NCERT Solutions Class 12 Maths Chapter 4 Exercise 4.3

Class 12 plays a very important role in the educational career of a student. The board examinations in Class 12 form the basis for admission into several colleges and universities. The marks obtained in the board examinations are an important criterion for securing admissions to various courses. Hence, students need to score good marks in the board examination. However, preparing for the board examination is not an easy task. It requires a lot of hard work and smart preparation. Especially, for a subject like Mathematics, students have to practice a lot of questions. That is why it is recommended by teachers to practice NCERT questions. The NCERT textbook is also recommended by the CBSE for its board examinations. Hence, students should solve the NCERT exercises.

Class 12 Mathematics Chapter 4 on Determinants is an interesting and challenging topic at the same time. It is an exciting chapter as students get to learn about Determinants, Properties of Determinants, Area of Triangle, Minor, and Cofactors. It can be challenging if students struggle to find the right resources to prepare.

Students usually struggle to find the apt resource to achieve their learning goals. The NCERT Solutions serve as the correct and necessary solution for their learning needs. Having access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 meets the needs of students in their preparation for board examinations.

Furthermore, one of the biggest merits of preparing from the NCERT textbook is that CBSE itself recommends it. The NCERT textbooks are prescribed by CBSE for their school curriculum. The textbooks contain several questions that are relevant for examinations and essential for practice. These questions often tend to puzzle students, making them feel underconfident about the topic and the subject. It is crucial to have a good grasp of the concepts from the very first topic.

Having access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 helps the students in understanding the questions given in the NCERT textbooks while having a strong hold on basic and advanced concepts. Students can build a solid foundational and conceptual understanding of the Maths Class 12 Chapter 4 Exercise 4.3 by solving it. In addition, they can compare their answers with the solutions for self-assessment and improvement.

NCERT Solutions Class 12 Maths Chapter 4 Exercise 4.3

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Access NCERT Solutions for Class 12 Maths Chapter 4 – Determinants

Having access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 helps students get a strong hold on basic and advanced concepts that support their learning even at the graduate level.

Students can learn more about the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 that are accessible on the Extramarks’ website prepared by subject experts. They offer curriculum mapping and full syllabus coverage to make students prepare holistically.

Introduction to Chapter 4 Determinants:

A determinant is a numerical value that is a function of the elements of a square matrix. The chapter on determinants is a high-scoring and crucial chapter from the point of view of the board examinations. Illustrations help students to understand the different sub-topics covered in the chapter. Students must practise the exercises given at the end of each chapter to analyse their understanding of the topic. Having access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 helps students self-evaluate their understanding, and identify their strengths and weaknesses.

Furthermore, the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 provide reviewed and analysed learning while also assisting students in the development of various skills, including logical and reasoning abilities. In addition to that, the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 help students prepare efficiently and effectively.

Area of a Triangle :

Students can find the area of a triangle with the help of determinants. They should solve the questions related to finding the area of a triangle from the exercises to get a good grasp of the concept. Moreover, the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 can guide students in finding the area of a triangle for a clear and better understanding.

An Overview of Exercise 4.3 of Class 12 Maths NCERT Solutions

Students learn to find the area of a triangle whose coordinates of vertices are given. From an examination point of view, it comes across as an essential exercise. The conceptual understanding and its application in this exercise play an important role. Access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 available on Extramarks, benefits students in multiple ways. The NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 assists students in solving difficult questions by providing easy and stepwise solutions to them. The NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 guides the students in developing an ability to solve questions with ease.

Area of Triangle in Determinant Form

The area of a triangle in determinant form is an essential exercise that helps the students to find the area of a triangle. It aids students to understand the ways to find the area of a triangle using determinants. It remains an easy and high-scoring exercise for students. The NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 available at Extramarks are detailed and genuine solutions. The NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 also help the students in achieving good results in the examination. Also, the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 reinforce their fundamentals and will be an important part of their overall preparation.

The Importance of Using NCERT Solutions for Class 12 Maths Part 1 Chapter 4- Determinants :

1) Help achieve positive results:

The NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 are made with appropriate illustrations and examples. Therefore, it increases students’ confidence and helps them excel in the final examination. It aids them to build a conceptual understanding of every topic by providing enough examples for practice.

2) Based on CBSE curriculum:

NCERT textbooks follow the CBSE curriculum completely. The majority of the board examination questions are picked from the NCERT textbooks. Students must practice the exercises rigorously to excel in their board examinations. Furthermore, they can check out the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 available on the Extramarks’ website. The curriculum and study material on the platform is brought together by the in-house experts of the industry. The information provided to the students is fact-checked and fact-based.

3) Detailed understanding of complex topics:

Mathematics requires breaking down complex topics into smaller parts to solve them with ease. Having access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3  will help you solve this problem. Practicing the NCERT textbook questions will boost students’ confidence and takes their understanding of the most difficult topic to another level. Students must compare their answers with the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 for self-assessment and improvement.

4) Preparation of students for entrance examinations:

The students who study with the help of NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 and other solutions can have more chances of cracking the JEE, IIT, BITSAT, and other entrance examinations with lesser effort. The NCERT books prepare students at various levels. The NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 explain the answers in easy language so that they can crack these difficult examinations with proper preparation.

Furthermore, Class 12 board examinations are designed in such a way to test the problem-solving skills of the students to the core. It is a challenging examination, and whoever can crack this one can clear competitive examinations.

5) Strengthens the basics of students:

It helps the students assess the gaps between the question paper and the content of the course. Be it the board examination or any competitive examination, the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 will help them in strengthening their basics and play an important role in their entire preparation period.

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Extramarks contains chapter-wise and topic-wise learning solutions for every subject, including Mathematics. The subject experts work tirelessly to make topics simple for examination preparation and make them easier to grasp. The syllabus covers all NCERT chapters along with the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 and many more.

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Extramarks provides detailed and genuine NCERT solutions, including the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3. Despite many platforms providing such solutions, there remains doubt regarding the credibility of those solutions. Board examinations are of utmost importance and students must use genuine study material. Detailed NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 and other exercises are explained in very simple language. Students learn the right approach to structure their answers. They provide the best and easiest methods to solve numerical problems in mathematics.

Extramarks also provides NCERT Solutions Class 12, NCERT Solutions Class 11, NCERT Solutions Class 10. NCERT Solutions Class 9. NCERT Solutions Class 8, NCERT Solutions Class 7, NCERT Solutions Class 6, NCERT Solutions Class 5, NCERT Solutions Class 4, NCERT Solutions Class 3, NCERT Solutions Class 2, and NCERT Solutions Class 1.

Chapter 4 – Determinants of  Other Exercises

Chapter 4 – Determinants of Other Exercises
Exercise 4.1
8 Questions & Solutions (3 Short Answers, 5 Long Answers)
Exercise 4.2
10 Questions & Solutions (4 Short Answers, 10 Long Answers)
Exercise 4.4
5 Questions & Solutions (2 Short Answers, 3 Long Answers)
Exercise 4.5
18 Questions & Solutions (4 Short Answers, 14 Long Answers)
Exercise 4.6
16 Questions & Solutions (3 Short Answers, 13 Long Answers)

Q.1 Find area of the triangle with vertices at the point given in each of the following:
(i) (1, 0), (6, 0), (4, 3)
(ii) (2, 7), (1, 1), (10, 8)
(iii) (–2, –3), (3, 2), (–1, –8)

Ans

(i) Theare a of triangle formed by given vertices (1, 0),(6, 0),(4, 3)        Δ=12|1 0 16 0 14 3 1|=12{1(0−3)−0(6−4)+1(18−0)}=12(−3−0+18)=12(15)=152 sq. units(ii) Theare a of triangle formed by given vertices (2, 7),(1,1),(10, 8)        Δ=12|2 7 11 1 110 8 1|=12{2(1−8)−7(1−10)+1(8−10)}=12(−14+63−2)=12(47)=472 sq. units(iii) The area of triangle formed by given vertices (−2,−3),          (3,2),(−1,−8)        Δ=12|−2 −3 1    3     2 1−1−8 1|=12{−2(2+8)−(−3)(3+1)+1(−24+2)}=12(−20+12−22)=12(−30)=−15 sq. unitsSince, area of triangle can never be negative. So, area of triangle = 15 square units.

Q.2 Show that points A(a, b + c), B(b, c + a), C(c, a + b) are collinear.

Ans

Theare a of triangle formed by given vertices A (a, b+c),B(b, c+a),C(c, a+b)        Δ=12|ab+c1bc+a1ca+b1|=12{a(c+a−a−b)−(b+c)(b−c)+1(ba+b2−c2−ca)}=12{ac−ab−(b2−c2)+(ba+b2−c2−ca)}=12(ac−ab−b2+c2+ba+b2−c2−ca)=0Since, area of triangle formed by A, B and C is zero, so thesepoints are collinear.

Q.3 Find the value of k if area of triangle is 4 sq. units and vertices are

(i) (k, 0), (4, 0), (0, 2) (ii) (–2, 0), (0, 4), (0, k)

Ans

(i) Given: area of triangle =±4  (∵Since area of triangle is absolute value of Δ)Then the area of triangle formed by given vertices (k,0),          (4,0),(0,2)        Δ=12|k 0 14 0 10 2 1|    ±4=12{k(0−2)−0(4−0)+1(8−0)}    ±4=12(−2k−0+8)    ±8=−2k+8⇒ −2k+8=8  or  −2k+8=−8⇒k=0​ or k=−16−2=8Thus, value of k is either 0 or 8.(ii)  Given: area of triangle =±4  (∵Since area of triangle is absolute value of Δ)Then the area of triangle formed by given vertices(−2,0),  (0, 4),(0, k)        Δ=12|−2 0 1   0 4 1   0 k 1|    ±4=12{−2(4−k)−0(0−0)+1(0−0)}    ±4=12(−8+2k−0+0)    ±8=2k−8⇒ 2k−8=8  or  2k−8=−8⇒k=162= 8​ or k=02=0Thus, value of k is either 8 or 0.

Q.4 (i) Find equation of line joining (1, 2) and (3, 6) using determinants.
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants.

Ans

(i)  Point A(1,2) and B(3,6) are joining each other and a point P(x,y) lieson it, so A, P and B are collinear and area of triangle formed by thesepoints is zero. ThenArea of triangle ABP = 0                    |1 2 13 6 1x y 1|=0⇒12{1(6−y)−2(3−x)+1(3y−6x)}=0⇒                            6−y−6+2x+3y−6x=0⇒                                                                2y−4x=0⇒           y=2xThus, the equation of line joining two points (1, 2) and (3,6)is​ y=2x.(ii)  Point A(3, 1) and B(9,3) are joining each other and a point P(x,y) lieson it, so A, P and B are collinear and area of triangle formed by thesepoints is zero. ThenArea of triangle ABP = 0                    |3 1 19 3 1x y 1|=0⇒12{3(3−y)−1(9−x)+1(9y−3x)}=0⇒                            9−3y−9+x+9y−3x=0⇒                                                                6y−2x=0⇒       3 y=xThus, the equation of line joining two points (1, 2) and (3,6)is​ x−3y=0.

Q.5 If area of triangle is 35 sq. units with vertices (2, -6), (5, 4) and (k, 4). Then k is
(A) 12 (B) –2 (C) –12, –2 (D) 12, –2

Ans

Given: area of triangle =± 35  (∵Since area of triangle is absolute value of Δ)Then the area of triangle formed by given vertices (2,−6),  (5, 4),(k, 4)        Δ=12|2 −6 15    4 1k    4 1|    ±35=12{2(4−4)−(−6)(5−k)+1(20−4k)}    ±35=12(0+30−6k+20−4k)    ±70=−10k+50⇒ −10k+50=70  or  −10k+50=−70⇒k=20−10=−2 or k=−120−10=12Thus, value of k is either 12 or −2.Thus, option (D) is correct.

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FAQs (Frequently Asked Questions)

1. Does access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 help students solve Mathematics question paper in the board examination in the allotted time duration?

Students must practice the NCERT textbook daily to help finish the examination paper in the allotted period. Lack of practice remains one of the biggest reasons for students’ inability to solve the exam in the allotted time frame. Furthermore, the exam paper contains questions from the NCERT textbook. Having access to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 helps students learn how to structure the answers to become efficient for timely completion of the question paper in the examination.

2. Does Extramarks provide NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 and other chapters for Class 12?

Yes, students can access the Class 12 Maths Chapter 4 Exercise 4.3 Solutions on the Extramarks’ website. The solutions are curated by academic experts, making them a reliable source of knowledge. The solutions are explained step-wise and are aligned with the answers mentioned in the NCERT textbooks to help students have an immersive experience. 

3. Do the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 and other exercises aid students to absorb the concept of the different chapters?

For senior classes like Class 12, NCERT solutions are vital for the preparation for the board examinations. Learning modules like the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 allows students to have ample practice before appearing for any examination or test. Answers are not enough because students need to understand the steps to reach the answer. The clarity of the process and steps can ensure that the students develop the ability to solve any question that comes their way.