**CBSE Class 12 Maths Chapter-1 Important Questions – Free PDF Download**

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**CBSE Class 12 Mathematics Important Questions Chapter 1 Relations and Functions**

**1 Mark Questions**

**1. A Relation R:A****à****A is said to be Reflexive if ——— for every a A where A is non**

**empty set.**

**Ans: **(a, a) R

**2. A Relation R:A****à****A is said to be Symmetric if ———- a,b,A**

**Ans:** (a, b) R, (b, a) R

**3. A Relation R:A****à****A is said to be Transitive if ————- a,b,c A**

**Ans:** (a, b)R, and (b, c)R (a, c) R.

**4. Define universal relation? Give example.**

**Ans:** A Relation R in a set A called universal relation if each element of A is related to every element of A. Ex. Let = {2,3,4}

R = (AA) = {(2,2),(2,3) (2,4) (3,2) (3,3) (3,4) (4,2) (4,3) (4,4) }

**5. What is trivial relation?**

**Ans:** Both the empty relation and the universal relation are some time called trivial relation.

**6. Prove that the function f: R ****à**** R, given by f(x) = 2x, is one – one.**

**Ans: **f is one – one as f(x_{1}) = f (x_{1})

2x_{1 }= 2x_{2}

x_{1} = x_{2}

Prove.

**7. State whether the function is one – one, onto or bijective f: R ****à**** R defined by f(x) = 1+ x ^{2}**

**Ans:**Let x

_{1}, x

_{2}x

If f(x

_{1}) = f(x

_{2})

Hence not one – one

**8. Let S = {1, 2, 3}**

**Determine whether the function f: S ****à**** S defined as below have inverse.**

**f = {(1, 2), (2, 1), (3, 1)}**

**Ans:** f(2) = 1 f(3) = 1,

f is not one – one, So that that f is not invertible.

**9. Find gof f(x) = |x|, g(x) = |5x + 1|**

**Ans:** gof (x) = g [f(x)]

= g [(x)]

=

**10. Let f, g and h be function from R to R show that (f + g) oh = foh + goh**

**Ans:** L.H.S = (f + g) oh

= {(f + g) oh} (x)

= (f + g) h (x)

= f [h (x)] + g [h (x)]

= foh + goh

**9. If a * b = a + 3b ^{2}, then find 2 * 4**

**Ans:**2 * 4 = 2 + 3 (4)

^{2}

= 2 + 3 16

= 2 + 48

= 50

**11. Show that function f: N ****à**** N, given by f(x) = 2x, is one – one.**

Ans: the function f is one – one, for

f(x_{1}) = f(x_{2})

2x_{1} = 2x_{2}

x_{1} = x_{2}

**12. State whether the function is one – one, onto or bijective f: R ****à**** R defined by f(x) = 3 – 4x**

**Ans:** is x_{1}, x_{2} R

f(x_{1}) = f(x_{2})

3 – 4x_{1} = 3 – 4x_{2}

x_{1} = x_{2}

Hence one – one

Y = 3 – 4x

= y

Hence onto also.

**13. Let S = {1, 2, 3}**

**Determine whether the function f: S ****à**** S defined as below have inverse.**

**f = {(1, 1), (2, 2), (3, 3)}**

**Ans:** f is one – one and onto, so that f is invertible with inverse f^{-1} = {(1, 1) (2, 2) (3, 3)}

**14. Find got f(x) = |x|, g(x) = |5x -2|**

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

= f{|5x – 2|)

= |5x – 2|

**15. Consider f: {1, 2, 3} ****à**** {a, b, c} given by f(1) = a, f(2) = b and f(3) = c find f ^{-1} and show**

**that (f**

^{-1})^{-1 }= f**Ans:**f = {(1, a) (2, b) (3, c)}

f

^{-1}= { (a, 1) (b, 2) (c, 3)}

(f

^{-1})

^{-1}= {(1, a) (2, b) (3, c)}

Hence (f

^{-1})

^{-1}= f.

**16. If f(x) = x + 7 and g(x) = x – 7, xR find (fog) (7)**

**Ans:** (fog) (x) = f[g(x)]

= f(x – 7)

= x – 7 + 7

= x

(fog) (7) = (7)

**17. What is a bijective function?**

**Ans:** A function f: X à Y is said to be one – one and onto (bijective), if f is both one – one and onto.

**18 Let f: R ****à**** R be define as f(x) = x ^{4} check whether the given function is one – one onto,**

**or other.**

**Ans:**Let x

_{1}, x

_{2}R

If f(x

_{1}) = f(x

_{2})

Not one – one

Not onto.

**19 Let S = {1, 2, 3}**

**Determine whether the function f: S ****à**** S defined as below have inverse.**

**f = {(1, 3) (3, 2) (2, 1)}**

**Ans:** f is one – one and onto, Ao that f is invertible with f^{-1} = {(3,1) (2, 3) (1, 2)}

**20 Find gof where f(x) = 8x ^{3}, g(x) = x^{1/3}**

**Ans:**gof (x) = g[f(x)]

= g (8x

^{3})

=

= 2x

**21. Let f, g and h be function from R + R. Show that (f.g) oh = (foh). (goh)**

**Ans:** (f. g) oh

(f. g) h (x)

f[h(x)]. g[h(x)]

foh. goh

**21. Let * be a binary operation defined by a * b = 2a + b – 3. find 3 * 4**

**Ans:** 3 * 4 = 2 (3) + 4-3 = 7

**22. show that a one – one function f: {1, 2, 3} ****à**** {1, 2, 3} must be onto.**

**Ans:** Since f is one – one three element of {1, 2, 3} must be taken to 3 different element of the co – domain {1, 2, 3} under f. hence f has to be onto.

**23. f: R ****à**** R be defined as f(x) = 3x check whether the function is one – one onto or other**

**Ans:** Let

**24. Let S = {1, 2, 3} Determine whether the function f: S à**

**S defined as below have inverse.**

f = { (1, 2) (2, 1) (3, 1) }

f = { (1, 2) (2, 1) (3, 1) }

**Ans:**f(2) = 1, f(3) =1

f is not one – one so that f is not invertible

Hence no inverse

**25. Find fog f(x) = 8x ^{3}, g(x) = x^{1/3}**

**Ans:**fog (x) = f(gx)

= 8x

**26. If f: R ****à**** R be given by f(x) = , find fof (x)**

**Ans:**

=

= x

**27. If f(x) is an invertible function, find the inverse of f(x) = **

**Ans:** Let f(x) = y

**4 Marks 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) – 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[f(x)]

= 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) + [(+f)] = (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 [g(x)]

= f (2x – 3)

=

= x

**(ii)** gof (x) = g [f(x)]

= 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.