Relations & Functions Exercise 1.1 NCERT Solutions Class 12 Math Chapter 1 free PDF Download

February 15, 2025January 7, 2025 by Neeraj Anand, Param Anand

Relation and Function are two ways of establishing links between two sets in mathematics. Relation and Function in maths are analogous to the relation that we see in our daily lives i.e., two persons are related by the relation of father-son, mother-daughter, brother-sister, and many more. On a similar pattern, two numbers can be related to each other as one number is the square of another number and many more. A function is a special kind of relation that is defined as a unique relation between two mathematical entities.

Class 12 NCERT Solutions- Mathematics Chapter 1 Relations And Functions – Exercise 1.1

Question 1. Determine whether each of the following relations is reflexive, symmetric or transitive:

(i) Relation R in the set A={ 1, 2,3, . . ., 13, 14} defined as R={ ( x , y):3x-y=0}

Solution:

A={1 ,2 ,3,…,13, 14}

R={(x,y): 3x-y=0}

Therefore R={(1,3),(2,6),(3,9),(4,12)}

R is not reflexive since (1,1),(2,2),(3,3),…,(14,14)∉R

Also, R is not reflexive since (1,3)∈R, but (3,1)∉R.[since 3(3)-1≠0]

Also, R is not transitive as (1,3), (3,9)∈R, but (1,9)∉R.[since 3(1)-9≠0]

Hence, R is not reflexive, nor symmetric nor transitive.

(ii) Relation R in the set N of natural numbers defined as R={ ( x, y) : y=x+5 and x<4}

Solution:

R={(x, y): y=x+5 and x<4}={(1,6), (2, 7), (3,8)}

It is seen that (1, 1)∉R. Therefore, R is not reflexive.

(1, 6)∈R. But, (6, 1)∉R so, R is not symmetric.

Now, since their is no pair in R such that (x, y) and (y, z)∈R, so (x, z) can not belong to R. Therefore, R is not transitive.

So, we can conclude that R is neither reflexive, nor symmetric, nor transitive.

(iii) Relation R in the set A= {1, 2, 3, 4, 5, 6} as R={(x, y): y is divisible by x}

Solution:

A={1, 2, 3, 4, 5, 6}

R={(x, y): y is divisible by x}

We know that any number is always divisible by itself.⇒ (x, x)∈ R. Therefore, R is reflexive.

We see, (2, 4)∈R [4 is divisible by 2]. But, (4, 2)∉R [2 is not divisible by 4]. Therefore, R is not symmetric.

Let’s assume (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y. Therefore, z is divisible by x.⇒ (x, z) ∈ R. Therefore, R is transitive.

(iv) Relation R in the set Z of all integers defined as R={(x, y): x-y is an integer}

Solution:

R={(x, y): x-y is an integer}

For every x ∈ Z, (x, x) ∈ R [x-x=0 which is an integer]. Therefore, R is reflexive.

For every x, y ∈ Z if (x, y) ∈ R, then x-y is an integer. ⇒-(x-y) is also an integer. ⇒ (y-x) is also an integer. Therefore, R is symmetric.

Let’s assume, (x, y) and (y, z) ∈ R, where x, y and z ∈ Z. ⇒ (x-y) and (y-z) are integers. ⇒ x-z=(x-y)+(y-z) is an integer. ⇒ (x, z) ∈ R. Therefore, R is transitive.

(v) Relation R in a set A of human beings in a town at a particular time, given by:

(a)R={(x,y) : x and y work at the same place.

Solution:

We can see (x,x) ∈ R .Therefore, R is reflexive.

If (x,y) ∈ R, then x and y work at the same place. So, (y,x) ∈ R. Therefore, R is symmetric.

Let, (x,y), (y,z) ∈ R.

⇒ x and y work mat the same place and y and z work at the same place.

⇒ x and z work at the same place.

Therefore, R is transitive.

(b) R={(x,y): x and y live in the same locality}.

Solution:

We can see (x,x) ∈ R. Therefore, R is reflexive.

If (x,y) ∈ R, then x and y live in same locality. So, (y,x) ∈ R. Therefore, R is symmetric.

Let (x,y) ∈ R and (y,z) ∈ R. So, x, y and z live in the same locality. So, (x,z) ∈ R. Therefore, R is transitive.

(c) R={(x,y): x is exactly 7 cm taller than y}.

Solution:

(x,x)∉R since, human being can not be taller than himself. So, R is not reflexive.

Let (x,y) ∈ R ,then x is exactly 7 cm taller than y. Then, y is not taller than x. Therefore, R is not symmetric.

Let (x,y), (y,z) ∈ R, then x is exactly 7 cm taller than y and y is exactly 7 cm taller than z which means x is 14 cm taller than z. So, (x,z)∉R . Therefore, R is not transitive.

(d) R={(x,y): x is wife of y}.

Solution:

(x,x) ∉ R. Since, x can not be the wife of herself. Therefore, R is not reflexive.

Let (x,y) ∈ R, then x is the wife of y. So, y is not the wife of x ,i.e., (y,x) ∉ R. Therefore, R is not symmetric.

Let (x,y), (y,z) ∈ R, then x is the wife of y and y is the wife of z which is not possible. So, R can not be transitive.

(e) R={(x,y): x is father of y}.

Solution:

(x,x) ∉ R. Since, x can not be the father of himself. Therefore, R is not reflexive.

Let (x,y) ∈ R, then x is the father of y. So, y can not be the father of x. So, (y,x) ∉ R. Therefore, R is not symmetric.

Let (x,y), (y,z) ∈ R, then x is the father of y and y is the father of z which means x is the grandfather of z. So, So, (x, z) ∉ R. Therefore, R is not transitive.

Question 2. Show that the relation R in the set R of real numbers, defined as R= {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive.

Solution:

It can be observed that (½, ½) ∉ R, since ½>(½)² =¼. Therefore, R is not reflexive.

(1,4) ∈ R as 1<4² .But, (4,1) ∉ R. Therefore, R is not symmetric.

(3,2), (2,1.5) ∈ R. But, 3> (1.5)² =2.25 . So, (3,1.5) ∉ R. Therefore, R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

Question 3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R={(a,b): b=a+1} is reflexive, symmetric or transitive}.

Solution:

Let the set {1, 2, 3, 4, 5, 6} be named A.

R={(1,2), (2, 3), (3, 4), (4, 5
), (5, 6)}

We can see (x, x) ∉ R. Since, x ≠ x+1. Therefore, R is not reflexive.

It is observed that (1,2) ∈ R but, (2,1) ∉ R. Therefore, R is not symmetric.

We can see, (1,2), (2, 3) ∈ R, but (1,3) ∉ R. Therefore, R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

Question 4. Show that the relation R in R defined as R={ (a, b): a≤ b}, is reflexive and transitive but not symmetric.

Solution:

Clearly, (a,a) ∈ R as a=a. Therefore, R is reflexive.

(2,4) ∈ R (as 2<4) but (4,2) ∉ R as 4 is greater than 2. Therefore, R is not symmetric.

Let (a,b), (b,c) ∈ R. Then, a≤ b and b≤ c.

⇒a ≤ c.

(a, c) ∈ R. Therefore, R is transitive.

Hence, R is reflexive and transitive but not symmetric.

Question 5. Check whether the relation R in R defined as R ={ (a, b): a ≤ b³ } is reflexive, symmetric or transitive.

Solution:

It is observed that (½, ½) ∉ R as ½ > (½)³ =(1/8). Therefore, R is not reflexive.

(1,2) ∈ R(as 1<8) but, (2,1) ∉ R. Therefore, R is not symmetric.

We have, (3, 3/2), (3/2, 6/5) ∈ R but, (3, 6/5) ∉ R. Therefore, R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

Question 6. Show that the relation R in the set {1, 2, 3} given by R={(1,2), (2,1)} is symmetric but neither reflexive nor transitive.

Solution:

Let the set {1, 2, 3} be named A.

It is seen that, (1, 1), (2,2), (3,3)∉ R. Therefore, R is not reflexive.

As (1, 2) ∈ R and (2, 1) ∈ R. Therefore, R is symmetric.

However, (1, 1)∉ R. Therefore, R is not transitive.

Hence, R is symmetric but neither reflexive nor transitive.

Question 7. Show that the relation R in the set A of all the books in a library of a college , given by R={(x,y): x and y have the same number of pages} is a equivalence relation.

Solution:

Set A is the set of all books in the library of a college.

R={(x,y):x and y have the same number of pages}

R is reflexive since (x,x) ∈ R as x and x have the same number of pages.

Let (x,y) ∈ R

⇒x and y have the same number of pages

⇒y and x have the same number of pages.

⇒(y,x)∈ R

Therefore , R is symmetric.

Let (x,y) ∈ R and (y,z)∈ R.

⇒x and y have the same number of pages and y and z have the same number of pages.

⇒x and z have the same number of pages.

⇒(x,z) ∈ R

Therefore, R is transitive.

Hence, R is an equivalence relation.

Solution:

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

R={(a,b):|a-b| is even }

It is clear that for any element a∈ A, we have |a-a|=0 (which is even).

Therefore, R is reflexive.

Let (a,b) ∈ R.

⇒|a-b| is even.

⇒|-(a-b)|=|b-a| is also even.

⇒(b,a)∈ R

Therefore, R is symmetric.

Now , let (a,b)∈ R and (b,c)∈ R.

⇒|a-b| is even and |b-c| is even.

⇒(a-b) is even and (b-c) is even.

⇒(a-c)=(a-b)+(b-c) is even. [Sum of two even integers is even]

⇒|a-c| is even.

Therefore, R is transitive.

Hence, R is an equivalence relation.

All elements of the set {1,2,3} are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even.

Similarly, all elements of the set {2,4} are related to each other as all the elements of this subset are even.

Also, no element of the subset {1,3,5} can be related to any element of {2,4} as all elements of {1,3,5} are odd and all elements of {2,4} are even . Thus, the modulus of the difference between the two elements (from each of these two subsets) will not be even.

Question 9: Show that each of the relation R in the set A={x∈ Z:0<=x<=12} , given by

(i) R={(a,b):|a-b| is a multiple of 4}

(ii) R={(a,b):a=b}

is an equivalence relation. Find the set of all elements related to 1 in each case.

Solution:

A={x∈ Z:0<=x<=12}={0,1,2,3,4,5,6,7,8,9,10,11,12}

(i) R={(a,b):|a-b| is a multiple of 4}

For any element a ∈A , we have (a,a)∈R as |a-a|=0 is a multiple of 4.

Therefore, R is reflexive.

Now , let (a,b)∈R ⇒|a-b| is a multiple of 4.

⇒|-(a-b)|=|b-a| is a multiple of 4.

⇒(b,a)∈R

Therefore, R is symmetric.

Let (a,b) ,(b,c) ∈ R.

⇒|a-b| is a multiple of 4 and |b-c| is a multiple of 4.

⇒(a-b) is a multiple of 4 and (b-c) is a multiple of 4.

⇒(a-c)=(a-b)+(b-c) is a multiple of 4.

⇒|a-c| is a multiple of 4.

⇒(a,c)∈R

Therefore, R is transitive.

Hence, R is an equivalence relation .

The set of elements related to 1 is {1,5,9} since |1-1|=0 is a multiple of 4,

|5-1|=4 is a multiple of 4,and

|9-1|=8 is a multiple of 4.

(ii) R={(a,b):a=b}

For any element a∈A, we have (a,a) ∈ R , since a=a.

Therefore, R is reflexive.

Now , let (a,b)∈R.

⇒a=b

⇒b=a

⇒(b,a)∈R

Therefore, R is symmetric.

Now, let (a,b)∈R and (b,c)∈R.

⇒a=b and b=c

⇒a=c

⇒(a,c)∈R

Therefore, R is transitive.

Hence, R is an equivalence relation.

The elements in R that are related to 1 will be those elements from set A which are equal to 1.

Hence, the set of elements related to 1 is {1].

Question 10: Give an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii)Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive.

Solution:

(i) Let A ={5,6,7}.

Define a relation R on A as R ={(5,6),(6,5)}.

Relation R is not reflexive as (5,5) , (6,6),(7,7)∉R.

Now, as (5,6)∈R and also (6,5)∈R, R is symmetric.

⇒(5,6),(6,5)∈R , but (5,5)∉R

Therefore , R is not transitive.

Hence, relation R is symmetric but not reflexive or transitive.

(ii) Consider a relation R in R defined as:

R= {(a,b): a

For any a∈R, we have (a,a)∉R since a cannot be strictly less than itself . In fact a=a.

Therefore, R is not reflexive. Now,

(1,2)∈R (as 1<2)

But, 2 is not less than 1.

Therefore, (2,1)∉R

Therefore, R is not symmetric.

Now, let (a,b),(b,c)∈R.

⇒a

⇒ (a,c)∈R

Therefore, R is transitive.

Hence, relation R is transitive but not reflexive and symmetric.

(iii) Let A={4,6,8}.

Define a relation R on A as:

A={(4,4),(6,6),(8,8),(4,6),(6,4),(6,8),(8,6)}

Relation R is reflexive since for every a∈A , (a,a)∈R i.e.,(4,4),(6,6),(8,8)∈R.

Relation R is symmetric since (a,b)∈R⇒(b,a)∈R for all a,b∈R.

Relation R is not transitive since (4,6),(6,8)∈R , but (4,8)∉R.

Hence, relation R is reflexive and symmetric but not transitive.

(iv) Define a relation R in R as:

R={ (a,b): a³ ≥ b³ }

Clearly (a,a)∈R as a^3=a^3

Therefore, R is reflexive.

Now, (2,1)∈R(as 2³>=1³)

But, (1,2)∉ R (as 1³< 2³)

Therefore, R is not symmetric

Let (a,b), (b,c) ∈ R.

⇒a³>= b³ and b³ >=c³

⇒a³>=c³

⇒(a, c)∈R

Therefore, R is transitive.

Hence, relation R is reflexive and transitive but not symmetric.

(v) Let A={-5,-6},

Define a relation R on A as:

R={(-5,-6), (-6,-5), (-5,-5)}

Relation R is not reflexive as (-6,-6)∉ R.

Relation R is symmetric as (-5,-6)∈ R and (-6,-5)∈ R.

It is seen that (-5,-6),(-6,-5)∈R . Also , (-5,-5)∈R.

Therefore, the relation R is transitive .

Hence, relation R is symmetric and transitive but not reflexive.

Solution:

We can see (P, P) ∈ R since, the distance of point P from the origin is always the same as the distance of the same point P from the origin. Therefore, R is reflexive.

Let (P,Q)∈ R.

⇒The distance of point P from the origin is the same as the distance of point Q from the origin.

⇒The distance of point Q from the origin is the same as the distance of point P from the origin.

So, (Q,P) ∈ R. Therefore, R is symmetric.

Let (P,Q), (Q,S) ∈ R.

⇒The distance of point P from the origin is the same as the distance of point Q from the origin and also, the distance of point P from the origin is the same.

⇒ The distance of points P and S from the origin is the same.

⇒(P, S) ∈ R. Therefore, R is transitive.

Therefore, R is an equivalence relation.

The set of all points related to P ≠ (0, 0) will be those points whose distance from the origin is the same as the distance of point P from the origin.

In other words, if O(0 0) is the origin and OP = k, then the set of all points related to P is at the same as k from the origin. Hence, this set of points forms a circle with the centre as the origin and the circle passes through point P.

Solution:

R is reflexive since, every triangle is similar to itself.

If (T₁, T₂) ∈ R, then T₁ is similar to T₂. So, (T₂, T₁) ∈ R. Therefore, R is symmetric.

Let (T₁, T₂), (T₂, T₃) ∈ R, then T₁ is similar to T₂ and T₂ is similar to T₃. So, T₁ is also similar to T₃. Therefore, (T₁, T₃) ∈ R so, R is transitive.

Thus, R is an equivalence relation.

We observe,

(3/6)=(4/8)=(5/10)=1/2

Therefore, the corresponding sides of triangles T₁ and T₃ are in the same ratio. Then, triangle T₁ is similar to triangle T₃.

Hence, T₁ is related to T₃.

Solution:

R is reflexive since(P₁, P₂) ∈ R as the same polygon has the same number of sides with itself.

Let (P₁, P₂) ∈ R, then P₁ and P₂ have the same number of sides. So, (P₂, P₁) ∈ R. Therefore, R is symmetric.

Let (P₁, P₂), (P₂, P₃) ∈ R, then P₁ and P₂ have the same number of sides. Also, P₂ and P₃ have the same number of sides. So, P₁ and P₃ have the same number of sides ,i.e., (P₁, P₃) ∈ R. Therefore, R is transitive.

Hence, R is an equivalence relation.

The elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons which have 3 sides(since, T is a polygon with 3 sides).

Hence, the set of all elements in A related to triangle T is the set of all triangles.

Solution:

R is reflexive as any line L₁ is parallel to itself i.e., (L₁, L₂) ∈ R.

Let (L₁, L₂) ∈ R, then L₁ is parallel to L₂. So, (L₂, L₁) ∈ R. Therefore, R is symmetric.

Let (L₁, L₂), (L₂, L₃) ∈ R, then L₁ is parallel to L₂ and L₂ is parallel to L₃. So, L₁ is parallel to L₃. Therefore, R is transitive.

Hence, R is an equivalence relation.

The set of all lines related to the line y=2x +4 is the set of all lines that are parallel to the line y=2x+4.Slope of the line is m=2.

It is known that parallel lines have the same slopes. The line parallel to the given line is of the form y=2x +c, where c ∈ R.

Hence, the set of all lines related to the given line is given by y=2x +c, where c ∈ R.

Question 15. Let R be the relation in the set {1, 2, 3, 4} given by R={(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2)}. Choose the correct answer.

(A) R is reflexive and symmetric but not transitive.

(B) R is reflexive and transitive but not symmetric.

(C) R is symmetric and transitive but not reflexive.

(D) R is equivalence relation.

Solution:

It is seen that (a,a) ∈ R, for every a ∈ {1, 2, 3, 4}. Therefore, R is reflexive.

It is seen that (1,2) ∈ R but (2,1) ∉ R. Therefore, R is not symmetric.

Also, it is observed that (a,b), (b,c) ∈ R⇒ (a,c) ∈ R for all a, b, c ∈ {1, 2, 3, 4}. Therefore, R is transitive.

Hence, R is reflexive and transitive but not symmetric. The correct answer is B.

Question 16. Let R be the relation in the set N given by R={(a,b):a=b-2; b>6}. Choose the correct answer.

(A) (2,4) ∈ R

(B) (3,8) ∈ R

(C) (6,8)

(D) (8,7) ∈ R

Solution:

Since b>a, (2,4) ∉ R also, as 3≠8-2, (3,8)∉R and as 8≠7-2. Therefore, (8,7)∉R

Consider (6,8). We have 8>6 and also, 6=8-2. Therefore, (6,8) ∈ R.

The correct answer is C.

Relations And Functions- FAQs

What is Relation?

Relation in mathematics is defined as the well-defined relationship between two sets. The relation connects the value of the first set with the value of the second set. We represent relation in mathematics using the ordered pair. If we are given two sets Set X and Set Y then the relation between them is represented using the ordered pair (x, y) such that, x ϵ X and y ϵ Y.

What is Reflexive Relation?

The Reflexive Relation means if element “a” is present in set A, then a relation “a” to “a” (aRa) should be present in relation R. If any such aRa is not present in R then R is not a reflexive relation.

What is Transitive Relation?

A relation R on a set A is called transitive relation if and only if

∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R,

where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

What is Symmetric Relation?

A relation R on a set A is called symmetric relation if and only if

∀ a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R and vice versa i.e.,

∀ a, b ∈ A, (a, b) ∈ R\ <=> (b, a) ∈ R,

where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

What is Equivalence Relation?

A relation R on a set A is called an equivalence relation if it is

Reflexive Relation: (a, a) ∈ R ∀ a ∈ A, i.e. aRa for all a ∈ A.

Symmetric Relation: ∀ a, b ∈ A, (a, b) ∈ R ↔ (b, a) ∈ R.

Transitive Relation: ∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.

where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

What is Function?

A function is a relation between two sets set A and set B. Such that every element of set A has an image in set B and no element in set A has more than one image in set B.

What are types of Functions?

There are several types of functions in maths. Some of the important types are:

One to One (Injective) function

Many to One function

Onto (Surjective) Function

Into Function

Weightage of chapter Relation and Function in Board Exam

The chapter Relation and Function consists of 8 Marks Weightage in Board Exams.

It is predicted according to previous year’s papers the chapter Relation and Function will be asked in 2 Marks which will be seen in MCQ.

And rest the marks can be distributed in any section of the Paper.

Neeraj Anand, Param Anand

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.

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CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution

The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.

CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24

Max Marks: 80

No.	Units	Marks
I.	Relations and Functions	08
II.	Algebra	10
III.	Calculus	35
IV.	Vectors and Three – Dimensional Geometry	14
V.	Linear Programming	05
VI.	Probability	08
	Total Theory	80
	Internal Assessment	20
	Grand Total	100

Unit-I: Relations and Functions

1. Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

2. Inverse Trigonometric Functions

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

Unit-II: Algebra

1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin^-1 x, cos^-1 x and tan^-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives

Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3. Integrals

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three – dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit-V: Linear Programming

1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.

Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.

Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26

Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?

Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.

How many units are discussed in the CBSE Class 12 Maths Syllabus?

In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.

How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?

About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.

Class 12 NCERT Solutions- Mathematics Chapter 1 Relations And Functions – Exercise 1.1

Table of Contents

Question 1. Determine whether each of the following relations is reflexive, symmetric or transitive:

(i) Relation R in the set A={ 1, 2,3, . . ., 13, 14} defined as R={ ( x , y):3x-y=0}

(ii) Relation R in the set N of natural numbers defined as R={ ( x, y) : y=x+5 and x<4}

(iii) Relation R in the set A= {1, 2, 3, 4, 5, 6} as R={(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as R={(x, y): x-y is an integer}

(v) Relation R in a set A of human beings in a town at a particular time, given by:

Question 2. Show that the relation R in the set R of real numbers, defined as R= {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.

Question 3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R={(a,b): b=a+1} is reflexive, symmetric or transitive}.

Question 4. Show that the relation R in R defined as R={ (a, b): a≤ b}, is reflexive and transitive but not symmetric.

Question 5. Check whether the relation R in R defined as R ={ (a, b): a ≤ b3 } is reflexive, symmetric or transitive.

Question 6. Show that the relation R in the set {1, 2, 3} given by R={(1,2), (2,1)} is symmetric but neither reflexive nor transitive.

Question 7. Show that the relation R in the set A of all the books in a library of a college , given by R={(x,y): x and y have the same number of pages} is a equivalence relation.

Question 9: Show that each of the relation R in the set A={x∈ Z:0<=x<=12} , given by

Question 10: Give an example of a relation. Which is

Question 13. Show that the relation R defined in the set A of all polygons as R={(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Question 14. Let L be the set of all lines in XY plane and R be the relation in L defined as R={(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y=2x+4.

Question 15. Let R be the relation in the set {1, 2, 3, 4} given by R={(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2)}. Choose the correct answer.

Question 16. Let R be the relation in the set N given by R={(a,b):a=b-2; b>6}. Choose the correct answer.

Relations And Functions- FAQs

What is Relation?

What is Reflexive Relation?

What is Transitive Relation?

What is Symmetric Relation?

What is Equivalence Relation?

What is Function?

What are types of Functions?

Weightage of chapter Relation and Function in Board Exam

CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution

CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24

Unit-I: Relations and Functions

Unit-II: Algebra

Unit-III: Calculus

Unit-IV: Vectors and Three-Dimensional Geometry

Unit-V: Linear Programming

Unit-VI: Probability

Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26

Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?

How many units are discussed in the CBSE Class 12 Maths Syllabus?

How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?

Anand Technical Publishers

Question 2. Show that the relation R in the set R of real numbers, defined as R= {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive.

Question 5. Check whether the relation R in R defined as R ={ (a, b): a ≤ b³ } is reflexive, symmetric or transitive.

Question 13. Show that the relation R defined in the set A of all polygons as R={(P₁, P₂): P₁ and P₂ have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Question 14. Let L be the set of all lines in XY plane and R be the relation in L defined as R={(L₁, L₂): L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y=2x+4.