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### Math Assignment Class XII | Relation and Functions

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# 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) = 9x^{2}
+ 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) = x^{2} + 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) = x ^{2} – 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

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) : |x ^{2} – y^{2}| < 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 = { (l_{1},
l_{2}) : 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 = **åœŸ 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 = { (l_{1}, l_{2}) : 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}**

**THANKS FOR YOUR VISIT**

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