Math Assignment Class XII | Relation and Functions

 Mathematics Assignment on  

RELATIONS & FUNCTIONS Class XII

Important and extra questions on Relations & Functions for class XII, This assignment is strictly based on previous years CBSE question papers.

Question 1

A function f : R₊ → R (where R₊ is the set of all non-negative real numbers) defined by 

f(x) = 4x + 3 is :

(A) one-one but not onto

(B) onto but not one-one

(C) both one-one and onto

(D) neither one-one nor onto

Answer  (A) one-one but not onto

Question 2

Let f : R₊ →[-5, ∞) be defined as f(x) = 9x² + 6x – 5, where R₊ is the set of all non-negative real numbers. Then f is :

A) One-One

B) Onto

C) Bijective

D) Neither one-one nor onto

Answer :  (c ) Bijective

Question 3

Let R₊ Denotes the set of all non-negative real numbers. Then the function f : R₊ → R₊ defined as f(x) = x² + 1 is :

A) one-one but not onto

B) onto but not one-one

C) both one one and onto

D) Neither one one nor onto

Answer :  (A) One -One but not onto

Question 4

A function f : R → R defined as f(x) = x² – 4x + 5 is :

A) injective but not surjective

B) surjective but not injective

C) both injective and surjective

D) neither injective nor surjective

Answer: (D) Neither Injective nor Surjective

Question 5

Show that the function f : R → R defined by   is neither one-one nor onto. Further, find set A so that given function f: R → A becomes an onto function

Answer: For the given function to become onto A = [-1, 1]

Question 6

A relation R is defined on N x N (where N is the natural number)  as :

(a, b) R (c, d) ⇔ a – c = b – d.  Show that R is an equivalence relation.

Answer: Yes R is an equivalence relation

Question 7

A function f is defined from R → R as f(x) = ax + b, such that f(1) = 1 and f(2) = 3. Find function f(x). Hence, check whether function f(x) is one-one and onto or not.

Answer: Yes f(x) is one-one and onto function.

Question 8

A relation R on the set A = {- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4} be defined as R = {(x, y) : x + y is an integer divisible by 2}. Show that R is an equivalence relation. Also, write the equivalence class [2].

Answer: Yes R is an equivalence relation.  Equivalence class [2]  = {- 4, - 2, 0, 2, 4}

Question 9:
Let A = R – {5} and B = R – {1}. Consider the function f : A→B defined by . Show that f is one-one and onto.

Answer: Yes f(x) is one-one and onto function.

Question 10

Check whether the relation S in the set of real numbers R defined by

 S = {(a, b) : Where a - b + √2 is an irrational number} is reflexive symmetric and transitive.

Answer: R is reflexive but neither symmetric nor transitive.

Question 11

A relation R on set A = {1, 2, 3, 4, 5} is defined as

R = {(x, y) : |x² – y²| < 8}.  Check whether the relation R is reflexive, symmetric and transitive.

Answer: R is reflexive, symmetric but not transitive

Question 12

Let    be a function defined as  . Show that f is a one-one function. Also, check whether f is an onto function or not.

Answer:  f(x) is one-one and not onto

Question 13

Let R be the relation defined by R = {(l₁, l₂) : l ₁ is perpendicular to l ₂. Check whether the relation R is an equivalence relation or not .

Answer: Relation R is not an Equivalence relation.(It is symmetric but neither reflexive nor transitive)

Question 14

A function f : A →B defined as f(x) = 2x is both one-one and onto. If A = {1, 2, 3, 4}, Then find the set B.

Solution: f(1) = 2, f(2) = 4, f(3) = 6, f(4) = 8    ∴ B = {2, 4, 6, 8}

Question 15

A relation R is defined on a set of real numbers R as  R = {(x, y) : x.y is an irrational number}. Check whether R is reflexive, symmetric and transitive or not.

Answer: R is symmetric but neither reflexive nor transitive

Question 16

A function f : [ - 4, 4] → [0, 4] is given by . Show that f is an onto function but not a one-one function. Further, find all possible values of a for which f(a) = √7

Answer: f is onto function but not one-one, a = 土 3

Question 17

Show that a function f : R → R defined as  is both one-one and onto.

Answer: Yes f(x) is both one-one and onto

Question 18 (Case study based question)

 Students of a school are taken to a railway museum to learn about railways heritage and its history.

An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by

R = {(l₁, l₂) : l ₁ is parallel to l ₂}

On the basis of the above information, answer the following questions :

(i) Find whether the relation R is symmetric or not.

(ii) Find whether the relation is transitive or not.

(iii) If one of the rail lines on the railway track is represented by the equation y = 3x + 2, then find the set of rail lines in R related to it.

Answer: (i) Yes R is symmetric        (ii) Yes R is transitive

              (iii) The set is {I : I is a line of type y = 3x + c, c ∈ R}

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