Updated On : April 3, 2023

Class 12 is a significant milestone in the life of a student. It is the result that the students attain in the 12th boards that play a major role in determining his/her future. Students who are interested in establishing a career in science fields must pay special attention to all the subjects in the science stream. The marks attained in these subjects will determine the kind of opportunities you receive in the future.

Mathematics for class 12 is definitely not easy. Expecting it to be easy is unwise. However, with the aid of the NCERT books, the students can understand the topics a little better. NCERT books combined with focus and determination can help you to achieve a grand score. The NCERT 12th Maths books are specifically curated by experts and professionals. They are highly involved in the educational system and are constantly trying to make changes that can help the students understand the topics a little better.

The **NCERT 12th Maths Book PDF **follows the CBSE syllabus. The most advantageous aspect of this book is that it stresses on building a strong foundation for the students. As such, the language used is simple. The examples and exercises given in the books are carefully selected so that the students are given the opportunity to practice easy, moderate, and hard math problems.

**NCERT 12th Maths Book PDF **consists of two parts. The first part has six chapters and the second part has seven chapters. There are an all total of 13 chapters. Understanding these thirteen chapters in details will not only help you to ace the CBSE board for the 12th but will aid you in clearing any other significant examinations. Check out the information given to know more about the NCERT maths book for class 12 in Hindi and English medium.

- The NCERT 12th Maths book has two parts.
- The simple language aids the students.
- It emphasizes on establishing clear foundations for each of the topics.
- The post elaborates on the chapters, marks allocated for each chapter, and the PDF files for English Medium as well as Hindi medium NCERT maths book of class 12.

The information given below elaborates the NCERT mathematics book for Class 12 in English medium school. The table provides detailed information about the name of the chapters, marks assigned to the chapters, and PDF files. Check out the table to know more about the **NCERT 12th Maths Book PDF **for the English Medium.

SL No. | Chapters | Marks allocated | |

NCERT Class 12 Maths Part 1 | |||

1 | Relations and Functions | 08 | |

2 | Inverse Trigonometric Functions | ||

3 | Matrices | 10 | |

4 | Determinants | ||

5 | Continuity and Differentiability | 20 | |

6 | Application of Derivatives | ||

NCERT Class 12 Maths Part 2 | |||

7 | Integrals | 35 | |

8 | Application of Integrals | ||

9 | Differential Equations | ||

10 | Vector Algebra | 14 | |

11 | Three Dimensional Geometry | ||

12 | Linear Programming | 05 | |

13 | Probability | 08 |

Given below are some questions from each of the chapters of the **NCERT 12th Maths Book PDF **in the English medium.

**Chapter 1 **

- Given, A= {2,3,4} ; B= {2,5,6,7} . Construct an example of each of the following:

(a) an injective mapping from a to B.

(b) a mapping from a to b which is not injective.

(c) a mapping from b to A.

- Let a ={1, 2, 3, ...,9} and R be the relation in A× a defined by (a , b) R (c , d) if a + d = b + c for (a , b), (c , d) in a × a Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].
- Let
*A =*[-1,1], then, discuss whether the following functions defined on*A*are one-one onto or bijective.

(1) *f(x) = x/2*

(2) *g(x) *= |x|

(3) *h(x) *= x |x|

(4) *k(x) *= x2

**Chapter 2**

- Find the value of cot (tan-1 α + cot-1 α)
- Determine the principal value of cos-1( -1/2)
- Find the value of tan-1 √3 – sec-1(–2) is equal to

(A) π (B) – π/3 (C) π/3 (D) 2π/3

**Chapter 3**

Questions could not be included due to the symbols not being copied correctly.

**Chapter 4**

- Show that the points A(a, b+c), B(b,c+a), C (c, a+b) are collinear.
- Find the equation of the line joining A(1, 3) and B(0, 0) using determinants and find k if D(k, 0) is a point such that area of triangle ABD is 3 sq units.
- The cost of 4 kg onion, 3 kg wheat, and 2 kg of rice is Rs 60. The cost of 2 kg onion, 4 kg wheat, and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find the cost of each item per kg by matrix method.

**Chapter 5**

- If f (x) = |cos x|, find f’(3π/4)
- Determine the points of continuity of the composite function y = f[f(x)], given that, f(x) = 1/x-1.
- Explain the continuity of the function f(x) = sin x . cos x

**Chapter 6**

- For the given curve: y = 5x – 2x3, when x increases at the rate of 2 units/sec, then how fast is the slope of curve changes when x = 3?
- What is the equation of the normal to the curve y = sin x at (0, 0)?

(a)x =0 (b) y=0 (c)x+y =0 (d)x-y=0

- A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at a rate of 0.05 cm per second. Find the rate at which its area is increasing if the radius is 3.2 cm.

**Chapter 7**

- Determine ∫tan8x sec4 x dx
- Integrate the given function using integration by substitution: 2x sin(x2+ 1) with respect to x:
- Write the anti-derivative of the following function: 3x2+4x3

**Chapter 8**

- Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant.
- Find the area of the region bounded by the two parabolas y = x2 and y2 = x.
- Find the area enclosed by the ellipse x2/a2 + y2/b2 =1.

**Chapter 9**

- The number of arbitrary constants in the general solution of a differential equation of fourth order is:

(A) 0 (B) 2 (C) 3 (D) 4

- Find the general solution of the differential equation dy/dx =1+y2/1+x2.
- Integrating factor of the differential equation (1 – x2)dy/dx – xy = 1 is

(A) -x

(B) x/ (1 + x2)

(C) √(1- x2)

(D) ½ log (1 – x2)

**Chapter 10**

- Find the unit vector in the direction of the sum of the vectors a⃗ =2i^−j^+2k^ and b⃗ =−i^+i^+3k^.
- Show that the points A, B, and C with position vectors a⃗ =3i^−4j^−4k^, b⃗ =2i^−j^+k^, and c⃗ =i^−3j^−5k^ form the vertices of a right-angled triangle.
- Find all vectors of magnitude 10√3 that are perpendicular to the plane of i^+2j^+k^ and −i^+3j^+4k^.

**Chapter 11**

- Find the direction cosines of the line passing through the two points (– 2, 4, – 5) and (1, 2, 3).
- Find the angle between the pair of lines given below.

(x + 3)/3 = (y -1)/5 = (z + 3)/4

(x + 1)/1 = (y – 4)/1 = (z – 5)/2

- Show that the lines (x – 5)/7 = (y + 2)/-5 = z/1 and x/1 = y/2 = z/3 are perpendicular to each other.

**Chapter 12**

- Solve the following linear programming problem graphically:

Minimise Z = 200 x + 500 y subject to the constraints:

x + 2y ≥ 10

3x + 4y ≤ 24

x ≥ 0, y ≥ 0

- A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food 'I' contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food 'II' contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food 'I' and Rs 70 per kg to purchase Food 'II'. Formulate this problem as a linear programming problem to minimize the cost of such a mixture.
- Solve the following LPP graphically:

Maximise Z = 2x + 3y, subject to x + y ≤ 4, x ≥ 0, y ≥ 0

**Chapter 13**

- The probability of solving the specific problem independently by persons A and B are 1/2 and 1/3 respectively. In case, if both the persons try to solve the problem independently, then calculate the probability that the problem is solved.
- A fair die is thrown double times. Assume that the event A is "odd number on the first throw" and B the event "odd number on the second throw". Compare the independence of the events A and B.
- A die is thrown twice and the sum of the numbers rising is noted to be 6. Calculate the is the conditional probability that the number 4 has arrived at least once?

Comprehensive information about **NCERT 12th Maths Book PDF **for the Hindi medium is given here. The table below provides detailed information regarding the chapters, marks allocation, and the PDF files of the NCERT maths book for class 12 in Hindi.

SL No. | Chapters | Marks allocated | |

NCERT Class 12 Maths Part 1 | |||

1 | संबंध् एवं फलन | 08 | |

2 | प्रतिलोम त्रिकोणमितीय फलन | ||

3 | आव्यूह | 10 | |

4 | सारणिक | ||

5 | सांतत्य तथा अवकलनीयता | 20 | |

6 | अवकलज वेफ अनुप्रयोग | ||

NCERT Class 12 Maths Part 2 | |||

7 | समाकलन | 35 | |

8 | समाकलनों वेफ अनुप्रयोग | ||

9 | अवकल समीकरण | ||

10 | सदिश बीजगणित | 14 | |

11 | त्रि-विमीय ज्यामिति | ||

12 | रैखिक प्रोग्रामन | 05 | |

13 | प्रायिकता | 08 |

The questions could not be copied from the PDF files as the Hindi alphabets translated into different symbols.

The simple way with which the **NCERT 12th Maths Book PDF **explains the various topics makes it vital for the students appearing for the 12th boards. Even though the NCERT books follow the CBSE syllabus, it is also beneficial for the ISC board students. They can use this book to solidify their foundation and ensure their success in any type of examination in the future. Take a look at the reasons why **NCERT 12th Maths Book PDF **is so important and popular.

- Various types of problems: NCERT maths books offer a wide range of math problems. Some are quite difficult to test the understanding of the students. You must also practice the examples given to enhance your knowledge of the particular chapter. When you begin to solve the questions and practice them repeatedly, it also acts as revision material for any upcoming examinations.
- Favourable for other examinations: Not only for the CBSE boards, but you will also find that solving the questions given in the
**NCERT 12th Maths Book PDF**will aid you in passing other examinations. It is because these questions are selected keeping in mind the type of questions that are presented in different national and state-level examinations. - Establishment of fundamental concepts: If you want to score high in any mathematics examinations, you need to make sure that your concepts are clear. It is useless to simply learn or memorize the formulas without understanding it. The NCERT books aid in clearing the concepts in a way that once you understand it, you are unlikely to forget it anytime soon.
- Easy to read: The language selected for the NCERT maths books are easy to read and understand. It is due to the fact that the experts have designed the book in a way that every student irrespective of his/her capability to understand complex words can read it.
- Widen the knowledge base: The
**NCERT 12th Maths Book PDF**offers extensive knowledge on each of the topics. The subjects have been researched thoroughly by the experts to ensure that you gain maximum knowledge about the topics. The information provided is authentic and is the reason why it is favoured by both teachers and students. The minute details in the book can assist in expanding your understanding of the chapters to a great extent.

Check out the list of NCERT books that you can refer for mathematics in class 12.

- Mathematics Exemplar Problem for Class XII, Published by NCERT
- Mathematics Part I - Textbook for Class XII, NCERT Publication
- Mathematics Part II - Textbook for Class XII, NCERT Publication

For the Hindi medium, NCERT also has books for mathematics of class 12 that you can refer to while preparing for the upcoming boards.

- Ganit Bhag I
- Ganit Bhag II

Apart from the **NCERT 12th Maths Book PDF**, the students can also refer to other books for solving mathematical problems in class 12. The more variety you encounter while solving the problems, the more prepared will you be for the upcoming examinations. Some reference books for mathematics of class 12 for all boards are listed below.

- Self Tutor ISC Mathematics for Class 12 by Anil Kr. Sharma
- Understanding ISC Mathematics Class XII by M. L. Aggarwal
- Mathematics Class XII (Set of 2 Volumes) by R. D. Sharma
- Senior secondary school maths – Class 12 by R. S. Aggarwal

April 3, 2023

Class 12 is a significant milestone in the life of a student. It is the result that the students attain in the 12th boards that play a major role in determining his/her future. Students who are interested in establishing a career in science fields must pay special attention to all the subjects in the science stream. The marks attained in these subjects will determine the kind of opportunities you receive in the future.

Mathematics for class 12 is definitely not easy. Expecting it to be easy is unwise. However, with the aid of the NCERT books, the students can understand the topics a little better. NCERT books combined with focus and determination can help you to achieve a grand score. The NCERT 12th Maths books are specifically curated by experts and professionals. They are highly involved in the educational system and are constantly trying to make changes that can help the students understand the topics a little better.

The **NCERT 12th Maths Book PDF **follows the CBSE syllabus. The most advantageous aspect of this book is that it stresses on building a strong foundation for the students. As such, the language used is simple. The examples and exercises given in the books are carefully selected so that the students are given the opportunity to practice easy, moderate, and hard math problems.

**NCERT 12th Maths Book PDF **consists of two parts. The first part has six chapters and the second part has seven chapters. There are an all total of 13 chapters. Understanding these thirteen chapters in details will not only help you to ace the CBSE board for the 12th but will aid you in clearing any other significant examinations. Check out the information given to know more about the NCERT maths book for class 12 in Hindi and English medium.

- The NCERT 12th Maths book has two parts.
- The simple language aids the students.
- It emphasizes on establishing clear foundations for each of the topics.
- The post elaborates on the chapters, marks allocated for each chapter, and the PDF files for English Medium as well as Hindi medium NCERT maths book of class 12.

The information given below elaborates the NCERT mathematics book for Class 12 in English medium school. The table provides detailed information about the name of the chapters, marks assigned to the chapters, and PDF files. Check out the table to know more about the **NCERT 12th Maths Book PDF **for the English Medium.

SL No. | Chapters | Marks allocated | |

NCERT Class 12 Maths Part 1 | |||

1 | Relations and Functions | 08 | |

2 | Inverse Trigonometric Functions | ||

3 | Matrices | 10 | |

4 | Determinants | ||

5 | Continuity and Differentiability | 20 | |

6 | Application of Derivatives | ||

NCERT Class 12 Maths Part 2 | |||

7 | Integrals | 35 | |

8 | Application of Integrals | ||

9 | Differential Equations | ||

10 | Vector Algebra | 14 | |

11 | Three Dimensional Geometry | ||

12 | Linear Programming | 05 | |

13 | Probability | 08 |

Given below are some questions from each of the chapters of the **NCERT 12th Maths Book PDF **in the English medium.

**Chapter 1 **

- Given, A= {2,3,4} ; B= {2,5,6,7} . Construct an example of each of the following:

(a) an injective mapping from a to B.

(b) a mapping from a to b which is not injective.

(c) a mapping from b to A.

- Let a ={1, 2, 3, ...,9} and R be the relation in A× a defined by (a , b) R (c , d) if a + d = b + c for (a , b), (c , d) in a × a Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].
- Let
*A =*[-1,1], then, discuss whether the following functions defined on*A*are one-one onto or bijective.

(1) *f(x) = x/2*

(2) *g(x) *= |x|

(3) *h(x) *= x |x|

(4) *k(x) *= x2

**Chapter 2**

- Find the value of cot (tan-1 α + cot-1 α)
- Determine the principal value of cos-1( -1/2)
- Find the value of tan-1 √3 – sec-1(–2) is equal to

(A) π (B) – π/3 (C) π/3 (D) 2π/3

**Chapter 3**

Questions could not be included due to the symbols not being copied correctly.

**Chapter 4**

- Show that the points A(a, b+c), B(b,c+a), C (c, a+b) are collinear.
- Find the equation of the line joining A(1, 3) and B(0, 0) using determinants and find k if D(k, 0) is a point such that area of triangle ABD is 3 sq units.
- The cost of 4 kg onion, 3 kg wheat, and 2 kg of rice is Rs 60. The cost of 2 kg onion, 4 kg wheat, and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find the cost of each item per kg by matrix method.

**Chapter 5**

- If f (x) = |cos x|, find f’(3π/4)
- Determine the points of continuity of the composite function y = f[f(x)], given that, f(x) = 1/x-1.
- Explain the continuity of the function f(x) = sin x . cos x

**Chapter 6**

- For the given curve: y = 5x – 2x3, when x increases at the rate of 2 units/sec, then how fast is the slope of curve changes when x = 3?
- What is the equation of the normal to the curve y = sin x at (0, 0)?

(a)x =0 (b) y=0 (c)x+y =0 (d)x-y=0

- A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at a rate of 0.05 cm per second. Find the rate at which its area is increasing if the radius is 3.2 cm.

**Chapter 7**

- Determine ∫tan8x sec4 x dx
- Integrate the given function using integration by substitution: 2x sin(x2+ 1) with respect to x:
- Write the anti-derivative of the following function: 3x2+4x3

**Chapter 8**

- Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant.
- Find the area of the region bounded by the two parabolas y = x2 and y2 = x.
- Find the area enclosed by the ellipse x2/a2 + y2/b2 =1.

**Chapter 9**

- The number of arbitrary constants in the general solution of a differential equation of fourth order is:

(A) 0 (B) 2 (C) 3 (D) 4

- Find the general solution of the differential equation dy/dx =1+y2/1+x2.
- Integrating factor of the differential equation (1 – x2)dy/dx – xy = 1 is

(A) -x

(B) x/ (1 + x2)

(C) √(1- x2)

(D) ½ log (1 – x2)

**Chapter 10**

- Find the unit vector in the direction of the sum of the vectors a⃗ =2i^−j^+2k^ and b⃗ =−i^+i^+3k^.
- Show that the points A, B, and C with position vectors a⃗ =3i^−4j^−4k^, b⃗ =2i^−j^+k^, and c⃗ =i^−3j^−5k^ form the vertices of a right-angled triangle.
- Find all vectors of magnitude 10√3 that are perpendicular to the plane of i^+2j^+k^ and −i^+3j^+4k^.

**Chapter 11**

- Find the direction cosines of the line passing through the two points (– 2, 4, – 5) and (1, 2, 3).
- Find the angle between the pair of lines given below.

(x + 3)/3 = (y -1)/5 = (z + 3)/4

(x + 1)/1 = (y – 4)/1 = (z – 5)/2

- Show that the lines (x – 5)/7 = (y + 2)/-5 = z/1 and x/1 = y/2 = z/3 are perpendicular to each other.

**Chapter 12**

- Solve the following linear programming problem graphically:

Minimise Z = 200 x + 500 y subject to the constraints:

x + 2y ≥ 10

3x + 4y ≤ 24

x ≥ 0, y ≥ 0

- A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food 'I' contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food 'II' contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food 'I' and Rs 70 per kg to purchase Food 'II'. Formulate this problem as a linear programming problem to minimize the cost of such a mixture.
- Solve the following LPP graphically:

Maximise Z = 2x + 3y, subject to x + y ≤ 4, x ≥ 0, y ≥ 0

**Chapter 13**

- The probability of solving the specific problem independently by persons A and B are 1/2 and 1/3 respectively. In case, if both the persons try to solve the problem independently, then calculate the probability that the problem is solved.
- A fair die is thrown double times. Assume that the event A is "odd number on the first throw" and B the event "odd number on the second throw". Compare the independence of the events A and B.
- A die is thrown twice and the sum of the numbers rising is noted to be 6. Calculate the is the conditional probability that the number 4 has arrived at least once?

Comprehensive information about **NCERT 12th Maths Book PDF **for the Hindi medium is given here. The table below provides detailed information regarding the chapters, marks allocation, and the PDF files of the NCERT maths book for class 12 in Hindi.

SL No. | Chapters | Marks allocated | |

NCERT Class 12 Maths Part 1 | |||

1 | संबंध् एवं फलन | 08 | |

2 | प्रतिलोम त्रिकोणमितीय फलन | ||

3 | आव्यूह | 10 | |

4 | सारणिक | ||

5 | सांतत्य तथा अवकलनीयता | 20 | |

6 | अवकलज वेफ अनुप्रयोग | ||

NCERT Class 12 Maths Part 2 | |||

7 | समाकलन | 35 | |

8 | समाकलनों वेफ अनुप्रयोग | ||

9 | अवकल समीकरण | ||

10 | सदिश बीजगणित | 14 | |

11 | त्रि-विमीय ज्यामिति | ||

12 | रैखिक प्रोग्रामन | 05 | |

13 | प्रायिकता | 08 |

The questions could not be copied from the PDF files as the Hindi alphabets translated into different symbols.

The simple way with which the **NCERT 12th Maths Book PDF **explains the various topics makes it vital for the students appearing for the 12th boards. Even though the NCERT books follow the CBSE syllabus, it is also beneficial for the ISC board students. They can use this book to solidify their foundation and ensure their success in any type of examination in the future. Take a look at the reasons why **NCERT 12th Maths Book PDF **is so important and popular.

- Various types of problems: NCERT maths books offer a wide range of math problems. Some are quite difficult to test the understanding of the students. You must also practice the examples given to enhance your knowledge of the particular chapter. When you begin to solve the questions and practice them repeatedly, it also acts as revision material for any upcoming examinations.
- Favourable for other examinations: Not only for the CBSE boards, but you will also find that solving the questions given in the
**NCERT 12th Maths Book PDF**will aid you in passing other examinations. It is because these questions are selected keeping in mind the type of questions that are presented in different national and state-level examinations. - Establishment of fundamental concepts: If you want to score high in any mathematics examinations, you need to make sure that your concepts are clear. It is useless to simply learn or memorize the formulas without understanding it. The NCERT books aid in clearing the concepts in a way that once you understand it, you are unlikely to forget it anytime soon.
- Easy to read: The language selected for the NCERT maths books are easy to read and understand. It is due to the fact that the experts have designed the book in a way that every student irrespective of his/her capability to understand complex words can read it.
- Widen the knowledge base: The
**NCERT 12th Maths Book PDF**offers extensive knowledge on each of the topics. The subjects have been researched thoroughly by the experts to ensure that you gain maximum knowledge about the topics. The information provided is authentic and is the reason why it is favoured by both teachers and students. The minute details in the book can assist in expanding your understanding of the chapters to a great extent.

Check out the list of NCERT books that you can refer for mathematics in class 12.

- Mathematics Exemplar Problem for Class XII, Published by NCERT
- Mathematics Part I - Textbook for Class XII, NCERT Publication
- Mathematics Part II - Textbook for Class XII, NCERT Publication

For the Hindi medium, NCERT also has books for mathematics of class 12 that you can refer to while preparing for the upcoming boards.

- Ganit Bhag I
- Ganit Bhag II

Apart from the **NCERT 12th Maths Book PDF**, the students can also refer to other books for solving mathematical problems in class 12. The more variety you encounter while solving the problems, the more prepared will you be for the upcoming examinations. Some reference books for mathematics of class 12 for all boards are listed below.

- Self Tutor ISC Mathematics for Class 12 by Anil Kr. Sharma
- Understanding ISC Mathematics Class XII by M. L. Aggarwal
- Mathematics Class XII (Set of 2 Volumes) by R. D. Sharma
- Senior secondary school maths – Class 12 by R. S. Aggarwal