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:
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

2.
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

Reflexive only
Reflexive and Transitive
Symmetric and Transitive
Symmetric only

View Solution

3.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of $p$ and $q$.

$p = -\frac{2}{3}, \, q = \frac{5}{3}$
$p = -\frac{8}{3}, \, q = \frac{20}{3}$
$p = \frac{20}{3}, \, q = -\frac{8}{3}$
$p = 0, \, q = 0$

View Solution

4.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at $x=0$, then find the value of $k$.

$0$
$-2$
$-1$
$2$

View Solution

5.
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is i/6, where i = 1, 2, 3.
Based on the above information, answer the following questions :

6.
If $ xy = e^{x - y} $, then find $ \frac{dy}{dx} $.

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:
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 \(A'\).

2.
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

3.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

4.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

6.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

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: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 \(A'\).

2.A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

3. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

4. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

6.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
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 \(A'\).

2.
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

3.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

4.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

6.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).