \iff a = x,\ b = y; n(A \times B) = n(A)\,n(B).
  2. A relation from A to B is any subset of A \times B; there are 2^{pq} of them.A से B में संबंध, A \times B का कोई भी उपसमुच्चय है; कुल 2^{pq}।A se B mein relation, A \times B ka koi bhi subset hai; kul 2^{pq}.
  3. Function: every element of the domain has exactly one image.फलन: प्रांत के हर अवयव का ठीक एक प्रतिबिंब।Function: domain ke har element ka theek ek image.
  4. Range ⊂ codomain. Range is what is actually reached.परिसर ⊂ सहप्रांत। परिसर = जहाँ तक वास्तव में पहुँचते हैं।Range ⊂ codomain. Range = jahan tak sach mein pahunchte hain.
  5. Domain of \sqrt{g(x)}: g(x) \ge 0; of \frac{1}{g(x)}: g(x) \ne 0.\sqrt{g(x)} का प्रांत: g(x) \ge 0; \frac{1}{g(x)} का: g(x) \ne 0।\sqrt{g(x)} ka domain: g(x) \ge 0; \frac{1}{g(x)} ka: g(x) \ne 0.
  6. |x| \ge 0; [x] \le x < [x] + 1; signum takes only −1, 0, 1.|x| \ge 0; [x] \le x < [x] + 1; चिह्न फलन केवल −1, 0, 1 लेता है।|x| \ge 0; [x] \le x < [x] + 1; signum sirf −1, 0, 1 leta hai.

⚠ Common mistakes⚠ आम गलतियाँ⚠ Common mistakes

  • Treating (2, 3) and (3, 2) as the same — in ordered pairs, order matters.(2, 3) और (3, 2) को एक मानना — क्रमित युग्म में क्रम मायने रखता है।(2, 3) aur (3, 2) ko same maanna — ordered pair mein order matter karta hai.
  • Saying range = codomain always. Range is only the images actually used.हमेशा परिसर = सहप्रांत मानना। परिसर केवल वास्तव में बने प्रतिबिंब हैं।Hamesha range = codomain maanna. Range sirf sach mein bane images hain.
  • Rejecting a function because two x have the same image. That is allowed; one x with two images is not.दो x का एक ही प्रतिबिंब होने पर फलन को नकारना। यह मान्य है; एक x के दो प्रतिबिंब मान्य नहीं।Do x ka same image hone par function reject karna. Ye allowed hai; ek x ke do images allowed nahi.
  • [-2.5] = -2 is wrong: the greatest integer ≤ −2.5 is −3.[-2.5] = -2 गलत है: −2.5 से छोटा या बराबर सबसे बड़ा पूर्णांक −3 है।[-2.5] = -2 galat hai: −2.5 se chhota ya barabar sabse bada integer −3 hai.
  • Forgetting to remove the zeros of g(x) from the domain of \frac{f}{g}.\frac{f}{g} के प्रांत से g(x) के शून्यकों को हटाना भूल जाना।\frac{f}{g} ke domain se g(x) ke zeros hatana bhool jaana.

✎ NCERT exercises — solutions✎ NCERT प्रश्नावली — हल✎ NCERT exercises — solutions

All 36 questions of the NCERT exercises. Tap a question to see the solution — try it yourself first!NCERT प्रश्नावली के सभी 36 प्रश्न। हल देखने के लिए प्रश्न पर टैप करें — पहले स्वयं हल करके देखें!NCERT exercises ke saare 36 questions. Hal dekhne ke liye question par tap kijiye — pehle khud try kijiye!

Exercise 2.1प्रश्नावली 2.1Exercise 2.1 · 10 Q

Q1. If \left(\frac{x}{3} + 1,\ y - \frac{2}{3}\right) = \left(\frac{5}{3},\ \frac{1}{3}\right), find x and y.यदि \left(\frac{x}{3} + 1,\ y - \frac{2}{3}\right) = \left(\frac{5}{3},\ \frac{1}{3}\right), तो x और y ज्ञात कीजिए।If \left(\frac{x}{3} + 1,\ y - \frac{2}{3}\right) = \left(\frac{5}{3},\ \frac{1}{3}\right), find x and y.
  1. \frac{x}{3} + 1 = \frac{5}{3} \Rightarrow \frac{x}{3} = \frac{2}{3} \Rightarrow x = 2
  2. y - \frac{2}{3} = \frac{1}{3} \Rightarrow y = 1

✔ x = 2,\ y = 1

Q2. If A has 3 elements and B = {3, 4, 5}, find the number of elements in A × B.यदि A में 3 अवयव हैं और B = {3, 4, 5}, तो A × B में अवयवों की संख्या ज्ञात कीजिए।If A has 3 elements and B = {3, 4, 5}, find the number of elements in A × B.
  1. n(A \times B) = 3 \times 3 = 9
Q3. If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.यदि G = {7, 8} और H = {5, 4, 2}, तो G × H और H × G ज्ञात कीजिए।If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
  1. G \times H = \{(7,5), (7,4), (7,2), (8,5), (8,4), (8,2)\}
  2. H \times G = \{(5,7), (5,8), (4,7), (4,8), (2,7), (2,8)\}
Q4. True or false? Correct the false ones. (i) If P = {m, n}, Q = {n, m}, then P × Q = {(m, n), (n, m)}. (ii) If A, B are non-empty, A × B is a non-empty set of ordered pairs (x, y) with x ∈ A, y ∈ B. (iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.सत्य या असत्य? असत्य कथनों को सही कीजिए। (i) यदि P = {m, n}, Q = {n, m}, तो P × Q = {(m, n), (n, m)}। (ii) यदि A, B अरिक्त हैं, तो A × B उन क्रमित युग्मों (x, y) का अरिक्त समुच्चय है जहाँ x ∈ A, y ∈ B। (iii) यदि A = {1, 2}, B = {3, 4}, तो A × (B ∩ φ) = φ।True or false? Correct the false ones. (i) If P = {m, n}, Q = {n, m}, then P × Q = {(m, n), (n, m)}. (ii) If A, B are non-empty, A × B is a non-empty set of ordered pairs (x, y) with x ∈ A, y ∈ B. (iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
  1. (i) False — P × Q has 2 × 2 = 4 pairs.(i) असत्य — P × Q में 2 × 2 = 4 युग्म हैं।(i) False — P × Q mein 2 × 2 = 4 pairs hain.
    P \times Q = \{(m, n), (m, m), (n, n), (n, m)\}
  2. (ii) True. (iii) True, since B ∩ φ = φ and A × φ = φ.(ii) सत्य। (iii) सत्य, क्योंकि B ∩ φ = φ और A × φ = φ।(ii) True. (iii) True, kyunki B ∩ φ = φ aur A × φ = φ.
Q5. If A = {−1, 1}, find A × A × A.यदि A = {−1, 1}, तो A × A × A ज्ञात कीजिए।If A = {−1, 1}, find A × A × A.
  1. \{(-1,-1,-1), (-1,-1,1), (-1,1,-1), (-1,1,1), (1,-1,-1), (1,-1,1), (1,1,-1), (1,1,1)\}

✔ 2^3 = 8 triplets.2^3 = 8 त्रिक।2^3 = 8 triplets.

Q6. If A × B = {(a, x), (a, y), (b, x), (b, y)}, find A and B.यदि A × B = {(a, x), (a, y), (b, x), (b, y)}, तो A और B ज्ञात कीजिए।If A × B = {(a, x), (a, y), (b, x), (b, y)}, find A and B.
  1. A = set of first elements, B = set of second elements.A = प्रथम अवयवों का समुच्चय, B = द्वितीय अवयवों का समुच्चय।A = pehle elements ka set, B = doosre elements ka set.

✔ A = {a, b}, B = {x, y}

Q7. A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}, D = {5, 6, 7, 8}. Verify (i) A × (B ∩ C) = (A × B) ∩ (A × C) (ii) A × C is a subset of B × D.A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}, D = {5, 6, 7, 8}। सत्यापित कीजिए (i) A × (B ∩ C) = (A × B) ∩ (A × C) (ii) A × C, B × D का उपसमुच्चय है।A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}, D = {5, 6, 7, 8}. Verify (i) A × (B ∩ C) = (A × B) ∩ (A × C) (ii) A × C is a subset of B × D.
  1. (i)\ B \cap C = \phi \Rightarrow A \times (B \cap C) = \phi
  2. A × B has second elements in {1, 2, 3, 4} and A × C in {5, 6}: no pair is common.A × B के द्वितीय अवयव {1, 2, 3, 4} में और A × C के {5, 6} में: कोई युग्म उभयनिष्ठ नहीं।A × B ke doosre elements {1, 2, 3, 4} mein aur A × C ke {5, 6} mein: koi pair common nahi.
    (A \times B) \cap (A \times C) = \phi \ \checkmark
  3. (ii)\ A \times C = \{(1,5), (1,6), (2,5), (2,6)\}
  4. Each of these pairs has first element in B (A ⊂ B) and second in D (C ⊂ D), so each is in B × D.इन सभी युग्मों का पहला अवयव B में (A ⊂ B) और दूसरा D में (C ⊂ D) है, इसलिए हर युग्म B × D में है।In sab pairs ka pehla element B mein (A ⊂ B) aur doosra D mein (C ⊂ D) hai, isliye har pair B × D mein hai.

✔ A \times C \subset B \times D ✓

Q8. A = {1, 2}, B = {3, 4}. Write A × B. How many subsets will A × B have? List them.A = {1, 2}, B = {3, 4}। A × B लिखिए। A × B के कितने उपसमुच्चय होंगे? उन्हें लिखिए।A = {1, 2}, B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
  1. A \times B = \{(1,3), (1,4), (2,3), (2,4)\},\quad 2^4 = 16 \text{ subsets}
  2. φ; the 4 singletons {(1,3)}, {(1,4)}, {(2,3)}, {(2,4)}; the 6 pairs {(1,3),(1,4)}, {(1,3),(2,3)}, {(1,3),(2,4)}, {(1,4),(2,3)}, {(1,4),(2,4)}, {(2,3),(2,4)}; the 4 triples (leave out one pair each time); and A × B itself.φ; 4 एकल {(1,3)}, {(1,4)}, {(2,3)}, {(2,4)}; 6 दो-युग्म वाले {(1,3),(1,4)}, {(1,3),(2,3)}, {(1,3),(2,4)}, {(1,4),(2,3)}, {(1,4),(2,4)}, {(2,3),(2,4)}; 4 तीन-युग्म वाले (हर बार एक युग्म छोड़कर); और स्वयं A × B।φ; 4 singletons {(1,3)}, {(1,4)}, {(2,3)}, {(2,4)}; 6 do-pair wale {(1,3),(1,4)}, {(1,3),(2,3)}, {(1,3),(2,4)}, {(1,4),(2,3)}, {(1,4),(2,4)}, {(2,3),(2,4)}; 4 teen-pair wale (har baar ek pair chhod kar); aur khud A × B.

✔ 1 + 4 + 6 + 4 + 1 = 16

Q9. n(A) = 3, n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B (x, y, z distinct).n(A) = 3, n(B) = 2। यदि (x, 1), (y, 2), (z, 1), A × B में हैं, तो A और B ज्ञात कीजिए (x, y, z भिन्न हैं)।n(A) = 3, n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B (x, y, z distinct).
  1. x, y, z ∈ A and A has 3 elements; 1, 2 ∈ B and B has 2 elements.x, y, z ∈ A और A में 3 अवयव; 1, 2 ∈ B और B में 2 अवयव।x, y, z ∈ A aur A mein 3 elements; 1, 2 ∈ B aur B mein 2 elements.

✔ A = {x, y, z}, B = {1, 2}

Q10. A × A has 9 elements, among them (−1, 0) and (0, 1). Find A and the remaining elements of A × A.A × A में 9 अवयव हैं, जिनमें (−1, 0) और (0, 1) हैं। A और A × A के शेष अवयव ज्ञात कीजिए।A × A has 9 elements, among them (−1, 0) and (0, 1). Find A and the remaining elements of A × A.
  1. n(A)² = 9 ⇒ n(A) = 3. From the pairs, −1, 0, 1 ∈ A.n(A)² = 9 ⇒ n(A) = 3। युग्मों से −1, 0, 1 ∈ A।n(A)² = 9 ⇒ n(A) = 3. Pairs se −1, 0, 1 ∈ A.
    A = \{-1, 0, 1\}

✔ Remaining: (−1, −1), (−1, 1), (0, −1), (0, 0), (1, −1), (1, 0), (1, 1)शेष: (−1, −1), (−1, 1), (0, −1), (0, 0), (1, −1), (1, 0), (1, 1)Baaki: (−1, −1), (−1, 1), (0, −1), (0, 0), (1, −1), (1, 0), (1, 1)

Exercise 2.2प्रश्नावली 2.2Exercise 2.2 · 9 Q

Q1. A = {1, 2, …, 14}, R = {(x, y) : 3x − y = 0; x, y ∈ A}. Write its domain, codomain and range.A = {1, 2, …, 14}, R = {(x, y) : 3x − y = 0; x, y ∈ A}। इसका प्रांत, सहप्रांत और परिसर लिखिए।A = {1, 2, …, 14}, R = {(x, y) : 3x − y = 0; x, y ∈ A}. Write its domain, codomain and range.
  1. y = 3x must stay ≤ 14, so x = 1, 2, 3, 4.y = 3x ≤ 14 होना चाहिए, अतः x = 1, 2, 3, 4।y = 3x ≤ 14 hona chahiye, isliye x = 1, 2, 3, 4.
    R = \{(1,3), (2,6), (3,9), (4,12)\}

✔ Domain {1, 2, 3, 4}; codomain A = {1, 2, …, 14}; range {3, 6, 9, 12}प्रांत {1, 2, 3, 4}; सहप्रांत A = {1, 2, …, 14}; परिसर {3, 6, 9, 12}Domain {1, 2, 3, 4}; codomain A = {1, 2, …, 14}; range {3, 6, 9, 12}

Q2. R on N: R = {(x, y) : y = x + 5, x a natural number less than 4}. Write R in roster form; find domain and range.N पर R = {(x, y) : y = x + 5, x, 4 से कम प्राकृत संख्या}। R को रोस्टर रूप में लिखिए; प्रांत और परिसर ज्ञात कीजिए।R on N: R = {(x, y) : y = x + 5, x a natural number less than 4}. Write R in roster form; find domain and range.
  1. R = \{(1,6), (2,7), (3,8)\}

✔ Domain {1, 2, 3}; range {6, 7, 8}प्रांत {1, 2, 3}; परिसर {6, 7, 8}Domain {1, 2, 3}; range {6, 7, 8}

Q3. A = {1, 2, 3, 5}, B = {4, 6, 9}, R = {(x, y) : the difference between x and y is odd}. Write R in roster form.A = {1, 2, 3, 5}, B = {4, 6, 9}, R = {(x, y) : x और y का अंतर विषम है}। R को रोस्टर रूप में लिखिए।A = {1, 2, 3, 5}, B = {4, 6, 9}, R = {(x, y) : the difference between x and y is odd}. Write R in roster form.
  1. Difference is odd when one is odd and the other even.अंतर विषम तभी होता है जब एक विषम और दूसरा सम हो।Difference odd tabhi hota hai jab ek odd aur doosra even ho.
    R = \{(1,4), (1,6), (2,9), (3,4), (3,6), (5,4), (5,6)\}
Q4. Fig 2.7 shows a relation between P = {5, 6, 7} and Q = {3, 4, 5} (5→3, 6→4, 7→5). Write it in (i) set-builder form (ii) roster form. What are its domain and range?चित्र 2.7 में P = {5, 6, 7} और Q = {3, 4, 5} के बीच संबंध है (5→3, 6→4, 7→5)। इसे (i) समुच्चय-निर्माण रूप (ii) रोस्टर रूप में लिखिए। प्रांत और परिसर क्या हैं?Fig 2.7 shows a relation between P = {5, 6, 7} and Q = {3, 4, 5} (5→3, 6→4, 7→5). Write it in (i) set-builder form (ii) roster form. What are its domain and range?
  1. (i)\ R = \{(x, y) : y = x - 2,\ x \in P\}\quad (ii)\ R = \{(5,3), (6,4), (7,5)\}

✔ Domain {5, 6, 7}; range {3, 4, 5}. See the arrow diagram.प्रांत {5, 6, 7}; परिसर {3, 4, 5}। तीर आरेख देखिए।Domain {5, 6, 7}; range {3, 4, 5}. Arrow diagram dekhiye.

Q5. A = {1, 2, 3, 4, 6}, R = {(a, b) : a, b ∈ A, b is exactly divisible by a}. (i) Roster form (ii) domain (iii) rangeA = {1, 2, 3, 4, 6}, R = {(a, b) : a, b ∈ A, b, a से पूर्णतः विभाज्य है}। (i) रोस्टर रूप (ii) प्रांत (iii) परिसरA = {1, 2, 3, 4, 6}, R = {(a, b) : a, b ∈ A, b is exactly divisible by a}. (i) Roster form (ii) domain (iii) range
  1. R = \{(1,1), (1,2), (1,3), (1,4), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (6,6)\}

✔ Domain = range = {1, 2, 3, 4, 6}प्रांत = परिसर = {1, 2, 3, 4, 6}Domain = range = {1, 2, 3, 4, 6}

Q6. Find the domain and range of R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}} का प्रांत और परिसर ज्ञात कीजिए।Find the domain and range of R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.

✔ Domain {0, 1, 2, 3, 4, 5}; range {5, 6, 7, 8, 9, 10}प्रांत {0, 1, 2, 3, 4, 5}; परिसर {5, 6, 7, 8, 9, 10}Domain {0, 1, 2, 3, 4, 5}; range {5, 6, 7, 8, 9, 10}

Q7. Write R = {(x, x³) : x is a prime number less than 10} in roster form.R = {(x, x³) : x, 10 से कम अभाज्य संख्या है} को रोस्टर रूप में लिखिए।Write R = {(x, x³) : x is a prime number less than 10} in roster form.
  1. R = \{(2, 8), (3, 27), (5, 125), (7, 343)\}
Q8. A = {x, y, z}, B = {1, 2}. Find the number of relations from A to B.A = {x, y, z}, B = {1, 2}। A से B में संबंधों की संख्या ज्ञात कीजिए।A = {x, y, z}, B = {1, 2}. Find the number of relations from A to B.
  1. n(A \times B) = 6 \Rightarrow 2^6 = 64
Q9. R on Z: R = {(a, b) : a, b ∈ Z, a − b is an integer}. Find its domain and range.Z पर R = {(a, b) : a, b ∈ Z, a − b एक पूर्णांक है}। प्रांत और परिसर ज्ञात कीजिए।R on Z: R = {(a, b) : a, b ∈ Z, a − b is an integer}. Find its domain and range.
  1. For any two integers a − b is an integer, so every pair of Z × Z is in R.किन्हीं दो पूर्णांकों का अंतर पूर्णांक होता है, अतः Z × Z का हर युग्म R में है।Kinhi do integers ka difference integer hota hai, isliye Z × Z ka har pair R mein hai.

✔ Domain = Z, range = Zप्रांत = Z, परिसर = ZDomain = Z, range = Z

Exercise 2.3प्रश्नावली 2.3Exercise 2.3 · 5 Q

Q1. Which are functions? If so, find domain and range. (i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)} (ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)} (iii) {(1,3), (1,5), (2,5)}कौन-से फलन हैं? यदि हाँ, तो प्रांत और परिसर बताइए। (i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)} (ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)} (iii) {(1,3), (1,5), (2,5)}Which are functions? If so, find domain and range. (i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)} (ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)} (iii) {(1,3), (1,5), (2,5)}
  1. (i) yes: no first element repeats. Domain {2, 5, 8, 11, 14, 17}, range {1}.(i) हाँ: कोई प्रथम अवयव दोहराया नहीं। प्रांत {2, 5, 8, 11, 14, 17}, परिसर {1}।(i) haan: koi pehla element repeat nahi. Domain {2, 5, 8, 11, 14, 17}, range {1}.
  2. (ii) yes. Domain {2, 4, 6, 8, 10, 12, 14}, range {1, 2, 3, 4, 5, 6, 7}.(ii) हाँ। प्रांत {2, 4, 6, 8, 10, 12, 14}, परिसर {1, 2, 3, 4, 5, 6, 7}।(ii) haan. Domain {2, 4, 6, 8, 10, 12, 14}, range {1, 2, 3, 4, 5, 6, 7}.

✔ (iii) Not a function: 1 has two images, 3 and 5.(iii) फलन नहीं: 1 के दो प्रतिबिंब 3 और 5 हैं।(iii) Function nahi: 1 ke do images 3 aur 5 hain.

Q2. Domain and range of (i) f(x) = -|x| (ii) f(x) = \sqrt{9 - x^2}प्रांत और परिसर ज्ञात कीजिए: (i) f(x) = -|x| (ii) f(x) = \sqrt{9 - x^2}Domain and range of (i) f(x) = -|x| (ii) f(x) = \sqrt{9 - x^2}
  1. (i) defined for all x; |x| \ge 0 so -|x| \le 0.(i) सभी x के लिए परिभाषित; |x| \ge 0 अतः -|x| \le 0।(i) saare x ke liye defined; |x| \ge 0 isliye -|x| \le 0.
    (i)\ \text{Domain } \mathbf{R},\ \text{range } (-\infty, 0](i)\ \text{प्रांत } \mathbf{R},\ \text{परिसर } (-\infty, 0](i)\ \text{Domain } \mathbf{R},\ \text{range } (-\infty, 0]
  2. (ii)\ 9 - x^2 \ge 0 \Rightarrow -3 \le x \le 3;\quad 0 \le 9 - x^2 \le 9 \Rightarrow 0 \le f(x) \le 3

✔ (ii) Domain [−3, 3], range [0, 3](ii) प्रांत [−3, 3], परिसर [0, 3](ii) Domain [−3, 3], range [0, 3]

Q3. f(x) = 2x − 5. Find (i) f(0) (ii) f(7) (iii) f(−3).f(x) = 2x − 5। (i) f(0) (ii) f(7) (iii) f(−3) ज्ञात कीजिए।f(x) = 2x − 5. Find (i) f(0) (ii) f(7) (iii) f(−3).
  1. f(0) = -5,\quad f(7) = 14 - 5 = 9,\quad f(-3) = -6 - 5 = -11
Q4. t(C) = \frac{9C}{5} + 32 converts °C to °F. Find (i) t(0) (ii) t(28) (iii) t(−10) (iv) C when t(C) = 212.t(C) = \frac{9C}{5} + 32, °C को °F में बदलता है। (i) t(0) (ii) t(28) (iii) t(−10) (iv) t(C) = 212 हो तो C ज्ञात कीजिए।t(C) = \frac{9C}{5} + 32 converts °C to °F. Find (i) t(0) (ii) t(28) (iii) t(−10) (iv) C when t(C) = 212.
  1. t(0) = 32,\quad t(28) = \frac{252}{5} + 32 = \frac{412}{5} = 82.4,\quad t(-10) = -18 + 32 = 14
  2. \frac{9C}{5} + 32 = 212 \Rightarrow \frac{9C}{5} = 180 \Rightarrow C = 100
Q5. Find the range: (i) f(x) = 2 - 3x, x ∈ R, x > 0 (ii) f(x) = x^2 + 2, x real (iii) f(x) = x, x realपरिसर ज्ञात कीजिए: (i) f(x) = 2 - 3x, x ∈ R, x > 0 (ii) f(x) = x^2 + 2, x वास्तविक (iii) f(x) = x, x वास्तविकFind the range: (i) f(x) = 2 - 3x, x ∈ R, x > 0 (ii) f(x) = x^2 + 2, x real (iii) f(x) = x, x real
  1. (i)\ x > 0 \Rightarrow -3x < 0 \Rightarrow 2 - 3x < 2 \Rightarrow (-\infty, 2)
  2. (ii)\ x^2 \ge 0 \Rightarrow x^2 + 2 \ge 2 \Rightarrow [2, \infty)\qquad (iii)\ \mathbf{R}

Miscellaneous Exercise on Chapter 2अध्याय 2 पर विविध प्रश्नावलीMiscellaneous Exercise on Chapter 2 · 12 Q

Q1. f(x) = x^2 for 0 \le x \le 3 and 3x for 3 \le x \le 10; g(x) = x^2 for 0 \le x \le 2 and 3x for 2 \le x \le 10. Show that f is a function and g is not.f(x) = x^2, 0 \le x \le 3 के लिए और 3x, 3 \le x \le 10 के लिए; g(x) = x^2, 0 \le x \le 2 के लिए और 3x, 2 \le x \le 10 के लिए। दिखाइए कि f फलन है और g नहीं।f(x) = x^2 for 0 \le x \le 3 and 3x for 3 \le x \le 10; g(x) = x^2 for 0 \le x \le 2 and 3x for 2 \le x \le 10. Show that f is a function and g is not.
  1. The only point covered by both rules is the meeting point. For f at x = 3 both rules agree:दोनों नियम केवल मिलन बिंदु पर एक साथ लागू होते हैं। f के लिए x = 3 पर दोनों नियम समान मान देते हैं:Dono rules sirf milan point par saath lagte hain. f ke liye x = 3 par dono rules same value dete hain:
    3^2 = 9 = 3 \times 3 \Rightarrow f(3) = 9 \text{ (one image)}3^2 = 9 = 3 \times 3 \Rightarrow f(3) = 9 \text{ (एक प्रतिबिंब)}3^2 = 9 = 3 \times 3 \Rightarrow f(3) = 9 \text{ (ek image)}
  2. For g at x = 2 the rules disagree, so 2 would have two images:g के लिए x = 2 पर नियम अलग मान देते हैं, अतः 2 के दो प्रतिबिंब होंगे:g ke liye x = 2 par rules alag value dete hain, isliye 2 ke do images honge:
    2^2 = 4 \ne 6 = 3 \times 2

✔ f is a function; g is not.f फलन है; g फलन नहीं।f function hai; g nahi.

Q2. If f(x) = x^2, find \frac{f(1.1) - f(1)}{1.1 - 1}.यदि f(x) = x^2, तो \frac{f(1.1) - f(1)}{1.1 - 1} ज्ञात कीजिए।If f(x) = x^2, find \frac{f(1.1) - f(1)}{1.1 - 1}.
  1. \frac{1.21 - 1}{0.1} = \frac{0.21}{0.1} = 2.1
Q3. Find the domain of f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}.f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12} का प्रांत ज्ञात कीजिए।Find the domain of f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}.
  1. x^2 - 8x + 12 = (x - 2)(x - 6) \ne 0 \Rightarrow x \ne 2, 6

✔ Domain = \mathbf{R} - \{2, 6\}प्रांत = \mathbf{R} - \{2, 6\}Domain = \mathbf{R} - \{2, 6\}

Q4. Find the domain and range of f(x) = \sqrt{x - 1}.f(x) = \sqrt{x - 1} का प्रांत और परिसर ज्ञात कीजिए।Find the domain and range of f(x) = \sqrt{x - 1}.
  1. x - 1 \ge 0 \Rightarrow x \ge 1;\quad \sqrt{x - 1} \ge 0

✔ Domain [1, \infty), range [0, \infty)प्रांत [1, \infty), परिसर [0, \infty)Domain [1, \infty), range [0, \infty)

Q5. Find the domain and range of f(x) = |x - 1|.f(x) = |x - 1| का प्रांत और परिसर ज्ञात कीजिए।Find the domain and range of f(x) = |x - 1|.

✔ Domain R; range [0, \infty) (it takes the value 0 at x = 1 and every positive value).प्रांत R; परिसर [0, \infty) (x = 1 पर 0 और हर धन मान लेता है)।Domain R; range [0, \infty) (x = 1 par 0 aur har positive value leta hai).

Q6. f = \left\{\left(x, \frac{x^2}{1 + x^2}\right) : x \in \mathbf{R}\right\}. Find the range of f.f = \left\{\left(x, \frac{x^2}{1 + x^2}\right) : x \in \mathbf{R}\right\}। f का परिसर ज्ञात कीजिए।f = \left\{\left(x, \frac{x^2}{1 + x^2}\right) : x \in \mathbf{R}\right\}. Find the range of f.
  1. y = \frac{x^2}{1 + x^2} = 1 - \frac{1}{1 + x^2}
  2. 1 + x^2 \ge 1, so 0 < \frac{1}{1 + x^2} \le 1; y = 0 at x = 0 and y never reaches 1.1 + x^2 \ge 1, अतः 0 < \frac{1}{1 + x^2} \le 1; x = 0 पर y = 0 और y कभी 1 नहीं होता।1 + x^2 \ge 1, isliye 0 < \frac{1}{1 + x^2} \le 1; x = 0 par y = 0 aur y kabhi 1 nahi hota.

✔ Range = [0, 1)परिसर = [0, 1)Range = [0, 1)

Q7. f(x) = x + 1, g(x) = 2x - 3. Find f + g, f - g and \frac{f}{g}.f(x) = x + 1, g(x) = 2x - 3। f + g, f - g और \frac{f}{g} ज्ञात कीजिए।f(x) = x + 1, g(x) = 2x - 3. Find f + g, f - g and \frac{f}{g}.
  1. (f + g)(x) = 3x - 2,\quad (f - g)(x) = -x + 4
  2. \left(\frac{f}{g}\right)(x) = \frac{x + 1}{2x - 3},\quad x \ne \frac{3}{2}
Q8. f = {(1, 1), (2, 3), (0, −1), (−1, −3)} with f(x) = ax + b. Determine a, b.f = {(1, 1), (2, 3), (0, −1), (−1, −3)}, f(x) = ax + b। a, b ज्ञात कीजिए।f = {(1, 1), (2, 3), (0, −1), (−1, −3)} with f(x) = ax + b. Determine a, b.
  1. f(0) = b = -1;\quad f(1) = a + b = 1 \Rightarrow a = 2

✔ a = 2,\ b = -1 (check: f(2) = 3, f(-1) = -3 ✓)a = 2,\ b = -1 (जाँच: f(2) = 3, f(-1) = -3 ✓)a = 2,\ b = -1 (check: f(2) = 3, f(-1) = -3 ✓)

Q9. R from N to N: R = {(a, b) : a = b²}. True? (i) (a, a) ∈ R for all a ∈ N (ii) (a, b) ∈ R ⇒ (b, a) ∈ R (iii) (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R. Justify.N से N में R = {(a, b) : a = b²}। क्या सत्य है? (i) सभी a ∈ N के लिए (a, a) ∈ R (ii) (a, b) ∈ R ⇒ (b, a) ∈ R (iii) (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R। कारण दीजिए।R from N to N: R = {(a, b) : a = b²}. True? (i) (a, a) ∈ R for all a ∈ N (ii) (a, b) ∈ R ⇒ (b, a) ∈ R (iii) (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R. Justify.
  1. (i) No: (2, 2) ∉ R since 2 ≠ 2². (ii) No: (4, 2) ∈ R but (2, 4) ∉ R since 2 ≠ 16.(i) नहीं: (2, 2) ∉ R क्योंकि 2 ≠ 2²। (ii) नहीं: (4, 2) ∈ R पर (2, 4) ∉ R क्योंकि 2 ≠ 16।(i) Nahi: (2, 2) ∉ R kyunki 2 ≠ 2². (ii) Nahi: (4, 2) ∈ R par (2, 4) ∉ R kyunki 2 ≠ 16.
  2. (iii) No: (16, 4) ∈ R and (4, 2) ∈ R but (16, 2) ∉ R since 16 ≠ 4.(iii) नहीं: (16, 4) ∈ R और (4, 2) ∈ R पर (16, 2) ∉ R क्योंकि 16 ≠ 4।(iii) Nahi: (16, 4) ∈ R aur (4, 2) ∈ R par (16, 2) ∉ R kyunki 16 ≠ 4.

✔ All three are false.तीनों असत्य।Teeno false.

Q10. A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16}, f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Is (i) f a relation from A to B (ii) f a function from A to B? Justify.A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16}, f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}। क्या (i) f, A से B में संबंध है (ii) f, A से B में फलन है? कारण दीजिए।A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16}, f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Is (i) f a relation from A to B (ii) f a function from A to B? Justify.

✔ (i) Yes — every pair is in A × B. (ii) No — 2 has two images, 9 and 11.(i) हाँ — हर युग्म A × B में है। (ii) नहीं — 2 के दो प्रतिबिंब 9 और 11 हैं।(i) Haan — har pair A × B mein hai. (ii) Nahi — 2 ke do images 9 aur 11 hain.

Q11. f ⊂ Z × Z, f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z?f ⊂ Z × Z, f = {(ab, a + b) : a, b ∈ Z}। क्या f, Z से Z में फलन है?f ⊂ Z × Z, f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z?
  1. a = 0, b = 1 gives (0, 1); a = 0, b = 2 gives (0, 2).a = 0, b = 1 से (0, 1); a = 0, b = 2 से (0, 2)।a = 0, b = 1 se (0, 1); a = 0, b = 2 se (0, 2).

✔ No — 0 has two different images.नहीं — 0 के दो अलग प्रतिबिंब हैं।Nahi — 0 ke do alag images hain.

Q12. A = {9, 10, 11, 12, 13}, f : A → N, f(n) = the highest prime factor of n. Find the range of f.A = {9, 10, 11, 12, 13}, f : A → N, f(n) = n का सबसे बड़ा अभाज्य गुणनखंड। f का परिसर ज्ञात कीजिए।A = {9, 10, 11, 12, 13}, f : A → N, f(n) = the highest prime factor of n. Find the range of f.
  1. f(9) = 3,\ f(10) = 5,\ f(11) = 11,\ f(12) = 3,\ f(13) = 13

✔ Range = {3, 5, 11, 13}परिसर = {3, 5, 11, 13}Range = {3, 5, 11, 13}

🎯 Quiz — 12 questions🎯 क्विज़ — 12 प्रश्न🎯 Quiz — 12 questions

Tap an option. The answer and a short reason appear at once.किसी विकल्प पर टैप करें। उत्तर और छोटा कारण तुरंत दिखेगा।Kisi option par tap kijiye. Answer aur chhota reason turant dikhega.

1. If n(A) = 3 and n(B) = 4, then n(A × B) isयदि n(A) = 3 और n(B) = 4, तो n(A × B) हैAgar n(A) = 3 aur n(B) = 4, to n(A × B) hai

2. If (x − 1, y + 3) = (4, 7), then (x, y) isयदि (x − 1, y + 3) = (4, 7), तो (x, y) हैAgar (x − 1, y + 3) = (4, 7), to (x, y) hai

3. How many relations are there from A = {a, b} to B = {1, 2}?A = {a, b} से B = {1, 2} में कितने संबंध हैं?A = {a, b} se B = {1, 2} mein kitne relations hain?

4. Which relation is a function?कौन-सा संबंध फलन है?Kaun-sa relation function hai?

5. The domain of f(x) = 1/(x − 3) isf(x) = 1/(x − 3) का प्रांत हैf(x) = 1/(x − 3) ka domain hai

6. The range of f(x) = |x| isf(x) = |x| का परिसर हैf(x) = |x| ka range hai

7. [−3.2] (greatest integer) equals[−3.2] (महत्तम पूर्णांक) बराबर है[−3.2] (greatest integer) barabar hai

8. The range of the signum function isचिह्न फलन का परिसर हैSignum function ka range hai

9. The domain of f(x) = √(4 − x²) isf(x) = √(4 − x²) का प्रांत हैf(x) = √(4 − x²) ka domain hai

10. If f(x) = x² and g(x) = x + 1, then (fg)(2) isयदि f(x) = x² और g(x) = x + 1, तो (fg)(2) हैAgar f(x) = x² aur g(x) = x + 1, to (fg)(2) hai

11. For R = {(1, 2), (2, 4), (3, 6)} from A = {1, 2, 3} to B = {1, 2, …, 6}, the range isA = {1, 2, 3} से B = {1, 2, …, 6} में R = {(1, 2), (2, 4), (3, 6)} का परिसर हैA = {1, 2, 3} se B = {1, 2, …, 6} mein R = {(1, 2), (2, 4), (3, 6)} ka range hai

12. A × B = B × A holds whenA × B = B × A कब सत्य हैA × B = B × A kab sach hai

← Ch 1Ch 3 →

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These are Simple Learning Hub's own study notes and solutions, written to follow the NCERT Class 11 Mathematics textbook (Reprint 2026-27) section by section; read them with the official NCERT chapter. Diagrams are original SLH drawings.ये Simple Learning Hub के अपने स्टडी नोट्स हैं, जो NCERT कक्षा 11 गणित की किताब (2026-27) के हर अनुभाग के साथ लिखे गए हैं; इन्हें NCERT के आधिकारिक अध्याय के साथ पढ़िए। डायग्राम SLH के अपने बनाए चित्र हैं।Ye Simple Learning Hub ke apne study notes hain, jo NCERT Class 11 Maths ki kitaab (2026-27) ke har section ke saath likhe gaye hain; inhe NCERT ke official chapter ke saath padhiye. Diagrams SLH ki apni drawings hain.