Question Set C

cbse 12th Standard Maths Relations and Functions Model Questions Papers Study Material - QB365

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Questions:

Multiple Choice Question

54 x 1 = 54

Let A = {1,2,3}, B = {4,5,6, 7) and let f = {(1, 4), (2, 5), (3,6)} be a function from A to B. Based on the given information f is best defined as

(a)

surjective function

(b)

injective function

(c)

bijective function

(d)

None of the above

The function f : R→ R defined as f(x) = x³ is

(a)

one-one but not onto

(b)

not one-one but onto

(c)

neither one-one nor onto

(d)

both one-one and onto

Let the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b): ab is a multiple of 4}. Then [1], the equivalence class containing 1, is

(a)

{1, 5, 9}

(b)

{0,1,2,5}

(c)

(d)

A relation R in set A = {1,2,3} is defined as R = {(1,1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?

(a)

(1, 1)

(b)

(1, 2)

(c)

(2, 2)

(d)

(3, 3)

Let A (3,5). Then, number of reflexive relations on A is

(a)

(b)

(c)

(d)

Select the correct option out of the four given options
Let R be a relation in the set N given by
R = {(a, b):ab - 2,b > 6}
Then,

(a)

(8, 7) ∈ R

(b)

(6,8) ∈ R

(c)

(3,8) ∈ R

(d)

(2,4) ∈ R

Let the function 'f' be defined by \(f(x)=5 x^{2}+2, \forall x \in R\) Then f' is

(a)

onto function

(b)

one-one, onto function

(c)

one-one, into function

(d)

many-one, into function

If f(x) = x³ and g(x) = cos 3x , then fog is

(a)

x³.cos 3x

(b)

cos 3x³

(c)

cos³ 3x

(d)

3cos x³

A relation defined in a non-empty set A, having n elements, has

(a)

n relations

(b)

2 relations

(c)

n² relations

(d)

2n² relations

10.

Let the function 'f' : N \(\rightarrow\) N be defined by \(f(x)= {2} x+3, \forall x \in N \text { . Then } f^{\prime \prime} \text { is }\)

(a)

not onto

(b)

bijective function

(c)

many-one, into function

(d)

none of these

11.

Let A = {a, b}. Then number of one-one functions from A to A possible are

(a)

(b)

(c)

(d)

12.

\(Let f(x)=\left\{\begin{array}{ll}-1, & x<0 \\ 0, & x=0 \text { and } g(x)=1+x-[x]\end{array}\right.\)
where [x] denotes the greatest integer less than or equal to x. Then, for all x, f(g(x)) is equal to

(a)

(b)

(c)

g(x)

(d)

f(x)

13.

If f: R \(\rightarrow\) R and g: R \(\rightarrow\) R are given by \(f(x)=\cos x \) and \( g(x)=3 x^{2}\), then

(a)

\( \operatorname{gof}(x)=\cos ^{2} x\)

(b)

\(\operatorname{fog}(x)=\cos x^{2}\)

(c)

\(gof \neq fog\)

(d)

\( f o g=g o f\)

14.

The greatest integer function : R ➝R, given by [(x) = [x] is

(a)

one-one

(b)

onto

(c)

both one-one and onto

(d)

neither one-one nor onto

15.

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

(a)

720

(b)

120

(c)

(d)

None of these

16.

Let A = {1, 2, 3, ... , n} and B = {a, b}. Then the number of surjections from A into B is

(a)

ⁿP₂

(b)

2ⁿ-2

(c)

2ⁿ-1

(d)

None of these

17.

The number of all one-one functions from set A = {1, 2, 3} to itself is

(a)

(b)

(c)

(d)

18.

f : X⟶Y is onto, if and only if

(a)

range of f = Y

(b)

range of f ≠ Y

(c)

range of f < Y

(d)

range of f ≥Y

19.

If A = {x ∈ Z : 0 ≤ x ≤ 12}and R is the relation in A given by R = {(a, b): a = b}. Then, the set of all elements related to 1 is

(a)

{1, 2}

(b)

{2, 3}

(c)

{1}

(d)

{2}

20.

For the set A = {1, 2, 3}, define a relation R in the set A as follows
R = {(1, 1), (2,2), (3, 3), (1, 3)}
Then, the ordered pair to be added to R to make it the smallest equivalence relation is

(a)

(1, 3)

(b)

(3, 1)

(c)

(2, 1)

(d)

(1, 2)

21.

The relation R in the set of natural numbers N defined as R = {(x, y) : y = x + 5 and x < 4} is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

None of these

22.

If a relation R on the set {1,2, 3} be defined by R = {(1, 2)}, then R is

(a)

reflexive

(b)

transitive

(c)

symmetric

(d)

None of these

23.

A relation f from C to R is defined by \(x f y \Leftrightarrow|x|=y\).Then, the correct option is

(a)

(2+i)f3

(b)

3f(-3)

(c)

i f 1

(d)

(2 +3i) f 13

24.

Let R be an equivalence relation on Z, the set of integers. R = { (a,b): a,b ∈ Z and a – b is a multiple of 3 } The Equivalence class of [1] is

(a)

{..-7,-4,2,5,8,.}

(b)

{.-4,-1,2,5,8,.}

(c)

{.-4,-1,2,5,8,.}

(d)

{…..-5,-2,1,4,7,.}

25.

Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. Then, R is

(a)

Symmetric only

(b)

An equivalence relation

(c)

Reflexive and symmetric only

(d)

Reflexive and transitive only

26.

Let R be a relation on N (set of natural numbers) such that (m, n) R (p, q)mq(n + p) = np(m + q). Then, R is

(a)

An Equivalence Relation

(b)

Only Reflexive

(c)

Symmetric and reflexive

(d)

Only Transitive

27.

Let R be a relation on a finite set A having n elements. Then, the number of relations on A is

(a)

n x n

(b)

2ⁿ

(c)

n²

(d)

2^nxn

28.

If R be a relation “less than” from set A = {1, 2, 3, 4} to B = {1, 3, 5}, i.e. (a, b) ∈ R if a < b, if (b,a) ∈ R^-1elements in R^-1 are

(a)

{(3, 3), (3, 5), (5, 3), (5, 5)}

(b)

{(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}

(c)

{(3, 3), (3, 4), (4, 5)}

(d)

{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

29.

Let R be a relation on N, set of natural numbers such that m R n ⇔ m divides n. Then R is

(a)

Reflexive and symmetric

(b)

Neither reflexive nor transitive

(c)

Reflexive and transitive

(d)

Symmetric and transitive

30.

Let A = {1,2,3,4} and B = {x,y,z}. Then R = {(1,x) , ( 2,z), (1,y), (3,x)} is

(a)

relation from B to A

(b)

Is not a relation

(c)

relation from A to B

(d)

relation from B to B

31.

Let C = {(a, b): a² + b² = 1; a, b ∈ R} a relation on R, set of real numbers. Then C is

(a)

Equivalence relation

(b)

Reflexive

(c)

Transitive

(d)

Symmetric

32.

Let R be a relation on set A of triangles in a plane. R = { (T₁ , T₂) : T₁, T₂element of A and T₁ is congruent to T₂} Then the relation R is ______

(a)

Equivalence relation

(b)

Transitive

(c)

Symmetric

(d)

Reflexive

33.

Let A = {1,2,3,4,5,6,7}. P={1,2}, Q = {3, 7}. Write the elements of the set R so that P, Q and R form a partition that results in equivalence relation

(a)

{4,5,6}

(b)

{0}

(c)

{1,2,3,4,5,6,7}

(d)

{ }

34.

Let R be the relation on the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3,3), (3,2)}. then R is

(a)

R is reflexive and symmetric but not transitive.

(b)

R is symmetric and transitive but not reflexive.

(c)

R is an equivalence relation.

(d)

R is reflexive and transitive but not symmetric

35.

If A = {1, 2, 3, 4} and B = {1, 3, 5} and R is a relation from A to B defined by (a, b) ∈ element of R ⇔ a < b. Then, R = ?

(a)

{(2, 3), (4, 5), (1, 3), (2, 5)}

(b)

{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

(c)

{(2, 3), (4, 5), (1, 3), (2, 5), (5, 3)}

(d)

{(5, 3), (3, 5), (5, 4), (4, 5)}

36.

In the set N x N the relation R is defined by (a, b) R (c, d) ⇔ ad = bc. Then R is

(a)

symmetric and transitive but not reflexive

(b)

reflexive and transitive but not symmetric

(c)

Equivalence relation

(d)

Partial order relation

37.

If A = {1,3,5,7} and define a relation, such that R = { (a,b) a,b ∈ A : |a+b| = 8}. Then how many elements are there in the relation R

(a)

(b)

(c)

(d)

38.

If A = {1,3,5,7} and we define a relation R = {(a,b), a,b ∈ A:|a - b| = 8} Then the number of elements in the relation R is

(a)

(b)

(c)

(d)

39.

Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then, R is

(a)

Symmetric

(b)

Transitive

(c)

Reflexive

(d)

Equivalence relation

40.

For real number x and y, we write xRy ⇔ x - y + \(\sqrt2\) irrational number. Then the relation R is

(a)

Reflexive

(b)

Symmetric

(c)

Transitive

(d)

Equivalence

41.

Let a relation T on the set R of real numbers be T = { (a,b) : 1 + ab < 0, a,∈R}. Then from among the ordered pairs (1,1) (1,2)(1,-2)(2,2), the only pair that belongs to T is________.

(a)

(2,2)

(b)

(1,1)

(c)

(1,-2)

(d)

(1,2)

42.

Let R = { (P,Q) : OP = OQ , O being the origin} be an equivalence relation on A . The equivalence class [( 1,2)] is

(a)

{(x, y): x² + y²= 5}

(b)

{(x, y): x² = y²}

(c)

{(x, y): x² + y²= 1}

(d)

{(x, y): x² + y²= 4}

43.

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is

(a)

(b)

(c)

(d)

44.

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

(a)

(b)

(c)

(d)

45.

Let f : R ➝ R be defined as f (x) = 3x. Choose the correct answer

(a)

f is one-one onto

(b)

f is many-one onto

(c)

f is one-one but not onto

(d)

f is neither one-one nor onto

46.

Let f : R ⟶ R be defined as f(x) = x⁴. Choose the correct answer

(a)

f is one-one onto

(b)

f is many-one onto

(c)

f is one-one but not onto

(d)

f is neither one-one nor onto

47.

Let R be the relation in the set N given by R = {(a, b): a = b − 2, b > 6}. Choose the correct answer.

(a)

(2, 4)∈ R

(b)

(3, 8) ∈ R

(c)

(6, 8)∈ R

(d)

(8, 7) ∈ R

48.

Given a function lf as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then

(a)

g(x) = 4x + 5

(b)

g(x) = \(\frac{5}{4x-5}\)

(c)

g(x) = \(\frac{x-4}{5}\)

(d)

g(x) = 5x – 4

49.

Set A has 3 elements and the set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is

(a)

144

(b)

(c)

(d)

50.

A relation S in the set of real numbers is defined as xSy ⇒ x – y+ \(\sqrt3\) is an irrational number, then relation S is

(a)

reflexive

(b)

reflexive and symmetric

(c)

transitive

(d)

symmetric and transitive

51.

Given set A = {a, b, c). An identity relation in set A is

(a)

R = {(a, b), (a, c)}

(b)

R = {(a, a), (b, b), (c, c)}

(c)

R = {(a, a), (b, b), (c, c), (a, c)}

(d)

R= {(c, a), (b, a), (a, a)}

52.

Given set A ={1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be

(a)

reflexive if (1, 1) is added

(b)

symmetric if (2, 3) is added

(c)

transitive if (1, 1) is added

(d)

symmetric if (3, 2) is added

53.

Given triangles with sides T₁ : 3, 4, 5; T₂ : 5, 12, 13; T₃ : 6, 8, 10; T₄ : 4, 7, 9 and a relation R in set of triangles defined as R = {(Δ₁, Δ₂) : Δ₁ is similar to Δ₂}. Which triangles belong to the same equivalence class?

(a)

T₁ and T₂

(b)

T₂ and T₃

(c)

T₁ and T₃

(d)

T₁ and T₄

54.

Let R be a relation on the set L of lines defined by l₁ R l₂ if l₁ is perpendicular to l₂, then relation R is

(a)

reflexive and symmetric

(b)

symmetric and transitive

(c)

equivalence relation

(d)

symmetric

Case Study Questions

4 x 4 = 16

55.

A general election of Lok Sabha is a gigantic exercise About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever.

Let l be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation R is defined on I as follows
R = {(V₁,V₂) : V₁,V₂ ∈ l and both use their voting right in general election-2019).
Answer the following questions using the above information.
(i) Two neighbours X and Y ∈ 1. X exercised his voting right while y did not cast her vote in general election-2019. Which of the following is true?
(a) (X, Y) ∈ R
(b) (Y, X) ∈ R
(c) (X, X) ∈ R
(d) (X, Y) ∈ R
(ii) Mr. X and his wife W both exercised their voting right in general election 2019. Which of the following is true?
(a) both (X, W) and (W,X) ∈ R
(b) (X, W) ∈ R but (W,X) ∉ R
(c) both (X, W) and (W,X) ∉ R
(d) (W,X) ∈ R but (X, W) ∉ R
(iii) Three friends F₁, F₂ and F₃ exercised their voting right in general election-2019, then which of the following is true?
(a) \(\left(F_1, F_2\right) \in R,\left(F_2, F_3\right) \in R\) and \(\left(F_1, F_3\right) \in R\)
(b) \(\left(F_1, F_2\right) \in R,\left(F_2, F_3\right) \in R\) and \(\left(F_1, F_3\right) \notin R\)
(c) \(\left(F_1, F_2\right) \in R,\left(F_2, F_2\right) \in R\) but \(\left(F_3, F_3\right) \notin R\)
(d) \(\left(F_1, F_2\right) \notin R,\left(F_2, F_3\right) \notin R\) and \(\left(F_1, F_3\right) \notin R\)
(iv) The above defined relation R is
(a) Symmetric and transitive but not reflexive z
(b) Universal relation
(c) Equivalence relation
(d) Reflexive but not symmetric and transitive
(v) Mr. Shyam exercised his voting right in General Election-2019, then Mr. Shyam is related to which of the following?
(a) All those eligible voters who cast their votes
(b) Family members of Mr.Shyam
(c) All citizens of India
(d) Eligible voters of India

56.

Students of a school are taken to a railway museum to learn about railways heritage and its history.

An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by
R = {(l₁, l₂) : l₁ is parallel to l₂}
On the basis of the above information, answer the following questions.
(i) Find whether the relation R is symmetric or not.
(ii) Find whether the relation R is transitive or not.
(iii) If one of the rail lines on the railway track is represented by the equation y = 3x + 2, then find the set of rail lines in R related to it.
Or
Let S be the relation defined by S= ((l₁,l₂) : l₁ is perpendicular to l₂) check whether the relation S is symmetric and transitive.

57.

Consider the mapping \(f: A \rightarrow B\) is defined by \(f(x)=\frac{x-1}{x-2}\) such that f is a bijection.
Based on the above information, answer the following questions.
(i) Domain of f is

(a) R - {2}

(b) R

(C) R-{1,2}

(d) R-{0}

(ii) Range of f is

(a) R

(b) R -{1}

(C) R-{0}

(d) R-{1,2}

(iii) If g: \(R-\{2\} \rightarrow R-\{1\}\) is defined by g(x) = 2f(x) - I, then g(x) in terms of x is

(a) \(\frac{x+2}{x}\)

(b) \(\frac{x+1}{x-2}\)

(d) \(\frac{x}{x-2}\)

(iv) The function g defined above, is

(a)

One-one

(b) Many-one

(c) into

(d) None of these

(v) A function J(x) is said to be one-one iff

(a) \(f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow-x_{1}=x_{2}\)

(b) \(f\left(-x_{1}\right)=f\left(-x_{2}\right) \Rightarrow-x_{1}=x_{2}\)

(d) None of these

58.

A relation R on a set A is said to be an equivalence relation on A iff it is
(a) Reflexive i.e.., \((a, a) \in R \ \forall \ a \in A\)
(b) Symmetric i.e., \((a, b) \in R \Rightarrow(b, a) \in R \ \forall \ a, b \in A\)
(c) Transitive i.e., \((a, b) \in R\) and \((b, c) \in R \Rightarrow(a, c) \in R\ \forall\ a, b, c \in A\)
Based on the above information, answer the following questions.
(i) If the relation R = {(1, 1), (1, 2), (1, 3), (2,2), (2, 3), (3,1), (3, 2), (3, 3)} defined on the set A = {1, 2, 3}, then R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

equivalence

(ii) If the relation R = {(1, 2), (2,1), (1, 3), (3, I)} defined on the setA = {1, 2, 3}, then R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

equivalence

(iii) If the relation R on the set N of all natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}, then R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

equivalence

(iv) If the relation R on the set A = {1, 2, 3, , 13, 14}defined as R = {(x, y) : 3x - y = 0}, then R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

equivalence

Assertion and reason

6 x 1 = 6

59.

Assertion: Let A = {-1, 1, 2, 3} and B = {1, 4, 9}, where f: A \(\rightarrow\)B given by f(x) = x², then f is a many-one function.
Reason: If x₁ \(\neq\)x₂ \(\Rightarrow\) f(x₁) \(\neq\)f(x₂), for every x₁,x₂ \(\in\)domain, then f is one-one or else many-one.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

60.

Assertion: If f(x) is odd function and g(x) is even function, then f(x) + g(x) is neither even nor odd.
Reason: \(f(x)=\left\{\begin{matrix} f(x),& f(x) is even\\ -f(x),& f(x) is odd\\ \end{matrix}\right.\)
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

61.

Assertion: Let L be the set of all lines in a plane and R be the relations in L defined as R = {(L₁, L₂): L₁ is perpendicular to L₂}.This relation is not equivalence relation.
Reason: A relation is said to be equivalence relation if it is reflexive,symmetric and transitive.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

62.

Assertion: If f is even function, g is odd function, then \(\frac{f}{g}\), (g \(\neq\)0) is an odd function.
Reason: If f(-x)=-f(x) for every x of its domain, then f(x) is called an odd function and if f(-x) = f(x) for every x of its domain, then f(x) is called an even function.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

63.

Assertion: f : R \(\rightarrow\) R is a function defined by f(x) = \(\frac{2x+1}{3}\). Then f^-1(x) =\(\frac{3x-1}{2}\).
Reason: f(x) is not a bijection.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

64.

Assertion: f : R \(\rightarrow\) R defined by f(x) = sin x is a bijection.
Reason: If f is both one-one and onto it is bijection.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.

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Answers Key:

Multiple Choice Question

54 x 1 = 54

(b)
injective function

(d)
both one-one and onto

(a)
{1, 5, 9}

(b)
(1, 2)

(b)
4

(b)
(6,8) ∈ R

(d)
many-one, into function

(d)
2n² relations

10.

(a)
not onto

11.

(a)
2

12.

We know that,
\(0 \leq x-[x]<1, \text { for all } x \in R \)
\(\Rightarrow 1 \leq 1+x-[x]<2 \text { for all } x \in R\)
\(\Rightarrow 15 g(x)<2 \text { for all } x \leq R\)
\(\Rightarrow f(g(x))=1, for \ all \ x \in R\)
\([\because f(x)=1, when x>0]\)

13.

\(\operatorname{gcg} f(x)=3 \cos ^{2} x_{i} f_{0}(x)=\cos 3 x^{2}\)
At X = 0, gof(x) ≠ fog(x)

14.

Range (f) = Integers ee Rand [2,3] = [2,4] = 2
⇒ f is not one-one.

15.

One-one onto mapping is possible only ifn(A) = n(B)

16.

Total number of functions = (n(B)^n(A) = 2ⁿ. Clearly a function will not be onto if all elements of A map to either a or b.

17.

If n(A) = x and n(B) = y, then number of one-one functions from A to B is given by ^y P_x, where x ≤y.

18.

A function f : A ➝ B is said to be onto, if for every b ∈ B, there exists an element a in A such that f(a) = b

19.

The set of all elements related to 1 is { a ∈ A : a = 1}.

20.

Clearly R is reflexive and transitive. For R to be symmetric we should add (3, 1) in R.

21.

R = {(I,6), (2, 7), (3, 8)}

22.

R does not have elements of the type (a,b) and (b, c).

23.

24.

(d)
{…..-5,-2,1,4,7,.}

25.

(d)
Reflexive and transitive only

26.

27.

(d)
2^nxn

28.

(b)
{(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}

29.

30.

31.

(d)
Symmetric

32.

(a)
Equivalence relation

33.

(a)
{4,5,6}

34.

(d)
R is reflexive and transitive but not symmetric

35.

(b)
{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

36.

37.

(d)
4

38.

(d)
0

39.

(b)
Transitive

40.

(b)
Symmetric

41.

42.

(a)
{(x, y): x² + y²= 5}

43.

(b)
2

44.

(a)
1

45.

(a)
f is one-one onto

46.

(d)
f is neither one-one nor onto

47.

48.

y = f(x)
⇒ 5x + 4
⇒ \(\frac{y-4}{5}\)
∴ f^-1(y) = \(\frac{y-4}{5}\)
or f^-1(x) = \(\frac{x-4}{5}\)

49.

Total injective mappings/functions
= ⁴ P₃ = 4! = 24.

50.

Reflexive, true as x s x ⇒ x - x + \(\sqrt3\)
= \(\sqrt3\) is an irrational number
Symmetric, fase e.g x =  \(\sqrt3\), y= 2
x S y ⇒ \(\sqrt3\) - 2 + \(\sqrt3\) = 2 \(\sqrt3\) -2 is an irrational number
but ySx ⇒ 2 - \(\sqrt3\) + \(\sqrt3\) = 2 \(\sqrt3\) -2 is not irrational number
transitive, false e.g = x = 1 + \(\sqrt3\), y = 5 z = 2 \(\sqrt3\)
x S y ⇒ 1 + \(\sqrt3\) - 5 + \(\sqrt3\) = 2 \(\sqrt3\) - 4 is an irrational number
y S z ⇒ 5 -2 \(\sqrt3\) + \(\sqrt3\) = 5 - \(\sqrt3\)  is an irrational number
But x S z ⇒ 1 + \(\sqrt3\) - 2 \(\sqrt3\) + \(\sqrt3\) = 1  is an irrational number

51.

A relation R is an identity relation in set A if for all a ∈ A, (a, a) ∈ R.

52.

Here (1,2) e R, (2,1) € R, if transitive (1,1) should belong to R.

53.

T₁ and T₃ are similar as their sides are proportional.

54.

Not reflexive, as l₁ R l₂
⇒ l₁ ⊥ l₁ Not true
Symmetric, true as l₁ R l₂ ⇒ l₂R h
Transitive, false as l₁ R l₂, l₂ R l₃
⇒ l₁ || l₃ . l₁ R l₂.

Case Study Questions

4 x 4 = 16

55.

(i) (d) Given, R = {(V₁,V₂) : V₁,V₂ ∈ l}and both use their voting right in general election-2019. Since, X,Y∈ I⋅X exercised his voting right while Y did not cast her vote in general election-2019.
∴ Clearly, (X,Y)∉R
(ii) (a) Relation is symmetric.
∴(X,W)∈R
⇒(W,X)∈R
(iii) (a) Since, (F,F)∈R,F ∈ I and F use their voting right.
⇒R is reflexive.
⇒(F1,F2)∈R
⇒(F2,F1)∈R
R is symmetric.
and (F₁,F₂) ∈ R
and (F₂,F₃) ∈ R
⇒ (F₁,F₃) ∈ R
(By transitive property)
(iv) (c) Given, relation RR is reflexive, symmetric and transitive.
∴R is equivalence relation.
(v) (a) Clearly, Mr. Shyam exercised his voting right in general election-2019, then Mr. Shyam is related to all those eligible voters who cast their votes.

56.

We have, R = {(l₁,l₂) :l₁ is parallel to l₂}
(i) If l₁ is parallel to l₂, then l₂ is parallel to l_1.
So, if (l₁, l₂) \(\in R,\) then (l₂, l₁) \(\in R\)
\(\therefore\) R is symmetric.
(ii) If l₁ is parallel to l₂ and l₂ is parallel to l₃, then l₁ is parallel to l₃.
So, if \(\left(l_1, l_2\right) \in R,\left(l_2, l_3\right) \in R\), then (l₁,l₃)\(\in R\)
\(\therefore\) R is transitive.
(iii) Let equation of line parallel to y = 3x + 2 be y = mx + c, where m is the slope of line. Since, y = 3x + 2 and y = mx + c are parallel. Slope of (y =3x + 2) = Slope of(y = mx + c)
\(\Rightarrow\) 3 = m i.e. m = 3
Hence, the required line is
y = 3x + c, where c \(\in R\)
Or
We have, S = {(I₁, I₂) : l₁ is perpendicular to l₂}
For Symmetric If I₁, is perpendicular to I₂, then l₂is perpendicular to l₁.
So, if (l₁,l₂) \(\in S\), then (l₂, l₁) \(\in S\)
\(\therefore\) S is symmetric.
For Transitive If I₁, is perpendicular to l₂, and l₂, is perpendicular to l₃, then I₁, is not perpendicular to l₃, it is parallel to I₃.
So, if \(\left(l_1, l_2\right) \in S,\left(l_2, l_3\right) \in S \text {, then }\left(l_1, l_3\right) \notin S \text {. }\)
\(\therefore\) S is not transitive.

57.

(i) (a) : For f(x) to be defined \(x-2 \neq 0\) i.e.,\(x \neq 2\)
\(\therefore\) Domain of f = R - {2}
(ii) (b) : Let y =J(x), then \(y=\frac{x-1}{x-2}\)
\(\Rightarrow x y-2 y=x-1 \Rightarrow x y-x=2 y-1 \Rightarrow x=\frac{2 y-1}{y-1}\)
Since, \(x \in R-\{2\}\),therefore \(y \neq 1\)
Hence, range of f = R-{1}
(iii) (d): We have,g(x) = 2f(x) - 1
\(=2\left(\frac{x-1}{x-2}\right)-1=\frac{2 x-2-x+2}{x-2}=\frac{x}{x-2}\)
(iv) (a) : We have, \(g(x)=\frac{x}{x-2}\) ,
Let \(g\left(x_{1}\right)=g\left(x_{2}\right) \Rightarrow \frac{x_{1}}{x_{1}-2}=\frac{x_{2}}{x_{2}-2}\)
\(\Rightarrow x_{1} x_{2}-2 x_{1}=x_{1} x_{2}-2 x_{2} \Rightarrow 2 x_{1}=2 x_{2} \Rightarrow x_{1}=x_{2}\)
Thus, \(g\left(x_{1}\right)=g\left(x_{2}\right) \Rightarrow x_{1}=x_{2}\)
Hence, g(x) is one-one.
(v) (c)

58.

(i) (a) : Clearly (1, 1), (2, 2), (3, 3), \(\in\) R. So, R is reflexive on A.
Since, \((1,2) \in R \text { but }(2,1) \notin R\) So, R is not symmetric on A.
Since, \((2,3), \in R\) and \((3,1) \in R\) but \((2,1) \notin R\) .So, R is not transitive on A.
(ii) (b) : Since, (1,1), (2, 2) and (3, 3) are not in R. So, R is not reflexive on A.
Now, \((1,2) \in R \Rightarrow(2,1) \in R\)
and \((1,3) \in R \Rightarrow(3,1) \in R\)
So, R is symmetric
Clearly,\((1,2) \in R \text { and }(2,1) \in R \text { but }(1,1) \notin R\)
So, R is not transitive on A.
(iii) (c) : We have, \(R=\{(x, y): y=x+5 \text { and } x<4\}\) ,where \(x, y \in N\) .
\(\therefore R=\{(1,6),(2,7),(3,8)\}\)
Clearly, (1, 1), (2, 2) etc. are not in R. So, R is not reflexive.
Since, \((1,6) \in R\) but \((6,1) \notin R\) So, R is not symmetric.
Since, \((1,6) \in R\) R and there is no order pair in R which has 6 as the first element. Same is the case for (2, 7) and (3, 8). So, R is transitive.
(iv) (d) : We have,R = {(x, y) : 3x - y = 0}, where \(x, y \in A=\{1,2, \ldots \ldots, 14\}\) .
\(\therefore\) R = {(I, 3), (2, 6), (3, 9), (4, 12)}
Clearly,\((1,1) \notin R\) So, R is not reflexive on A.
Since, \((1,3) \in R\) but \((3,1) \notin R\) .So, R is not symmetric on A.
Since, \((1,3) \in R\) and \((3,9) \in R\) but \((1,9) \notin R\) So, R is not transitive on A.
(v) (d) : Clearly, (1, 1), (2, 2), (3, 3) ∈ R. So, R is reflexive on A.
We find that the ordered pairs obtained by interchanging the components of ordered pairs in R are also in R. So, R is symmetric on A.
For \(1,2,3 \in A\) such that (1, 2) and (2, 3) are in Rimplies that (1, 3) is also, in R. So, R is transitive on A. Thus, R is an equivalence relation.