NCERT Solutions for Class 12 Maths Free PDF Download Chapterwise

Q: Question 1. Find the rate of change of the area of a circle with respect to its radius r when (i) r=3 cm (ii) r=4 cm A. 6π, 8π B. 5π,8π C. 4π,10π D. 2π,8π

Answer: The area of a circle (A) with radius (r) is given by, A=πr2 Now, the rate of change of the area with respect to its radius is given by, $\frac{dA}{dr}$ = $\frac{d}{dr}$ (πr2)=2πr At r=3 cm, $\frac{dA}{dr}$ =2π(3) = 6π At r=4 cm, $\frac{dA}{dr}$ = 2π(4) =8π

Q: Question 2. Let f:R→R be defined as f(x)=3x. Choose the correct answer. A. f is one-one onto B. f is many-one onto C. f is one-one but not onto D. f is neither one-one nor onto

Answer: f:R→R is defined as f(x)=3x. Let x,y ∈ R such that f(x)=f(y) ⇒3x=3y ⇒x=y ∴f is one-one Also, for any real number (y) in co-domian R, there exists $\frac{y}{3}$ in R such that f($\frac{y}{3}$) =3($\frac{y}{3}$)=y ∴f is onto. Hence, function f is one-one and onto The correct answer is A.

Q: Question 3: Give an example of a relation. Which is Reflexive and symmetric but not transitive.

Answer: 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)∈Ri.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) $\notin$ R. Hence, relation R is reflexive and symmetric but not transitive.

Q: Question 4: 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 an equivalence relation.

Answer: R={(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)} It is seen that (a,a)∈R, for every a∈{1,2,3,4}. ∴ R is reflexive. It is seen that (1,2)∈R, but (2,1) $\notin$ R. ∴ 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} ∴ R is transitive. Hence,R is reflexive and transitive but not symmetric. The correct answer is B.

NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Maths free PDF format is available to download from the links below. NCERT stands for National Council of Education Research and Training is the most preferred curriculum by all the boards like CBSE, Gujarat board, Madhya Pradesh board, etc. The NCERT Solutions for Class 12 Maths will help you to solve the answer the faster and it will clear each and every concept very clearly. NCERT Solutions also helps in cracking many entrance exams like JEE Main, JEE Advanced, NEET, etc. Questions and answers in these solutions will help in scoring better marks in exams.

Download NCERT solutions for other subjects here.

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NCERT Solutions for Class 12 Maths Chapterwise

Chapter 1 – Relations and Functions

Chapter 2 – Inverse Trigonometric Functions

Chapter 3 – Matrices

Chapter 4 – Determinants

Chapter 5 – Continuity and Differentiability

Chapter 6 – Application of Derivatives

Chapter 7 – Integrals

Chapter 8 – Application of Integrals

Chapter 9 – Differential Equations

Chapter 10 – Vector Algebra

Chapter 11 – Three Dimensional Geometry

Chapter 12 – Linear Programming

Chapter 13 – Probability

NCERT Solutions for Class 12 maths

Class 12 Maths Chapterwise NCERT Solutions Free PDF Download

NCERT Solutions of Class 12 Maths Chapter 1 Relations and Functions

In this chapter, you will learn about the Relations between two items and other topics like the Cartesian product of set. You will get more than 40 problems solved in this chapter in this free NCERT solutions PDF.

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

We covered trigonometric functions in Class 11. Now it is time to learn about Inverse Trigonometric functions. In this chapter, you will learn how to measure the angles between the two sides of the right angle and properties of angle. We have solved more than 40 problems in NCERT solutions for inverse trigonometric functions.

NCERT Solutions for Class 12 Maths Chapter 3 Matrices

Matrix is one of the most important applications of Maths. Understand matrices will also help you in further courses. In this chapter, you will learn different topics on matrices like transpose of matrices, multiplication of matrices, etc. Download NCERT Solutions for Matrices here.

NCERT Solutions for Class 12 Maths Chapter 4 Determinants

Learning matrices is not complete without learning Determinants. The inverse of the matrix depends upon the Determinants. In this chapter, Determinant of a matrix, properties of determinants and more. We have solved more than 100 problems for this chapter. Download NCERT solution for Class 12 Maths Determinants here.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

Continuity and differentiability are all about derivates and its algebra. You will get more than 100 problems solved in this NCERT Solutions for Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives

Tangents and normal, Maxima and minima and some more applications of derivatives you will learn in this chapter. Download NCERT Solutions free PDF for applications of derivates here.

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

Integrals are the most important and toughest topic in mathematics. It is very important to understand integrals very clearly and NCERT solutions will definetly help you in that. Our team of experts has solved more than 200 problems in this chapter for you. Download NCERT Solutions for Class 12 Integrals here.

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

Integrals have a lot of applications in Engineering. In this chapter, we will learn how to measure how to find area under simple curves, area between two curves, area bounded by curve and line. We have solved more than 50 problems in this chapter. Download NCERT Solutions for this chapter here.

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations

In this chapter, you will learn about Linear and homogenous Differential equations and how to find the order of differential equations. We have solved more than 50 problems solved in this chapter. Download NCERT Solutions for Differential equations here.

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

Vector algebra is a really important topic for cracking exams like JEE Main and Advanced. This topic can be helpful in real life scenarios too. In this chapter, you will learn different types of vectors, product, and addition of vectors. We have solved more than 50 problems solved in this chapter. Download NCERT Solutions of Vector algebra here.

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

Three Dimensional Geometry is an interesting topic. In this chapter, you will learn to measure the distance between parallel lines, the distance between skew lines, etc. Download NCERT Solutions for three-dimensional geometry here.

NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming

Linear programming problems have wide applications in business mathematics, economics, and statistics. In this chapter, you will learn about types of linear programming, graphical method of linear programming and more. Download NCERT Solutions for Linear programming here.

NCERT Solutions for Class 12 Maths Chapter 13 Probability

Probability is one of the most important topics in Mathematics which helps in real life applications as well. In this chapter, we will learn about Baye’s theorem, Bernoulli trials and more. Download NCERT Solutions for this chapter here.

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Solved Questions For You:

Question 1. Find the rate of change of the area of a circle with respect to its radius $r$ when

(i) $r = 3$ cm
(ii) $r = 4$ cm

A. 6π, 8π

B. 5π,8π

C. 4π,10π

D. 2π,8π

Answer:

The area of a circle

(A)

with radius

(r)

is given by,

A = π r 2

Now, the rate of change of the area with respect to its radius is given by,

$\frac{dA}{dr}$ = $\frac{d}{dr}$ (

π r^{2}) = 2 π r

r = 3

cm,

$\frac{dA}{dr}$

= 2 π (3) = 6 π

r = 4

cm,

$\frac{dA}{dr}$ =

2 π (4) = 8 π

Question 2. Let $f : R \to R$ be defined as $f (x) = 3 x$ . Choose the correct answer.

A. $f$ is one-one onto

B. $f$ is many-one onto

C. $f$ is one-one but not onto

D. $f$ is neither one-one nor onto

Answer:

$f : R \to R$ is defined as $f (x) = 3 x$ .

Let $x, y \in R$ such that $f (x) = f (y)$

$\Rightarrow 3 x = 3 y$
$\Rightarrow x = y$
$∴ f$ is one-one
Also, for any real number $(y)$ in co-domian $R$ ,
there exists $\frac{y}{3}$ in $R$ such that
$f ($\frac{y}{3}$) = 3 ($\frac{y}{3}$) = y$

$∴ f$ is onto.

Hence, function $f$ is one-one and onto
The correct answer is $A$ .

Question 3: Give an example of a relation. Which is Reflexive and symmetric but not transitive.

Answer:

Let $A = {4, 6, 8}$

Define a relation

R

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 \in A, (a, a) \in R i . e ., (4, 4), (6, 6), (8, 8)} \in R

Relation

R

is symmetric since

(a, b) \in R \Rightarrow (b, a) \in R

for all

$a, b \in R$ .

Relation $R$ is not transitive since $(4, 6), (6, 8) \in R$ , but $(4, 8) $\notin$ R$ .

Hence, relation

R

is reflexive and symmetric but not transitive.

Question 4: 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 an equivalence relation.

Answer:

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

It is seen that

(a, a) \in R

, for every

a \in {1, 2, 3, 4}

∴

R

is reflexive.
It is seen that

(1, 2) \in R

, but

(2, 1) \(\notin\) R

∴

R

is not symmetric.
Also, it is observed that

(a, b), (b, c) \in R \Rightarrow (a, c) \in R

for all

a, b, c \in {1, 2, 3, 4}

∴

R

is transitive.
Hence,

R

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

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NCERT Solutions for Class 12 Maths Free PDF Download

NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Maths Chapterwise

Class 12 Maths Chapterwise NCERT Solutions Free PDF Download

NCERT Solutions of Class 12 Maths Chapter 1 Relations and Functions

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

NCERT Solutions for Class 12 Maths Chapter 3 Matrices

NCERT Solutions for Class 12 Maths Chapter 4 Determinants

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming

NCERT Solutions for Class 12 Maths Chapter 13 Probability

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