NCERT Solutions for Class 12 Maths Chapter 4 Determinants

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NCERT Solutions for Class 12 Maths Chapter 4 Determinants covers important concepts of Determinants of a matrix and inverse of a matrix. A determinant of the matrix is a scalar value that is calculated using a square matrix. To every square matrix, we can associate a number that is real or complex. The determinant is denoted by det A or |A|. The NCERT Solutions of Chapter 4 Determinants deals with properties of determinants, area of a triangle, minors and cofactors, and applications of matrices and determinants.

The unit algebra comprising Chapter 3 Matrices and Chapter 4 Determinants has a weightage of 10 marks in the final CBSE Board examination. The questions asked from the chapter generally include adjoint and inverse matrices, finding the determinants of a given matrix, and solving a system of linear equations in two or three variables

Download PDF: NCERT Solutions for Chapter 4 Determinants

NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants

Important Topics in Class 12 Mathematics Chapter 4 Determinants

Each square matrix of the order n can associate a number known as determinants of the square matrix A. It can be of orders one, two, and three.

Determinant of order one: Consider a matrix A = [a], the determinant of this matrix is equal to a. Determinant of order two: If the order of the matrix is 2, and the given matrix A is- \([ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22}\\ \end{matrix} ]\) Determinant of A, \|A\| = \(\| \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22}\\ \end{matrix} \|\), = a₁₁.a₂₂- a₂₁.a₁₂ Determinant of order three: If the order of the matrix is 2, and the given matrix A is- \([ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{matrix} ]\) Determinant of A, \|A\| = a₁₁ a₂₂ a₃₃ – a₁₁ a₂₃ a₃₂ – a₁₂ a₂₁ a₃₃ + a₁₂ a₂₃ a₃₁ + a₁₃ a₂₁ a₃₂ – a₁₃ a₃₁ a₂₂

The area of a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is given by – A = ½ [x₁(y₂–y₃) + x₂(y₃–y₁) + x₃(y₁–y₂)].

We can find the area of a triangle using determinants by \(\begin{array}{l}\Delta = \frac{1}{2}\begin{vmatrix} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{vmatrix}\end{array}\)

Minors and Cofactors: Suppose \(\begin{array}{l}\Delta = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}\end{array}\)

Minor = \(\begin{array}{l}M_{f}=\begin{vmatrix} a & b\\ g & h \end{vmatrix}\end{array}\), M_f = a.h – b.g Cofactor of an element a_ij in determinant is defined as A_ij= (-1)^i+jM_ij

NCERT Solutions For Class 12 Maths Chapter 4 Exercises:

The detailed solutions for all the NCERT Solutions for Chapter 4 Determinants under different exercises are as follows:

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CBSE CLASS XII Related Questions

1.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

\(-\frac{\pi}{2}\)
\(-\frac{\pi}{4}\)
\(\frac{\pi}{4}\)
\(\frac{\pi}{2}\)

View Solution

2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

View Solution

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

View Solution

4.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

\(\begin{bmatrix} \frac{1}{3} & 3 \\ 0 & \frac{4}{3} \end{bmatrix}\)
\(\begin{bmatrix} \frac{1}{3} & 0 \\ 3 & \frac{4}{3} \end{bmatrix}\)
\(\begin{bmatrix} \frac{4}{3} & 3 \\ 1 & 0 \end{bmatrix}\)
\(\begin{bmatrix} \frac{4}{3} & 0 1 & \frac{3}{3} \end{bmatrix}\)

View Solution

5.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

View Solution

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

View Solution

NCERT Solutions for Class 12 Maths Chapter 4 Determinants

NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants

Important Topics in Class 12 Mathematics Chapter 4 Determinants

NCERT Solutions For Class 12 Maths Chapter 4 Exercises:

CBSE CLASS XII Related Questions

1.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

5.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

Similar Mathematics Concepts

Comments

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NCERT Solutions for Class 12 Maths Chapter 4 Determinants

NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants

Important Topics in Class 12 Mathematics Chapter 4 Determinants

NCERT Solutions For Class 12 Maths Chapter 4 Exercises:

CBSE CLASS XII Related Questions

1. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

2.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

5.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

2.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

5.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).