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Class – XI – CBSE - Mathematics Relations and Functions
CBSE NCERT Solutions for Class 11 Mathematics Chapter 02
Back of Chapter Questions
Exercise 2.1
1. If find the values of x and y.
Solution: Step
1:
Given,
By comparing the above step we get,
Overall hint: When two coordinates are equal their abscissa and ordinate are also equal
2. If the set A has elements and the set , then find the number of elements in
and
Step 2:
Therefore,
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Solution: Step
1:
Given that, Set
Number of elements in set
Step 2:
Overall hint: The total elements in product of two sets is The product of elements in individual
sets.
3. If
Solution: Step1:
Given that,
As we know that the Cartesian product of two non-empty sets and is defined as
Step 2:
4. State whether each of the following statement is true or false. If the statement is false, rewrite the
given statement correctly
(i) If
Number of elements in
Therefore, the number of elements in is 9
and , find
Therefore,
Overall hint:
and then
has elements and
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Solution:
Step 1:
False
If
5. State whether each of the following statement is true or false. If the statement is false, rewrite the
given statement correctly Solution:
(ii) If and are non-empty sets, then is a non-empty set of ordered pairs such
that and
Step1:
True
As we know that, If and are non-empty sets, then is a non-empty set of ordered
pairs such that and
Overall hint: If and are non-empty sets, then is a non-empty set of ordered pairs such that and
6. State whether each of the following statement is true or false. If the statement is false, rewrite the
given statement correctly
(iii) If .
Overall Hint: If
then
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Solution:
Step1: True
As,
7. If
Solution:
8. If
Solution:
Step 1:
Given that,
As we know, the Cartesian product of two non-empty sets and is defined as
is the set of all second elements.
find
Find and
Step 2:
Therefore, is the set of all first elements and
Hence, and
Overall hint:
Overall hint:
Step 1:
We know that fo r any non - empty set is defined as
Given,
Overall hint:
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9. Let . Verify that
(i)
Solution:
Step 1;
…(i)
10. Let . Verify that
(ii)
Solution:
Step 1:
Given that, and
Step 2:
…..(ii)
Overall Hint: is Associative property of 3 Sets
To prove is a subset of
and
and
is a subset of
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Overall Hint: For a set to be subset all its elements must be present in another set which may
consist of other elements also.
11. Let . Write . How many subsets will have? List them.
Solution: Step 1:
12. Let and be two sets such that
are in
find and , where
Solution:
Step2:
By this, it is clear that all the elements of are the elements of . Thus, is a
subset of
and
Given that,
Step 2:
It is known that if C is a set with n then n
Thus, the set has subsets. These are
Over all hint: If C is a set with n then n
and . If
a nd are distinct elements.
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Step 1:
Given, are in
It is known that,
Set of first elements of the ordered pair elements of
Set of second elements of the ordered pair elements of .
Step 2:
and z are the elements of
the elements of that are
. Since,
Overall hint: The first elements are in first set and the second elements are in the second set
13. The Cartesian product has elements among which are found . Find the
set and the remaining elements of .
Solution:
Step 2:
and ; and
and
Step 1:
We know that, if and , then .
.
is
that is and and
and .
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Exercises 2.2
1. Let . Define a relation from to by
. Write down its domain, co-domain and range.
Solution:
Step 2:
The domain of will be the set of all first elements of the ordered pairs in the relation
Therefore, Domain of
Step 3:
The whole set
Therefore, Co-domain of
The range of R is the set of all second elements of the ordered pairs in the relation.
Therefore, Range of
The ordered pair and are two of the nine elements of .
It is known that
So, the set A is
The other elements of are and
O verall Hint:
Step 1:
Given relation from to is given as
i.e.,
is the co - domain of relation
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Overall Hint: The domain of will be the set of all first elements of the ordered pairs in the
relation and the range of R is the set of all second elements of the ordered pairs in the relation.
. Depict this relationship using roster form. Write down the domain and the range.
Solution:
The domain of is the set of all first elements of the ordered pairs in the relation
Therefore, Domain of
Step 3:
The Range of is the set of all second elements of the ordered pairs in the relation
Therefore, Range of
Overall hint: The domain of is the set of all first elements of the ordered pairs in the relation
and the Range of is the set of all second elements of the ordered pairs in the relation
3.
. Write in roster
form.
2. Define a relation on the set of natural n umbers by
Step 1
Given,
The natural numbers less than are and .
Therefore,
Step 2:
and Define a relation from to by
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Solution:
Step 1:
Therefore, R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}
Overall Hint: x-y=n/2 +1
4. The figure shows a relationship between the sets and Write this relation
(i) In set-builder form (ii) Roster form.
What is it’s domain and range?
Solution:
Domain of
Range of
Overall hint: In Set builder form we write the generalized equation but in roster form we write all
the elements
Given, and
Step 1:
From the given figure,
( i ) From the figure we can write the relation in set builder form as
Step 2:
( ii )
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5. Let defined by
.
(ii) Find the
domain of
(iii) Find the range of
Solution:
(iii) The Range of is the set of all second elements of the ordered pairs in the relation
Therefore, Range of
Overall hint: The domain of is the set of all first elements of the ordered pairs in the relation
and the Range of is the set of all second elements of the ordered pairs in the relation
6. Determine the domain and range of the relation defined by
Solution:
Step 1:
Given relation,
Therefore,
Step 2:
Step 1:
Given,
( i )
Step 2:
( ii ) The domain of is the set of all first elements of the or dered pairs in the relation
Therefore, Domain of
Step 3:
Let be the relation on
( i) Wri te in roster form
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and
and
and
The domain of is the set of all first elements of the ordered pairs in the relation
Therefore, Domain of
Step 3:
The Range of is the set of all second elements of the ordered pairs in the relation
Therefore Range of
Overall Hint: The domain of is the set of all first elements of the ordered pairs in the relation
and the Range of is the set of all second elements of the ordered pairs in the relation
7. Write the relation in roster form.
Solution:
Step 1:
Given relation is,
We know that
Therefore,
Overall hint: The prime number below 10 makes up domain and the co domain is made up of the
cube of prime numbers below 10
8. Let Find the number of relations from
Solution:
Step 1:
Given that, .
We know that, If are non-empty sets, then is a non-empty set of ordered pairs
such that Step
2:
Therefore,
From the above step, . So, the number of subsets of .
and to
is
is a prime number less than
and are the prime numbers less than
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Thus, the number of relations from to is
Overall hint: , So, the number of subsets of .
9. Let be the relation on defined by . Find the
domain and range of Solution:
Step 1:
Given relation is
We know that the difference between any two integers is always an integer.
Therefore, Domain of
And, Range of
Overall hint: The domain of is the set of all first elements of the ordered pairs in the relation
and the Range of is the set of all second elements of the ordered pairs in the relation
Exercise 2.3
1. Is the following relation are function? Give reasons. If it is a function, determine its domain and
range.
(i)
Solution:
Step 1:
Are the elements of the domain of the given relation.
Step 2:
The elements of the domain of the relation have unique images. So, this relation is function
From the given relation, Domain and Range
is
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and
Overall Hint For a relation to be function it should have a unique image
Is the following relation are function? Give reasons. If it is a function, determine its domain and
range.
(ii)
Solution:
Step 1:
are the elements of the domain of the given relation.
Step 2:
The elements of the domain of the relation have unique images. So, this relation is a function
From the given relation we get, Domain and Range
Overall Hint: For a relation to be function it should have a unique image
Is the following relation are function? Give reasons. If it is a function, determine its domain and
range.
(iii)
Solution:
Step 1:
has two images and . So, this relation is not function.
Whereas for a relation to be a function, the domain elements should have unique images Overall
hint: For a relation to be function it should have a unique image
2. Find the domain and range of the following real function:
(i)
The element
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Solution: Step 1:
(i) Given,
We can also write
is all real numbers except positive real numbers.
Overall hint: Domain is R and range is the where the function exists
Find the domain and range of the following real function:
(ii)
Step2:
Therefore,
As is defined for , the domain of is
It is observed that the range of
Therefore, the range of is
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Solution:
will always lie between and
Overall hint: Domain is R and range is the where the function exists
3. A function is defined by Write down the values of
(i)
Solution:
Step 1:
Overall hint: Substitute x values in given f(x) equation
is defined by Write down the values of
(ii)
Step 1:
Given,
Step 2:
For any value of such that ,
Therefore, the range of
Given function is
Putting in
A function
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Solution:
Step 1:
Overall hint: Substitute x values in given f(x) equation
Write down the values of
Overall hint: Substitute x values in given f(x) equation
4. The function which maps temperature in degree Celsius into temperature in degree Fahrenheit
is defined by Find
Given function is
Putting in
A fun ction is defined by
( iii)
Given function is
Putting in
( i)
( ii)
( iii)
( iv) The value of when
Solution:
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Overall hint: Substitute c values in t(c) equation
Step 1:
Given,
Overall hint: Substitute c values in t(c) equation
Solution:
Step 1:
Given,
Step 1 :
Given,
= 32
The function which maps temperature in degree Celsius into temperature in degree Fahrenheit
is defined by Find
( ii)
Solution:
The function which maps temperature in degree Celsius into temperature in degree Fahrenheit
is defined by Find
( ii )
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Find
when
Overall hint: Substitute c values in t(c) equation
The function which maps temperature in degree Celsius into temperature in degree Fahrenheit
is defined by
(iv) The value of
Solution:
Step 1:
Given,
Given,
Therefore, the value of C when
Overall hint: Substitute t(c) values in c equation
5. Find the range of each of the following functions:
(i)
, is
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Solution:
Overall Hint: Range is where the function exists
Find the range of each of the following functions:
(ii) is a real number
Solution:
Therefore, Range of
Overall Hint: Range is where the function exists
Find the range of each of the following functions: (iii) is a real number
Solution:
Step 1:
Step 1:
If
[ by adding 2 both the sides ]
Therefore, Range of
Step 1:
is a real number
Let be any real number
⇒ [ by adding 2 both the sides ]
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is a real number
From the given function, the range of is the set of all real numbers.
Therefore, Range of
Overall Hint: Range is where the function exists
Miscellaneous Exercise
Solution:
1. The relation is defined by, .
Show that is a function and is not a function.
Step 1:
The relation is defined by
For
For
And at or
We can observe that for , the images of are unique. Therfore, the given relation
is a function.
The relation is defined by
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Step 3:
The element as it’s images
Therefore, the given relation is not a function.
Overall Hint: A function must have unique images
2. If .
Solution:
Overall hint: Take x=1.1 and x=1
3. Find the domain of the function
Solution:
Step 1:
For
For
And at or
has
find
Step 1:
Given,
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is defined for all real numbers except for
.
Overall hint: Domain is where f(x) can be defined
4. Find the domain and the range of the real function defined by Solution:
Step 1:
The given function is
Thus, the domain of is the set of all real numbers greater than or equal to .
Therefore, Domain of
is the set of all real numbers greater than or equal to
.
Overall hint: Domain is where f(x) can be defined and range is where we get f(x) values
Given function is
It i s seen that function
Therefore, Domain of
and
Function
Step 2:
As
Thus, the range of
Therefore, Range of
is defined only when
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5. Find the domain and the range of the real function defined by Solution:
Step 1:
is . The given function
given, | is defined for all real values of . We know that from
Therefore, Domain
of Step 2:
Also, for | assumes all the real values,
Therefore, the range of is the set of all non-negative real numbers.
Overall Hint: Domain is where f(x) can be defined and range is where we get f(x) values
6. Let be a function from into Determine the range of
Solution:
7. Let be defined, respectively by
Step 1:
Giv en function is
For
The range of is the set of all second elements. It can be observed that all these elements are
greater than or equal to but less than .
Therefore, range of
Overall hint: Range of f is where the function exists
Find
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and Solution:
Step 1:
8. Let be a function from defined by
Given, is defined as
Therefore,
Step 2:
Therefore,
Therefore,
Overall hint: (f - g)(x)=f(x) - g(x) and (f+g)(x)=f(x)+g(x)
for some integers Determine t o
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Solution:
in equation (1)
Therefore, the values of and are and respectively.
Overall Hint: Take a,b values from the relation using f(x)=ax+b
from defined by . Are the 9. Let be a relations
following true?
(i) for all
Solution:
Step 1:
is not true.
Overall hint: If one (a,a) isn’t present in the relation then the given statement is false
9. Let be a relations from defined by
. Are the following true?
(ii) implies
Step 1:
Given function is
……….. (1)
…………..(2)
Step 2:
Put the value of
to
As, ; however,
Thus, the statement for all
to
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Solution:
Step 1:
Mark1:1
DL1: E
Overall Hint: when (a,b) is there (b,a) need not be present
10. Let
. Are
the
(iii)
Solution:
We see that because
Now,
implies is not true
Overall hint: If (a,b),(b,c) exists (a,c) may or may not exist
11. Let are the
(i)
We see that because and
We know that, Therefore,
Thus, the statement , implies is not true.
and and
be a relations from to defined by
following true?
implies
Thus, .
Thus, the statement
and
following true?
is a relations from to
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Justify your answer Solution:
Step 1:
Given,
Overall Hint: A relation from a non-empty set to a non-empty set is a subset of the
Cartesian product of
12. Let are the
(ii)
Justify your answer Solution:
Step 1:
As the same first element i.e., has two different images i.e., and
The relation is not a function.
Overall Hint: A pre image must have only one image
and
A relation from a non - empty set to a non - empt y set
is a subset of the Cartesian product of
Therefore,
It is seen that is a subset of
Thus, f is a relation from A to B.
and
following true?
is function from to
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13. Let be the subset of defined by a function
from to
? Justify your answer.
Solution:
Overall hint: function must have single pre image for a image
14. Let and let be defined by the highest prime factor of
Solution:
is defined as the highest prime factor of
Therefore, The highest prime factor of
Is
Step 1:
The relation f is defined as
We know that a relation from se t to set is said to be a function if every element of set
has unique image in set
As i.e., and .
The element has two different images i.e., and .
Therefore, the given relation is not a function.
Find the range of
Step 1:
Given, and
.
Prime factor of
Prime factor of
Prime fact or of
Prime factor of
Prime factor of
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Step 2:
The highest prime factor of
The highest prime factor of
The highest prime factor of
The highest prime factor of
The range of is the set of all where
Therefore, Range of
Overall hint: The range of is the set of all where
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