Relations and Functions: Important Questions

Relations and Functions carry a total of 4-6 marks in the CBSE Class XII Board Examination. Questions are expected in the various sections of the question paper corresponding to

(i) Very Short Answer Type (VSA) Questions: 1 Mark,

(ii) Short Answer Type (SA) Questions: 2 Marks,

(iii) Long Answer Type I (LA-I) Questions: 4 Marks, and

(iv) Long Answer Type II (LA-II) Questions: 6 Marks.

Use Eckovation App to learn all the concepts of this unit through Video Lectures and Quiz. After completing all the topics, try solving the following questions and you will easily score full marks from this chapter. Answers to these questions are given in the end. You can verify your answers and check your level of understanding. If you face any difficulty solving these questions, you can watch the Video Lectures and strengthen your weak areas using Eckovation App.

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)

1. If A is the set of students of a school then write, which of following relations are. (Universal, Empty or neither of the two).
R1 = {(a, b) : a, b are ages of students and |a – b| ≥ 0}
R2 = {(a, b) : a, b are weights of students, and |a – b| < 0}
R3 = {(a, b) : a, b are students studying in same class}

2. Is the relation R in the set A = {1, 2, 3, 4, 5} defined as R = {(a, b) : b = a + 1} reflexive?

3. If R, is a relation in set N given by R = {(a, b) : a = b – 3, b > 5}, then does elements (5, 7) ∈ R?

4. If f : {1, 3} → {1, 2, 5} and g : {1, 2, 5} → {1, 2, 3, 4} be given by f = {(1, 2), (3, 5)}, g = {(1, 3), (2, 3), (5, 1)} Write down gof.

5. Let g, f : R → R be defined by g(x)=(x+2)/3, f(x)= 3x-2. Write fog.

6. If f : R →R defined by f(x)= (2x-1)/5 be an invertible function, write f–1(x).

7. If f(x)= x/(x+1) ∀ x ≠ -1, Write fof(x).

8. Let * is a Binary operation defined on R, then if
(i) a * b = a + b + ab, write 3 * 2
(ii) a*b = (a+b)2/3 , Write (2 * 3 )* 4.

9. If n(A) = n(B) = 3, Then how many bijective functions from A to B can be formed?

10. If f (x) = x + 1, g(x) = x – 1, Then (gof) (3) = ?

11. Is f : N → N given by f(x) = x2 is one-one? Give reason.

12. If f : R → A, given by f(x) = x2 – 2x + 2 is onto function, find set A.

13. If f : A → B is bijective function such that n (A) = 10, then n (B) = ?

14. If n(A) = 5, then write the number of one-one functions from A to A.

15. R = {(a, b) : a, b ∈ N, a ≠ b and a divides b}. Is R reflexive? Give reason.

16. Is f : R → R, given by f(x) = |x – 1| is one-one? Give reason.

17. f : R → B given by f(x) = sin x is onto function, then write set B.

18. If f(x) = log[(1+x)/(1-x)], show that f[2x/(1+x2) = 2f(x).

19. If ‘*’ is a binary operation on set Q of rational numbers given by a*b = ab/5 then write the identity element in Q.

20. If * is Binary operation on N defined by a * b = a + ab ∀ a, b ∈ N. Write the identity element in N if it exists.

SHORT ANSWER TYPE QUESTIONS (4 Marks)

21. Check the following functions for one-one and onto.
(a) f : R→R, f(x) = (2x-3)/7
(b) f : R → R, f(x) = |x + 1|
(c) f : R – {2} → R, f(x) = (3x-1)/(x-2)
(d) f : R → [–1, 1], f(x) = sin2x

22. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a* b = H.C.F. of a and b. Write the operation table for the operation *.

23. Let f : R – {-4/3} → R – {4/3} be a function given by f(x) = 4x/(3x+4). Show that f is invertible with f-1(x) = 4x/(4-3x).

24. Let R be the relation on set A = {x : x ∈ Z, 0 ≤ x ≤ 10} given by R = {(a, b) : (a – b) is multiple of 4}, is an equivalence relation. Also, write all elements related to 4.

25. Show that function f : A → B defined as f(x) = (3x+4)/(5x-7) where A = R – {7/5}, B = R – {3/5} is invertible and hence find f–1 .

26. Let * be a binary operation on Q. Such that a * b = a + b – ab.
(i) Prove that * is commutative and associative.
(ii) Find identity element of * in Q (if it exists).

27. If * is a binary operation defined on R – {0} defined by a * b = 2a/b2, then check * for commutativity and associativity.

28. If A = N × N and binary operation * is defined on A as (a, b) * (c, d) = (ac, bd).
(i) Check * for commutativity and associativity.
(ii) Find the identity element for * in A (If it exists).

29. Show that the relation R defined by (a, b) R(c, d) ⇔ a + d = b + c on the set N × N is an equivalence relation.

30. Let * be a binary operation on set Q defined by a * b = ab/4,  show that
(i) 4 is the identity element of * on Q.
(ii) Every non zero element of Q is invertible with a-1 = 16/a , a ∈ Q – {0}.

31. Show that f : R+ → R+ defined by f(x) = 1/2x is bijective where R+ is the set of all non-zero positive real numbers.

32. Consider f : R+ → [–5, ∞) given by f(x) = 9x2 + 6x – 5 show that f is invertible with f-1 = [√(x+6) – 1]/3.

33. If ‘*’ is a binary operation on R defined by a * b = a + b + ab. Prove that * is commutative and associative. Find the identity element. Also show that every element of R is invertible except –1.

34. If f, g : R → R defined by f(x) = x2 – x and g(x) = x + 1 find (fog) (x) and (gof) (x). Are they equal?

35. f : [1,∞ ) → [2,∞) is given by f(x) = x +(1/x) , find f-1(x)

36. f : R → R, g : R → R given by f(x) = [x], g(x) = |x| then find (fog)(-2/3) and (gof)(-2/3).

ANSWERS

1. R1 : is universal relation.
R2 : is empty relation.
R3 : is neither universal nor empty.

2. No, R is not reflexive.

3. (5, 7) ∉ R

4. gof = {(1, 3), (3, 1)}

5. (fog)(x) = x ∀ x ∉ R

6. f-1(x) = (5x+1)/2

7. fof(x) = x/(2x+1), x ≠ -1/2

8. (i) 3 * 2 = 11
(ii) 1369/27

9. 6

10. 3

11. Yes, f is one-one since ∀ x1, x2 ∈ N ⇒ x12 = x22.

12. A = [1, ∞) because Rf = [1, ∞)

13. n(B) = 10

14. 120.

15. No, R is not reflexive since (a, a) ∉ R ∀ a ∈ N.

16. f is not one-one function since f(3) = f (–1) = 2 but 3 ≠ – 1 i.e. distinct elements have same images.

17. B = [–1, 1]

19. e = 5

20. Identity element does not exist.

21. (a) Bijective
(b) Neither one-one nor onto.
(c) One-one, but not onto.
(d) Neither one-one nor onto.

22. 

24. Elements related to 4 are 0, 4, 8.

25. f-1(x) = (7x+4)/(5x-3)

26. 0 is the identity element.

27. Neither commutative nor associative.

28. (i) Commutative and associative.
(ii) (1, 1) is identity in N × N

33. 0 is the identity element.

34. (fog) (x) = x2 + x, (gof) (x) = x2 – x + 1. Clearly, they are unequal.

35. f-1(x) = [x + √(x2 – 4)]/2

36. (fog)(-2/3) = 0, (gof)(-2/3) = 1.

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