Plus Two Maths Chapter Wise Previous Questions Chapter 1 Relations and Functions are part of Plus Two Maths Chapter Wise Previous Year Questions and Answers. Here we have given Plus Two Maths Chapter Wise Previous Chapter 1 Relations and Functions.

## Kerala Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 1 Relations and Functions

Question 1.

If

a. Find fof (x)

b. Find the inverse of f. **[March-2018] **Answer:

Question 2.** **Let A=N x N and be a binary operation on A defined by (a,b)*(c,d) = (a+c, b+d)

**[March-2018]**

a. Find (1,2)*(2,3)

b. Prove that ‘*’ is commutative.

c. Prove that ‘*’ is associative.

Answer:

a. (a,b) * (c,d)=(a+c, b+d)

(1,2)*(2,3) = (1+2, 2+3) = (3,5)

b. (a,b) * (c,d) = (a+c, b+d)

(c,d) *(a,b) = (c+a, d+b) = (a+c, b+d)

(a,b) * (c,d) = (c,d) * (a,b)

* is commutative

c. Let (a,b), (c,d), (e,f) ∈ A (a,b) * [ (c,d) * (e,f)]

= (a,b)*[(c+ d), (d + f)]

= (a+c+e, b+d+f)

[(a,b)*(c,d)*(e,f) = (a+c, b+d)*(e,f)

= (a+c+e. b+d+f)

i.e (a,b)*[(c,d)*(e,f)] = [(a,b) * (c,d)]*(e,f)

∴ * is associative

Question 3.

a. Let R be a relation defined on A = (1, 2, 3} by R = {(1, 3), (3, 1), (2, 2)}. R is

a. Reflexive

b. Symmetric

c. Transitive

d. Reflexive but not transitive

b. Find fog and gof if f(x) = |x + 1| and g(x) = 2x – 1.

c. Let * be a binary operation defined on N × N by

(a, b) * (c, d) = (a + c, b + d).

Find the identity element for * if it exists. ** [March-2017] **Answer:

a. Symmetric

b. f(x) = |x + 1| g(x) = 2x – 1

fog = f(g(x)) = f(2x – 1)

= |2x – 1 + 11 = |2x| = 2x

gof = g(f(x) = g(|x + 1|)

= 2 |x + 11 – 1 |.

c. Let (e, f) be the identify function, then (a, b)* (e, f) = (a + e, b + f)

For identity function, a = a+e ⇒ e = 0 and b + f= b ⇒ f=0

Also 0 ∴ N∴ Identity element does not exist

Question 4.

a. The function f: N → N, given by f(x)=2x is

(i) one-one and onto

(ii) one-one but not onto

(iii) not one-one and not onto

(iv) onto, but not one-one

b. Find gof(x),if f(x) = 8x^{3 }and g(x) = _{X1/3 }c. Let * be an operation such that a * b = LCM of a and b defined on the set A = {1, 2, 3, 4, 5}. Is *a binary operation? Justify your answer.** [March-2016] **Answer:

a. i. one-one but not onto

b. f(x) = 8x^{3}, g(x) = x >3^{1/3}

gof(x) = g(f(x)) = (8x^{3})^{1/3} = 2x

c. a*b = LCM of a and b

A = {1,2, 3,4, 5}

3*2 = 2*3 = 6 ∉ A, 5*2=2*5=10∉ A

5*4=4*5 = 20∉ {1,2, 3,4, 5}

Question 5.

a. What is the minimum number of ordered pairs to form a non-zero reflexive relation on a set of n elements?

b. On the set R of real numbers, S is a relation defined as S = {(x,y) I x∈R, y ∈R, x + y = xy}.Find a∈R such that ‘a’ is never the first element of an ordered pair in S. Also find b ∈ R such that ‘b’ is never the second element of an ordered pair in S.

c. Consider the function

Find a function g (x) on a suitable domain such that (gof) (x)= x = (fog) (x). **[March-2015] **Answer:

a. n

Question 6.

a. Let R be the relation on the set N of natural numbers given by R = {(a,b): a-b>2, b>3}. Choose the correct answer. **[March-2014] **(A) (4,1)∈ R

(B) (5,8) ∈ R

(C) (8,7)∈ R

(D) (10,6) ∈ R

b. If / (x) = 8x

^{3}and g(x) = x , find g(x) and g(f(x)) and f(g(x)).

c. Let * be a binary operation on the set Q of rational numbers defined by . Check whether* is commutative and associative?

Answer:

a. (D) (10,6)∈ R

Question 7.

Consider f: R→ R given by/(x)= 5x + 2

a. Show that f is one-one.

b. Is f is invertible? Justify your answer.

c. Let * be a binary operation on N defined by a * b = H.C.F of a and b.

i. Is * commutative?

ii. Is * associative?** [March-2013] **Answer:

a. y = 5x + 2; f(x

_{1}) =f(x

_{2})

⇒ 5x

_{1}+ 2 = 5x

_{2}+ 2; 5x

_{1}= 5x

_{2}

x

_{1}= x

_{2}i.e, f(x) is one-one.

b.

i.e. f(x) = y

Thus/(x) is onto.Thus ‘f” is invertible

c. a * b = H.C.F of a and b

= H.C.F of b and a = b * a

i.e, * is commutative.

a * (b * c) = a * (HCF of b and c)

= HCF of (a, b, c)

(a * b) * c = (HCF of a and b) * c

= HCF of (a, b, c) i.e, * is associative.

Question 8.

(i) * : R x R → R is given by a * b = 3a^{2} – b. Find the value of 2 * 3. Is *commutative? Justify your answer.

(ii) If f:R → R is defined by f(x) = x^{2}– 3x+ 2. Find (fof)(x) and (fof).

Answer:

(i) a * b = 3a^{2} – b

2*3 = 3.2^{2} -3=9

b * a = 3b^{2} – a ≠ a * b

.’. * is not commutative

(ii) (fof)(x)=f(f(x))= f( x^{2} – 3x + 2)

= (x^{2} – 3x + 2)^{2} -3(x^{2} – 3x + 2) + 2

= x^{4} + 9x^{2} + 4 – 6x^{3} – 12x + 4x^{2} – 3x^{2}+ 9x-6 + 2 = x^{4}– 6x^{3} + 10x^{2} – 3x

(fof)(l)= 1-6+ 10-3 = 2.

Question 9.

a. Give an example of a relation on set A = {1, 2, 3, 4} which is reflexive and symmetric but not transitive.

b. Show that f: [-1, 1] → R given by is one-to -one.

c. Let * be a binary operation on defined by .Find the inverse of 9 with respect to*.**[March-2012] **Answer:

a. A = {1, 2, 3, 4}

R= {(1, 1), (2, 2),(3, 3), (4, 4),

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

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

⇒ x_{1}(x_{2} + 2) = x_{2}(x, + 2)

⇒ x_{1} x_{2} + 2x_{1} = x_{1}x_{2} + 2x_{2 }⇒2x_{1} = 2x_{1}⇒ x_{1} = x_{2} i.e. f(x) is one-to-one.

c. Let ‘e’ be the identity then a * e = a

i. e,

If b is the inverse of 9 then 9 * b = e

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