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# NCERT Solutions for Class 12 Maths Chapter 4 - Exercise Last updated date: 30th Nov 2023
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## NCERT Solutions Class 12 Maths Chapter 4 Exercise 4.3

Vedantu provides students with exclusive access to NCERT Solutions Class 12 Maths Chapter 4. Experienced teachers of Vedantu have formulated the study material on the third chapter of Class 12 Mathematics. The fourth chapter of Class 12 Maths deals with Determinants. On this page, you will easily be able to find all Maths Class 12 Chapter 4 Question Answers in PDF form, which you can easily download as per your convenience.

## Access NCERT Solutions for Class 12 Maths Chapter 4 - Determinants

### Exercise- 4.3

1. Find the area of the triangle with vertices at the point given in each of the following:

i. $\left( \text{1,0} \right)\text{,}\left( \text{6,0} \right)\text{,}\left( \text{4,3} \right)$

Ans: Given vertices,$\left( \text{1,0} \right)\text{,}\left( \text{6,0} \right)\text{,}\left( \text{4,3} \right)$

Thus, the area of the triangle is given by,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{1} & \text{0} & \text{1} \\ \text{6} & \text{0} & \text{1} \\ \text{4} & \text{3} & \text{1} \\ \end{matrix} \right|$

$\Rightarrow \Delta =\dfrac{\text{1}}{\text{2}}\left[ \text{1}\left( \text{0-3} \right)\text{-0}\left( \text{6-4} \right)\text{+1}\left( \text{18-0} \right) \right]$

$\Rightarrow \Delta \text{=}\dfrac{\text{1}}{\text{2}}\left[ \text{-3+18} \right]$

$\Rightarrow \Delta \text{=}\dfrac{\text{15}}{\text{2}}$ square units

$\therefore$Area of the triangle with vertices $\left( \text{1,0} \right)\text{,}\left( \text{6,0} \right)\text{,}\left( \text{4,3} \right)$ is $\dfrac{\text{15}}{\text{2}}$ square units.

ii. $\left( \text{2,7} \right)\text{,}\left( \text{1,1} \right)\text{,}\left( \text{10,8} \right)$

Ans: Given vertices,$\left( \text{2,7} \right)\text{,}\left( \text{1,1} \right)\text{,}\left( \text{10,8} \right)$

The area of the triangle is given by,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{2} & \text{7} & \text{1} \\ \text{1} & \text{1} & \text{1} \\ \text{10} & \text{8} & \text{1} \\ \end{matrix} \right|$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ \text{2}\left( \text{1-8} \right)\text{-7}\left( \text{1-10} \right)\text{+1}\left( \text{8-10} \right) \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ \text{2}\left( \text{-7} \right)\text{-7}\left( \text{-9} \right)\text{+1}\left( \text{-2} \right) \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ \text{-16+63} \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{47}}{\text{2}}$ square units

$\therefore$Area of the triangle with vertices $\left( \text{2,7} \right)\text{,}\left( \text{1,1} \right)\text{,}\left( \text{10,8} \right)$ is $\dfrac{47}{\text{2}}$ square units.

iii. $\text{(-2,-3),}\left( \text{3,2} \right)\text{,(-1,-8)}$

Ans: Given vertices,$\text{(-2,-3),}\left( \text{3,2} \right)\text{,(-1,-8)}$

The area of the triangle with vertices $\text{(-2,-3),}\left( \text{3,2} \right)\text{,(-1,-8)}$ is given by,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{-2} & \text{-3} & \text{1} \\ \text{3} & \text{2} & \text{1} \\ \text{-1} & \text{-8} & \text{1} \\ \end{matrix} \right|$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ \text{-2}\left( \text{2+8} \right)\text{+3}\left( \text{3+1} \right)\text{+1}\left( \text{-24+2} \right) \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ \text{-20+12-22} \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ }=-\dfrac{\text{30}}{\text{2}}$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =-15}$

$\therefore$The area of the triangle with vertices $\left( \text{2,7} \right)\text{,}\left( \text{1,1} \right)\text{,}\left( \text{10,8} \right)$is $\left| \text{-15} \right|\text{=15}$ square units.

2. Show that points $\text{A}\left( \text{a,b+c} \right)\text{,B}\left( \text{b,c +a} \right)\text{,C}\left( \text{c,a +b} \right)$ are collinear.

Ans: To show that the points $\text{A}\left( \text{a,b+c} \right)\text{,B}\left( \text{b,c +a} \right)\text{,C}\left( \text{c,a +b} \right)$ are collinear, the area of the triangle formed by these points as vertices should be zero.

$\therefore$ Area of $\text{ }\!\!\Delta\!\!\text{ ABC}$ is given by,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{a} & \text{b+c} & \text{1} \\ \text{b} & \text{c+a} & \text{1} \\ \text{c} & \text{a+b} & \text{1} \\ \end{matrix} \right|$

Applying the row operations, $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$

$\Rightarrow \Delta \text{=}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{a} & \text{b+c} & \text{1} \\ \text{b-a} & \text{a-b} & \text{0} \\ \text{c-a} & \text{a-c} & \text{0} \\ \end{matrix} \right|$

Taking out $\left( a-b \right)$ and $\left( c-a \right)$ common from $R_2$ and $R_3$ respectively,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left( \text{a-b} \right)\left( \text{c-a} \right)\left| \begin{matrix} \text{a} & \text{b+c} & \text{1} \\ \text{-1} & \text{1} & \text{0} \\ \text{1} & \text{-1} & \text{0} \\ \end{matrix} \right|$

Applying the row operation $R_3\to R_3\text{+}R_2$

$\Rightarrow \Delta \text{=}\dfrac{\text{1}}{\text{2}}\left( \text{a-b} \right)\left( \text{c-a} \right)\left| \begin{matrix} \text{a} & \text{b+c} & \text{1} \\ \text{-1} & \text{1} & \text{0} \\ \text{0} & \text{0} & \text{0} \\ \end{matrix} \right|$

We know that when all the elements of a row or a column in a determinant are zero then the value of the determinant is always zero.

$\therefore \Delta \text{=0}$

Thus, the area of the triangle formed by points $\text{A}$ , $\text{B}$ and $\text{C}$ is zero.

Hence, the points $\text{A}\left( \text{a,b+c} \right)\text{,B}\left( \text{b,c +a} \right)\text{,C}\left( \text{c,a +b} \right)$ are collinear.

3. Find values of $\text{k}$ if area of triangle is $\text{4}$ square units and vertices are:

i. $\left( \text{k,0} \right)\text{,}\left( \text{4,0} \right)\text{,}\left( \text{0,2} \right)$

Ans: Given vertices are $\left( \text{k,0} \right)\text{,}\left( \text{4,0} \right)\text{,}\left( \text{0,2} \right)$.

We know, the area of the triangle is given by,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{k} & \text{0} & \text{1} \\ \text{4} & \text{0} & \text{1} \\ \text{0} & \text{2} & \text{1} \\ \end{matrix} \right|$

$\Rightarrow \Delta \text{=}\dfrac{\text{1}}{\text{2}}\left[ \text{k}\left( \text{0-2} \right)\text{-0}\left( \text{4-0} \right)\text{+1}\left( \text{8-0} \right) \right]$

$\Rightarrow \Delta \text{=}\dfrac{\text{1}}{\text{2}}\left[ \text{-2k+8} \right]$

$\therefore \Delta =-k+4$

Since the area is given to be $\text{4}$ square units, thus

$-k+4=\pm 4$

When $-k+4=-4$

$\therefore k=8$.

When $-k+4=4$

$\therefore k=0$.

Hence, $\text{k=0,8}$.

ii. $\text{(-2,0),}\left( \text{0,4} \right)\text{,}\left( \text{0,k} \right)$

Ans: Given vertices are $\text{(-2,0),}\left( \text{0,4} \right)\text{,}\left( \text{0,k} \right)$.

The area of the triangle is given by,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{-2} & \text{0} & \text{1} \\ \text{0} & \text{4} & \text{1} \\ \text{0} & \text{k} & \text{1} \\ \end{matrix} \right|$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ -2\left( 4-k \right) \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ }=k-4$

Since the area is given to be $\text{4}$ square units, thus

$k-4=\pm 4$

When $k-4=-4$

$\therefore k=0$.

When $k-4=4$

$\therefore k=8$.

Hence, $k=0,8$.

4. Determine the following:

i. Find the equation of line joining $\left( \text{1,2} \right)$ and $\left( \text{3,6} \right)$ using determinants.

Ans: Let us assume a point, $\text{P}\left( \text{x, y} \right)$ on the line joining points $\text{A}\left( \text{1,2} \right)$ and $\text{B}\left( \text{3,6} \right)$ .

Then, the points $\text{A}$,$B$ and $P$ are collinear.

Thus, the area of the triangle $\text{ABP}$ will be zero.

$\therefore \dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{1} & \text{2} & \text{1} \\ \text{3} & \text{6} & \text{1} \\ \text{x} & \text{y} & \text{1} \\ \end{matrix} \right|\text{=0}$

$\Rightarrow \dfrac{\text{1}}{\text{2}}\left[ \text{1}\left( \text{6-y} \right)\text{-2}\left( \text{3-x} \right)\text{+1}\left( \text{3y-6x} \right) \right]\text{=0}$

$\Rightarrow \text{6-y-6+2x+3y-6x=0}$

$\Rightarrow \text{2y-4x=0}$

$\Rightarrow \text{y=2x}$

$\therefore$ The equation of the line joining the given points is $y=2x$.

ii. Find the equation of line joining $\left( \text{3,1} \right)$ and $\left( \text{9,3} \right)$ using determinants.

Ans: Let us assume a point, $\text{P}\left( \text{x, y} \right)$ on the line joining points $\text{A}\left( \text{3,1} \right)$ and $\text{B}\left( \text{9,3} \right)$.

Then, the points $\text{A}$,$B$ and $P$ are collinear.

Thus, the area of the triangle $\text{ABP}$ will be zero.

$\therefore \dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{3} & \text{1} & \text{1} \\ \text{9} & \text{3} & \text{1} \\ \text{x} & \text{y} & \text{1} \\ \end{matrix} \right|\text{=0}$

$\Rightarrow \dfrac{\text{1}}{\text{2}}\left[ \text{3}\left( \text{3-y} \right)\text{-1}\left( \text{9-x} \right)\text{+1}\left( \text{9y-3x} \right) \right]\text{=0}$

$\Rightarrow \text{9-3y-9+x+9y-3x=0}$

$\Rightarrow \text{6y-2x=0}$

$\Rightarrow \text{x-3y=0}$

$\therefore$ The equation of the line joining the given points is $\text{x-3y=0}$ .

5. If the area of triangle is $\text{35}$ square units with vertices $\text{(2,-6)}$ , $\left( \text{5,4} \right)$ and $\left( \text{k,4} \right)$ . Then $\text{k}$ is

1. $\text{12}$

2. $\text{-2}$

3. $\text{-12,-2}$

4. $\text{12,-2}$

Ans: Given vertices, $\text{(2,-6)}$,$\left( \text{5,4} \right)$ and $\left( \text{k,4} \right)$

The area of the triangle is given by,

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left| \begin{matrix} \text{2} & \text{-6} & \text{1} \\ \text{5} & \text{4} & \text{1} \\ \text{k} & \text{4} & \text{1} \\ \end{matrix} \right|$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ \text{2}\left( \text{4-4} \right)\text{+6}\left( \text{5-k} \right)\text{+1}\left( \text{20-4k} \right) \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =}\dfrac{\text{1}}{\text{2}}\left[ \text{50-10k} \right]$

$\Rightarrow \text{ }\!\!\Delta\!\!\text{ =25-5k}$

Given, the area of the triangle is $\text{35}$ square units .

Thus, we have:

$\Rightarrow 25-5k=\pm 35$

$\Rightarrow 5\left( 5-k \right)=\pm 35$

$\Rightarrow 5-k=\pm 7$.

When $5-k=7$

$\therefore k=-2$.

When $5-k=-7$

$\therefore k=12$.

Hence, $k=12,-2$ .

Thus, D. $12,-2$ is the correct option.

## Introduction to Chapter 4 Determinants:

You must have learned in the earlier classes about matrix and matrices. For every square matrix, you can assign a real number. It is called the Determinant.

Determinant of Matrices of Order 1:

If A = [a ] be the matrix of order 1, then Determinant of A is said to be equal to a.

Determinant of Matrices of Order 2 ☓ 2:

For matrices with order 2, the Determinant is obtained through the multiplication of matrices.

Determinant of Matrices of Order 3 ☓ 3:

For matrices with order 3 ☓ 3, the Determinant is obtained by a series of steps explained in the NCERT solutions for Class 12 Maths Part 1 Chapter 4 Determinants.

Properties of Determinants :

Chapter 4, Determinants, gives us a few properties.

• Property 1: If you interchange the rows and columns, the value of Determinants will not change.

• Property 2: If two rows or 2 columns are interchanged, then the value of Determinants will change concerning sign.

• Property 3:  If two-row or columns are the same, then the value of Determinants is zero.

• Property 4: If each element in matrix is multiplied by a constant k, then the value of Determinants will also be multiplied by k.

• Property 5: If the sum of two terms expresses elements, then the value of Determinants will also be expressed as the sum

### Area of a Triangle :

As we have seen in earlier classes, the area of a triangle is half into base into height.

Here, the value of the area of the triangle is expressed as half of the value of the matrix of order 3 ☓ 3.

Exercise 4.3 has five questions based on these concepts.

For detailed and correct solutions for Exercise 4.3, you can download our NCERT Solutions for Class 12 Maths Part 1 Chapter 4 Determinants.

### An Overview of Exercise 4.3 of Class 12 Maths NCERT Solutions

Area of Triangle in Determinant Form

When the coordinates of the triangle's vertices are given, the area of the triangle in determinant form is calculated in coordinate geometry. One of the most important applications of determinants is finding the area of a triangle in determinant form. The area of a triangle is usually calculated using the formula half the product of the triangle's base and altitude. However, if the triangle's height is unknown but its vertices are known, the determinant formula can be used to calculate the triangle's area.

The Formula for Finding the Area of a Triangle in Determinant Form

In determinant form, the formula for the area of a triangle gives a scalar value that can be positive or negative. However, because the area of a triangle can never be negative, we consider the determinant's absolute value as the triangle's area. If we have the vertices of a triangle ABC as A(x1, y1), B(x2, y2), and C(x3, y3) in a cartesian plane, then we can calculate the area of the triangle in determinant form with the below-given formula:

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Vedantu is responsible for carving the bright futures of many Class 12 students. The language of the NCERT solutions is kept simple and lucid for better understanding. All our solutions are in pdf versions. This makes it easy to download and accessible widely by the students. With step by step illustration of the solution for exercise 4.3, you are sure to score better marks in Maths Part 1 Class 12. For those students who would not want to miss any single chance of studying this year, we have devised our student-friendly app to view the solutions on the go.

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### Access Other Exercises of Class 12 Maths Chapter 4

 Chapter 4 Determinants  All Exercises in PDF Format Exercise 4.1 8 Questions and Solutions Exercise 4.2 16 Questions and Solutions Exercise 4.4 5 Questions and Solutions Exercise 4.5 18 Questions and Solutions Exercise 4.6 16 Questions and Solutions

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## FAQs on NCERT Solutions for Class 12 Maths Chapter 4 - Exercise

1. Can I download NCERT Solutions Class 12 Maths Chapter 4 from Vedantu?

Yes, you can download the Class 12 Maths NCERT Solutions Chapter 4 by signing into the Vedantu’s website. These solutions are solved by our expert teachers in a very easy and step by step method. We have also added class 12 Maths Chapter 4 exercise 4.3 pdf on this site. Regular solving of these solutions will help the students to secure better marks.

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Yes, the pdfs provided by Vedantu for various NCERT Solutions and notes are free of cost. Vedantu is an online learning platform which provides free PDF downloads of solved NCERT Solutions and sample papers along with previous years solutions for various state boards and CBSE exams. These solutions are solved by expert teachers on Vedantu.com.

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3. Which books shall I refer along with NCERT Class 12 Maths Textbook?

Although NCERT textbooks provide detailed information and solutions for every chapter, it is advisable to the students to refer to R.S. Agarwal and R.D. Sharma textbooks for better understanding. There are proper chapter-wise solutions provided in the textbook which are categorised as very short, short, long, multiple-type and high order thinking type questions. Therefore, CBSE students can refer to both R.S. Agarwal and R. D. Sharma books for gaining better concepts regarding any topic. The students can also download free PDFs of solved R.S.Agarwal's questions from the Vedantu’s website. These solutions are solved by our expert teachers who are having years of experience in this subject.

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Class 12 final examination holds an important part in every student's life. Therefore, they need to solve the NCERT solutions and practice papers regularly. Candidates aiming for JEE should have a good grasp over the maths subject. They are advised to revise the important mathematical formulas and attempt maximum mock tests for better understanding and marks.

Before starting their preparation, students must have a deep understanding of the Class 12 maths syllabus and should focus on his/her weak areas. He/She must also follow a regular time table and segregate the syllabus accordingly.