Relations and Functions Class 12 Important Previous Year Questions

1. Let T be the set of all triangles in a plane with R a relation in T given by
R = {(T1, T2): T1 is congruent to T2}.
Show that R is an equivalence relation.
Ans. R is reflexive, since every  is congruent to itself.
(T1T2)R similarly  (T2T1) R
since T1 T2
(T1T2) R, and (T2,T3) R
(T1T3)R Since three triangles are
congruent to each other.

2. Show that the relation R in the set Z of integers given byR={(a, b) : 2 divides a-b}. is equivalence relation.
Ans. R is reflexive , as 2 divide a-a = 0
((a,b)R ,(a-b) is divide by 2
 (b-a) is divide by 2 Hence (b,a) R hence symmetric.
Let a,b,c   Z
If  (a,b) R
And (b,c)  R
Then  a-b and b-c is divided by 2
 a-b +b-c is even
 (a-c is even
 (a,c) R
Hence it is transitive.

3. Let L be the set of all lines in    plane and R be the relation in L define if R = {(l1, L2 ): L1 is  to L2 } . Show that R is symmetric but neither reflexive nor transitive.
Ans. R is not reflexive , as a line L1 cannot be  to itself  i.e  (L1,L1 ) R
 L1 L2
L2 L1
 (L2,L1)R
L1  L2 and L2 L3

 
Then L1 can never be   to L3 in fact L1 || L3
i.e (L1,L2) R, (L2,L3)  R.
But (L1, L3) R

4. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} asR = {(a, b): b = a+1} is reflexive, symmetric or transitive.
Ans. R = {(a,b): b= a+1}
Symmetric or transitive
R = {(1,2) (2,3) (3,4) (4,5) (5,6) }
R is not reflective , because (1,1) R
R is not symmetric because (1,2) R but (2,1) R
(1,2) R and (2,3) R
But (1,3) R Hence it is not transitive

5. Let L be the set of all lines in xy plane and R be the relation in L define as R = {(L1, L2): L1 || L2} Show then R is on equivalence relation.
Find the set of all lines related to the line y=2x+4.
Ans. L1||L1    i.e (L1, L1)  R  Hence reflexive
L1||L2 then L2 ||L1   i.e (L1L2) R
 (L2,L) R Hence symmetric
We know the
L1||L2  and L2||L3
Then L1|| L3
Hence Transitive .  y = 2x+K
When K is real number.

6. Show that the relation in the set R of real no. defined R = {(a, b) : a b3 }, is neither reflexive nor symmetric nor transitive.
Ans. (i) (a, a)   Which is false R is not reflexive.
(ii)  Which is false R is not symmetric.
(iii)  Which is false

7. Let A = NN and * be the binary operation on A define by (a, b) * (c, d) = (a + c, b + d)Show that * is commutative and associative.
Ans. (i) (a, b) * (c, d) = (a + c, b + d)
= (c + a, d + b)
= (c, d) * (a, b)
Hence commutative
(ii) (a, b) * (c, d) * (e, f)
= (a + c, b + d) * (e, f)
= (a + c + e, b + d + f)
= (a, b) * (c + e, d + f)
= (a, b) * (c, d) * (e, f)
Hence associative.

8. Show that if f:  is defining by f(x) = and g: is define by
g(x) = then fog = IA and gof = IB  when ; IA (x) = x, for all xA, IB(x) = x, for all xB are called identify function on set A and B respectively.
Ans. gof (x) = 


Which implies that gof = IB
And Fog = IA

9. Let f: N  à N be defined by f(x) = 
Examine whether the function f is onto, one – one or bijective
Ans. 

f is not one – one
1 has two pre images 1 and 2
Hence f is onto
f is not one – one but onto.

10. Show that the relation R in the set of all books in a library of a collage given by R ={(x, y) : x and y  have same no of pages}, is an equivalence relation.
Ans. (i)  (x, x)  R, as x and x have the same no of pages for all xR  R is reflexive.
(ii)  (x, y) R
x and y have the same no. of pages
y and x have the same no. of pages
(y, x) R
(x, y) = (y, x) R is symmetric.
(iii)  if (x, y)  R, (y, y)  R
(x, z) R
 R is transitive.

11. Let * be a binary operation. Given by a * b = a – b + abIs * :
(a) Commutative
(B) Associative
Ans. (i)  a * b = a – b + ab
b * a = b – a + ab
a * b  b * a
(ii)  a * (b * c) = a * (b – c + bc)
= a – (b – c + bc) + a. (b – c + bc)
= a – b + c – bc + ab – ac + abc
(a * b) * c = (a – b + ab) * c
= (�–�+��)–� + ( a – b + ab)
= a- b + ab – c + ac – bc + abc
a * (b * c)   (a * b) * c.

12. Let f: R à R be f (x) = 2x + 1 and g: R à R be  g(x) = x2 – 2 find (i) gof (ii) fog
Ans. (i)  gof (x) = g�(�)
= g (2x + 1)
= (2x + 1)2 – 2
(ii)  fog (x) = f (fx)
= f (2x + 1)
= 2(2x + 1) + 1
= 4x + 2 + 1 = 4x + 3

13. Let A = R – {3} and B = R- {1}. Consider the function of f: A à B defined by
f(x) =  is f one – one and onto.
Ans. Let x1 x2  A
Such that f(x1) = f(x2)

f is one – one


Hence onto

14. Show that the relation R defined in the set A of all triangles asR = { is similar to T2 }, is an equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5. T2 with
sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
Ans. (i)  Each triangle is similar to at well and thus (T1, T1)  R
  R is reflexive.
(ii)  (T1, T2)  R
 T1 is similar to T2
T2 is similar to T1
(T2, T1)  R
R is symmetric
(iii)  T1 is similar to T2 and T2 is similar to T3
T1 is similar to T3
(T1, T3)  R
R is transitive.
Hence R is equivalence
(II) part  T1 = 3, 4, 5
T2 = 5, 12, 13
T3 = 6, 8, 10
T1 is relative to T3.

15. Determine which of the following operation on the set N are associative and which are commutative.
(a) a * b = 1 for all a, b N
(B) a * b =  for all a, b, N
Ans. (a)  a * b = 1
b * a = 1
for all a, b  N also
(a * b) * c = 1 * c = 1
a * (b * c) = a * (1) = 1 for all, a, b, c R N
Hence R is both associative and commutative
(b)  a * b = ,  b * a = 
Hence commutative.
(a * b) * c = 
=
= 

* is not associative.

17. Let A and B be two sets. Show that f: A B à B A such that f(a, b) = (b, a) is a bijective function.
Ans. Let (a1 b1) and (a2, b2)  A  B
(i)  f(a1 b1) = f(a2, b2)
b1 = b2 and a1 = a2
(a1 b1) = (a2, b2)
Then f(a1 b1) = f(a2, b2)
(a1 b1) = (a2, b2) for all
(a1 b1) = (a2, b2)  A  B
(ii)  f is injective,
Let (b, a) be an arbitrary
Element of B  A. then b  B and a  A
(a, b) )  (A  B)
Thus for all (b, a)  B  A their exists (a, b) )  (A  B)
Hence that
f(a, b) = (b, a)
So f: A  B à B  A
Is an onto function.
Hence bijective

18. Show that the relation R defined by (a, b) R (c, d) a + b = b + c on the set N N is an equivalence relation.
Ans. (a, b) R (c, d)  a + b = b + c where a, b, c, d  N
(a, b ) R (a, b)  a + b = b + a (a, b)  N  N
R is reflexive
(a, b) R (c, d) a + b
= b + c (a, b ) (c, d)  N  N
d + a = c + b
c + b = d + a
 (c, d) R (a, b) (a, b), (c, d)  N  N
Hence reflexive.
(a, b) R (c, d)  a + d = b + c   (1) (a, b), (c, d)  N  N
(c, d) R (e, f)  c + f = d + e   (2) (c, d), (e, f)  N  N
Adding (1) and (2)
(a + b) + (+�) = (b + c) + (d + e)
a + f = b + e
(a, b) R (e, f)
Hence transitive
So equivalence

19. Let * be the binary operation on H given by a * b = L. C. M of a and b. find
(a) 20 * 16
(b) Is  * commutative
(c) Is * associative
(d) Find the identity of * in N.
Ans. (i)  20 * 16 = L. C.M of 20 and 16
= 80    
 
(ii)  a * b = L.C.M of a and b
= L.C.M of b and a
= b * a
(iii)  a * (b * c) = a * (L.C.M of b and c)
= L.C.M of (a and L.C.M of b and c)
= L.C.M of a, b and c
Similarity
(a * b) * c = L. C.M of a, b, and c
(iv)  a * 1 = L.C.M  of a and 1= a
=1

20. If the function f: R à  R is given by f(x) =  and g: R à R is given by g(x) = 2x – 3, Find 
(i) fog  
(ii) gof. Is f-1 = g
(iii)  fog = gof = x
Ans. (i)  fog (x) = f �(�)
= f (2x – 3)
= 
= x
(ii)  gof (x) = g �(�)

= x
(iii)  fog = gof = x
Yes,

21. Let L be the set of all lines in Xy plane and R be the relation in L define as R = {(L1, L2): L1 || L2} Show then R is on equivalence relation.
Find the set of all lines related to the line y=2x+4.
Ans. L1||L1    i.e (L1, L1)  R  Hence reflexive
L1||L2 then L2 ||L1   i.e (L1L2) R
 (L2, L) R Hence symmetric
We know the
L1||L2 and L2||L3
Then L1|| L3
Hence Transitive.  y = 2x+K
When K is real no.

FAQs

Q1: Are relations and functions difficult to understand?

A1: Relations and functions might seem challenging at first, but with proper guidance and practice, students can grasp these concepts effectively.

Q2: How can solving previous year questions help me prepare for the exams?

A2: Solving previous year questions gives you an insight into the exam pattern, the type of questions asked, and helps you identify your weak areas.

Q3: Can you suggest some additional resources to study relations and functions?

A3: Apart from your textbook, you can refer to online educational platforms that offer interactive lessons and practice problems.

Q4: Can you explain the difference between a one-to-one function and an onto function?

A4: A one-to-one function ensures that each element of the domain maps to a unique element in the codomain, while an onto function covers the entire codomain without leaving any element unmapped.

Q5: Is it essential to solve all the previous year questions?

A5: While it's not mandatory, solving a good number of previous year questions helps you get accustomed to the exam format and boosts your confidence.