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## Relations and Functions Class 12 Maths MCQs Pdf

1. Let R be a relation on the set L of lines defined by l_{1} R l_{2} if l_{1} is perpendicular to l_{2}, then relation R is

(a) reflexive and symmetric

(b) symmetric and transitive

(c) equivalence relation

(d) symmetric

**Answer/Explanation**

Answer: d

Explaination: (d), not reflexive, as l_{1} R l_{2}

⇒ l_{1} ⊥ l_{1} Not true

Symmetric, true as l_{1} R l_{2} ⇒ l2R h

Transitive, false as l_{1} R l_{2}, l_{2} R l_{3}

⇒ l_{1} || l_{3} . l_{1} R l_{2}.

2. Given triangles with sides T_{1} : 3, 4, 5; T_{2} : 5, 12, 13; T_{3} : 6, 8, 10; T_{4} : 4, 7, 9 and a relation R in set of triangles defined as R = {(Δ_{1}, Δ_{2}) : Δ_{1} is similar to Δ_{2}}. Which triangles belong to the same equivalence class?

(a) T_{1} and T_{2}

(b) T_{2} and T_{3}

(c) T_{1} and T_{3}

(d) T_{1} and T_{4}

**Answer/Explanation**

Answer: c

Explaination: (c), T_{1} and T_{3} are similar as their sides are proportional.

3. 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

**Answer/Explanation**

Answer: c

Explaination: (c), here (1,2) e R, (2,1) € R, if transitive (1,1) should belong to R.

4. 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)}

**Answer/Explanation**

Answer: b

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

5. A relation S in the set of real numbers is defined as xSy ⇒ x – y+ √3 is an irrational number, then relation S is

(a) reflexive

(b) reflexive and symmetric

(c) transitive

(d) symmetric and transitive

**Answer/Explanation**

Answer: a

Explaination:

6. 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) 12

(c) 24

(d) 64

**Answer/Explanation**

Answer: c

Explaination: (c), total injective mappings/functions

= ^{4} P_{3} = 4! = 24.

7. 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) = \(\frac{5}{4 x-5}\)

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

(d) g(x) = 5x – 4

**Answer/Explanation**

Answer: c

Explaination:

8. Let Z be the set of integers and R be a relation defined in Z such that aRb if (a – b) is divisible by 5. Then R partitions the set Z into ______ pairwise disjoint subsets.

**Answer/Explanation**

Answer:

Explaination: Five, as remainder can be 0, 1, 2, 3, 4.

9. Consider set A = {1, 2, 3 } and the relation R= {(1, 2)}, then? is a transitive relation. State true or false.

**Answer/Explanation**

Answer:

Explaination: True, as there is no situation

(a, b) ∈ R, (b, c) ∈ R Hence, transitive. We can also say, a relation containing only one element is transitive.

10. Every relation which is symmetric and transitive is reflexive also. State true or false.

**Answer/Explanation**

Answer:

Explaination: False,e.g.if R is arelationinset A = {2,3,4} defined as {(2, 3), (3, 2), (2, 2)} is symmetric and transitive but not reflexive.

11. Let R be a relation in set N, given by R = {(a, b): a = b – 2, b > 6} then (3, 8) ∈ R. State true or false with reason.

**Answer/Explanation**

Answer:

Explaination: False, as in (3, 8), b = 8

⇒ a = 8 – 2

⇒ a = 6, but here a = 3.

12. Let R be a relation defined as R = {(x, x), (y, y), (z, z), (x, z)} in set A = {x, y, z} then R is (reflexive/symmetric) relation.

**Answer/Explanation**

Answer:

Explaination: Reflexive, as for all a ∈ A, (a, a) ∈ R.

13. Let R be a relation in the set of natural numbers N defined by R = {(a, b) ∈ N × N: a < b}. Is relation R reflexive? Give a reason.

**Answer/Explanation**

Answer:

Explaination:

Given R = {(a, b) ∈ N × N: a < b}.

Not reflexive, as for (a, a) × R

⇒ a< a, not true.

14. Let A be any non-empty set and P(A) be the power set of A. A relation R defined on P(A) by X R Y ⇔ X ∩ Y = X, X, Y ∈ P(A). Examine whether ? is symmetric.

**Answer/Explanation**

Answer:

Explaination: X R Y ⇔ X ∩ Y = X ⇒ Y ∩ X = X ⇒ Y R X.

Hence, symmetric.

15. State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive. [NCERT; Delhi 2011]

**Answer/Explanation**

Answer:

Explaination: (1, 2) ∈ R, (2, 1) ∈ R, but (1, 1) ∉ R.

16. Show that the relation R in the set {1,2,3} given by R = {(1,1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. [NGERT]

**Answer/Explanation**

Answer:

Explaination:

Given R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} defined on R: {1, 2, 3} → {1, 2, 3}

For reflexive: As (1, 1), (2,2), (3, 3) ∈ R. Hence, reflexive

For symmetric: (1, 2) ∈ R but (2, 1) ∉ R. Hence, not symmetric.

For transitive: (1, 2) ∈ R and (2, 3) ∈R but (1, 3) ∉ R. Hence, not transitive.

17. Let A = {3, 4, 5} and relation R on set A is defined as R = {(a, b) e A x A : a – b – 10). Is relation an empty relation?

**Answer/Explanation**

Answer:

Explaination: We notice for no value of a, b s A, a-b = 10. Hence, (a, b) £ R for a, b e A. Hence, empty relation.

18. Given set A = {a, b} and relation R on A is defined as R = {(a, a), (b, b)}. Is relation an identity relation?

**Answer/Explanation**

Answer:

Explaination: Yes, as (a, a) ∈ R, for all a ∈ A..

19. Let set A represents the set of all the girls of a particular class. Relation R on A is defined as R = {(a, b) ∈ A × A : difference between weights of a and b is less than 30 kg}. Show that relation R is a universal relation.

**Answer/Explanation**

Answer:

Explaination: Let a, b ∈ A then a – b < 30 kg, always true for students of a particular class, i.e. aRb ∀ a, b ∈ A. Hence, universal relation.

20. If A = {1, 2, 3} and relation R = {(2, 3)} in A. Check whether relation R is reflexive, symmetric and transitive.

**Answer/Explanation**

Answer:

Explaination:

Not reflexive, as (1, 1) ∉ R.

Not symmetric, as (2, 3) ∈ R but (3,2) ∉ R.

Transitive, as relation R in a non empty set containing one element is transitive.

21. State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive. [Delhi]

**Answer/Explanation**

Answer:

Explaination: As (1, 2) ∈ R, (2, 1) ∈ R, but (1, 1) ∉ R.

22. Consider the set A containing n elements, then the total number of injective functions from set A onto itself is _____ .

**Answer/Explanation**

Answer:

Explaination: Total number of injective functions from set containing n elements to a set containing n elements is ^{n} P_{n} = n!

23. The domain of the function *f* : R → R defined by f(x) = \(\sqrt{4-x^{2}}\) is ______ .

**Answer/Explanation**

Answer:

Explaination:

[-2, 2]. For domain 4 – x² ≥ 0

⇒ 4 ≥ x²

⇒ x² ≤ 4

⇒ x² ≤ (2)²

⇒ -2 ≤ x ≤ 2, i.e. [-2, 2].

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

(a) 2

(b) 4

(c) 1

(d) 3

**Answer/Explanation**

Answer:

Explaination: (a), as if n(A) = m, then possible one-one functions from A to A are m!

25. Let A = {1, 2, 3, 4} and B = {a, b, c}. Then number of one-one functions from A to B are ______.

**Answer/Explanation**

Answer:

Explaination: 0, as n(A) > n(B)

26. If n(A) = p, then number of bijective functions from set A to A are ______ ..

**Answer/Explanation**

Answer:

Explaination: p!, as for bijective functions from A to B, n(A) = n(B) and function is one-one onto.

27. The function *f *: R → R defined as *f*(x) = [x], where [x] is greatest integer ≤ x, is onto function. State true or false.

**Answer/Explanation**

Answer:

Explaination: False, as range of *f* is set of integers, i.e.

Z and range of f ⊆ co-domain R. Hence,not onto e.g. for \(\frac{1}{2}\) ∈ R (co-domain) there is no x ∈ R (domain) such that y = *f*(x) or \(\frac{1}{2}\) e∈ R has no pre-image.

28. If \(*f*(x)=\frac{x-1}{|x-1|}, x(\neq 1) \in R\) then range of ‘f’ is _______ .

**Answer/Explanation**

Answer:

Explaination:

29. If *f* : R → R be defined by *f*(x) = (3 – x^{3})^{1/3}, then find *fof*(x). [NCERT]

**Answer/Explanation**

Answer:

Explaination:

30. If f is an invertible function defined as *f*(x) = \(\frac{3x-4}{5}\), write *f*^{-1}(x).

**Answer/Explanation**

Answer:

Explaination:

31. 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. State whether *f* is one-one or not. [AI 2011] *f*

**Answer/Explanation**

Answer:

Explaination: One-one, as for x_{1} ≠ x_{2}

⇒ *f*(x_{1}) ≠ *f*(x_{2}).

32. Let *f* : R → R is defined by *f* (x) = | x |. Is function f onto? Give a reason. [HOTS]

**Answer/Explanation**

Answer:

Explaination: *f* is not onto, as for some y ∈ R from co-domain, there is no x ∈ R from domain such that y = f(x), e.g. for -2 ∈ R (co-domain) there is no x ∈ R (domain) such that *f*(x) = -2, i.e. |x| = -2. Hence, not onto.

33. If f : R → R and g : R → R are given by f (x) = sin x and g(x) = 5x², find gof(x).

**Answer/Explanation**

Answer:

Explaination: gof(x) = g(f(x)) = g(sin x) = 5 sin² x.

34. Let *f* : {1, 3,4} → {1,2, 5} and g: {1,2, 5} → {1, 3} be given by *f* = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. [NCERT]

**Answer/Explanation**

Answer:

Explaination:

go*f*: {1, 3, 4} → {1, 3}.

go*f*(1) = g(*f*(1)) = g(2) = 3.

go*f*(3) = g(K3)) = g(5) = 1

go*f*(4) = g(*f*(4)) = g(1) = 3

go*f* = {(1, 3), (3, 1), (4, 3)}.

35. Prove that *f* : R → R given by *f*(x) = x^{3} + 1 is one-one function.

**Answer/Explanation**

Answer:

Explaination:

Given *f*(x) = x^{3} + 1

For x_{1} ≠ x_{2}

⇒ x_{1}^{3} ≠ x_{2}^{3}

⇒ x_{1}^{3} + 1 ≠ x_{2}^{3} + 1

⇒ *f*(x_{1}) ≠ *f*(x_{2}). Hence, one-one

36. Show that the Signum Function *f* : R → R,

one-one nor onto.

**Answer/Explanation**

Answer:

Explaination:

Range of function is {-1, 0, 1} and co-domain is set of real numbers R.

⇒ Range ⊆ co-domain.

There is at least one element in R(codomain) which is not image of any element of the domain, e.g. for 2 e R(co-domain), there is no x in domain such that *f*(x) = 2, x ∈ R.

Hence, function is not onto.

Also, let x_{1} = 2 and x_{2} = 3 then *f*(x_{1}) = 1 and *f*(x_{2}) = 1

i.e., x_{1} ≠ x_{2} ⇒ *f*(x_{1}) = *f*(x_{2}).

So, function is not one-one.

37. Given *f*(x) = sin x check if function f is one-one for (i) (0, π) (ii) (-\(\frac{π}{2}\), \(\frac{π}{2}\)).

**Answer/Explanation**

Answer:

Explaination:

38. If *f* : R → R is defined by *f*(x) = 3x + 2, define *f* (*f*(x)). [Foreign]

**Answer/Explanation**

Answer:

Explaination: *f*(*f*(x)) = *f*(3x + 2) = 3(3x + 2) + 2

= 9x+ 8.

39. Write fog, if *f* : R → R and g : R → R are given by *f*(x) = |x| and *g*(x) = |5x – 2|. [Foreign]

**Answer/Explanation**

Answer:

Explaination: (*fog*)(x) =*f*(*g*(x))

=*f*(|5x-2|) = ||5x-2||.

40. Write fog, if *f* : R → R and g : R → R are given by *f*(x) = 8x^{3} and g(x) = x^{1/3}. [Foreign]

**Answer/Explanation**

Answer:

Explaination: (*fog*)(x) = *f*(*g*(x)) = *f*(x^{1/3}) = 8(x^{1/3})^{3} = 8x.

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