Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line ๐‘ฆ = ๐‘ฅ − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Students of Grade 9 Planned to Plant - Teachoo.jpg


Answer the following using the above information.

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Question 1

Let relation R be defined by R = {(L 1 , L 2 ) : L 1 โˆฅ L 2 where L 1 , L 2 ∈ L}
then R is______ relation
(a) Equivalence
(b) Only reflexive
(c) Not reflexive
(d) Symmetric but not transitive

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Question 2

Let R = {(L 1 , L 2 ) โˆถ L 1 ⊥ L 2 where L 1 , L 2 ∈ L} which of the following is true?
(a) R is Symmetric but neither reflexive nor transitive
(b) R is Reflexive and transitive but not symmetric
(c) R is Reflexive but neither symmetric nor transitive
(d) R is an Equivalence relation

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Question 3

The function f: R R defined by ๐‘“(๐‘ฅ) = ๐‘ฅ − 4 is___________
(a) Bijective
(b) Surjective but not injective
(c) Injective but not Surjective
(d) Neither Surjective nor Injective

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For Proof, Check Example 2 – Chapter 1 Class 12

Question 4

Let ๐‘“: ๐‘… → ๐‘… be defined by ๐‘“(๐‘ฅ) = ๐‘ฅ − 4. Then the range of ๐‘“(๐‘ฅ) is ________
(a) R
(b) Z
(c) W
(d) Q

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Question 5

Let R = {(L 1 , L 2 ) : L 1 is parallel to L 2 and L 1 : y = x – 4} then which of the following can be taken as L 2 ?
(a) 2x – 2y + 5 = 0
(b) 2x + y = 5
(c) 2x + 2y + 7 = 0
(d) x + y = 7

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  1. Chapter 1 Class 12 Relation and Functions (Term 1)
  2. Serial order wise

Transcript

Question Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line ๐‘ฆ = ๐‘ฅ โˆ’ 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L. Answer the following using the above information. Question 1 Let relation R be defined by R = {(L1, L2) : L1 โˆฅ L2 where L1, L2 โˆˆ L} then R is______ relation (a) Equivalence (b) Only reflexive (c) Not reflexive (d) Symmetric but not transitive R = {(L1, L2) : L1 โˆฅ L2 where L1, L2 โˆˆ L} Check Reflexive Since L1 and L1 are always parallel to each other So, (L1, L1) โˆˆ R for all L1 โˆด R is reflexive Check symmetric If L1 and L2 are parallel to each other Then, L2 and L1 are also parallel to each other Thus (L1, L2) โˆˆ R , and (L2, L1) โˆˆ R โˆด R is symmetric Check transitive To check whether transitive or not, If (x, y) โˆˆ R & (y, z) โˆˆ R , then (x, z) โˆˆ R If L1 and L2 are parallel to each other, And L2 and L3 are parallel to each other Then, L1 and L3 will also be parallel to each other Thus, for all values of L1 , L2 , L3 (L1, L2) โˆˆ R & (L2, L3) โˆˆ R , then (L1, L3) โˆˆ R โˆด R is transitive Since R is reflexive, symmetric and transitive โˆด R is an Equivalence relation So, the correct answer is (a) Question 2 Let R = {(L1, L2) โˆถ L1 โŠฅ L2 where L1, L2 โˆˆ L} which of the following is true? (a) R is Symmetric but neither reflexive nor transitive (b) R is Reflexive and transitive but not symmetric (c) R is Reflexive but neither symmetric nor transitive (d) R is an Equivalence relation R = {(L1, L2) : L1 โŠฅ L2 where L1, L2 โˆˆ L} Check Reflexive Since a line can never be perpendicular to itself โˆด (L1, L1) โˆ‰ R for all L1 โˆด R is not reflexive Check symmetric If L1 and L2 are perpendicular to each other Then, L2 and L1 are also perpendicular to each other Thus (L1, L2) โˆˆ R , and (L2, L1) โˆˆ R โˆด R is symmetric Check transitive To check whether transitive or not, If (x, y) โˆˆ R & (y, z) โˆˆ R , then (x, z) โˆˆ R If L1 and L2 are perpendicular to each other, And L2 and L3 are also perpendicular to each other Then, L1 and L3 are not perpendicular to each other โˆด R is not transitive Thus, R is Symmetric but neither reflexive nor transitive So, the correct answer is (b) Question 3 The function f: R โ†’ R defined by ๐‘“(๐‘ฅ) = ๐‘ฅ โˆ’ 4 is___________ (a) Bijective (b) Surjective but not injective (c) Injective but not Surjective (d) Neither Surjective nor Injective A linear function, defined from R to R is always one-one and onto โˆด ๐‘“(๐‘ฅ) is Bijective So, the correct answer is (a) Question 4 Let ๐‘“: ๐‘… โ†’ ๐‘… be defined by ๐‘“(๐‘ฅ) = ๐‘ฅ โˆ’ 4. Then the range of ๐‘“(๐‘ฅ) is ________ (a) R (b) Z (c) W (d) Q For ๐‘“(๐‘ฅ) = ๐‘ฅ โˆ’ 4 For all real values of x, we can get a real number ๐‘“(๐‘ฅ) โˆด Range of ๐‘“(๐‘ฅ) is R So, the correct answer is (a) Question 5 Let R = {(L1, L2 ) : L1 is parallel to L2 and L1 : y = x โ€“ 4} then which of the following can be taken as L2 ? (a) 2x โ€“ 2y + 5 = 0 (b) 2x + y = 5 (c) 2x + 2y + 7 = 0 (d) x + y = 7 Since L2 must be parallel to L1, their slope must be equal Slope of L1: y = x โ€“ 4 is = 1 Checking Slope of given options Slope of Part (a): 2x โ€“ 2y + 5 = 0 Slope = 1 Slope of Part (b): 2x + y = 5 Slope = โˆ’2 Slope of Part (c): 2x + 2y + 7 = 0 Slope = โˆ’1 Slope of Part (d): x + y = 7 Slope = โˆ’1 Since slope of option (a) is same as slope of L1 So, the correct answer is (a)

About the Author

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.