Important Questions for CBSE Class 12 Maths Chapter 4

Important Questions for CBSE Class 12 Maths Chapter 4 – Determinants

We have already learned that algebraic equations can be expressed as matrices. The determinant is used to determine the uniqueness of a solution associated with such matrices. It has numerous applications in engineering, economics, Science and other fields. With real entries, we will learn the determinants up to order 3. Determinants for Class 12 Maths also covers the properties of matrices, cofactors and applications of matrices and determinants, as well as finding the area of a triangle.

The topics covered in determinants for Class 12 Maths include the following:

Introduction
Determinant Definition
Properties
Area of a Triangle
Minors and Cofactors
Adjoint and the inverse of a matrix
Determinants and Matrix Applications

Extramarks Revision Notes for Class 12 Maths Chapter 4 are useful before examinations due to the concise and comprehensive manner in which the content is written.. It is easier to understand and memorise the concepts due to the simple language. These notes are available on the website for better access to students.

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You can also find CBSE Class 12 Maths Important Questions Chapter-by-Chapter Important Questions here:

CBSE Class 12 Maths Important Questions
Sr No	Chapter No	Chapter Name
1	Chapter 1	Relations and Functions
2	Chapter 2	Inverse Trigonometric Functions
3	Chapter 3	Matrices
4	Chapter 4	Determinants
5	Chapter 5	Continuity and Differentiability
6	Chapter 6	Application of Derivatives
7	Chapter 7	Integrals
8	Chapter 8	Application of Integrals
9	Chapter 9	Differential Equations
10	Chapter 10	Vector Algebra
11	Chapter 11	Three Dimensional Geometry
12	Chapter 12	Linear Programming
13	Chapter 13	Probability

CBSE Class 12 Maths Chapter-4 Important Questions

Q.1-

$If value of determinant |\begin{matrix} \sin 80 ° & \cos 80 ° \\ \sin 10 ° & \cos 10 ° \end{matrix}| is$

Opt-

0.
$\frac{1}{\sqrt{2}}$
1/2.
1.

Ans –

$\begin{array}{l} We have, \\ = |\begin{matrix} \sin 80 ° & \cos 80 ° \\ \sin 10 ° & \cos 10 ° \end{matrix}| \\ = \sin 80 ° \cos 10 ° + \cos 80 ° \sin 10 ° \\ = \sin (80 ° + 10 °) = \sin 90 ° = 1 . \end{array}$

Q.2-

$\begin{array}{l} Using properties of determinants, show that \\ |\begin{array}{l} 1 + a 1 1 \\ 1 1 + b 1 \\ 1 1 1 + c \end{array}| = abc (1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = abc + bc + ca + ab . \end{array}$

Opt-

$\begin{array}{l} Let = |\begin{array}{l} 1 + a 1 1 \\ 1 1 + b 1 \\ 1 1 1 + c \end{array}| \\ By applying c_{1} c_{2} and c \end{array}$

Ans-

$\begin{array}{l} Let = |\begin{array}{l} 1 + a 1 1 \\ 1 1 + b 1 \\ 1 1 1 + c \end{array}| \\ By applying c_{1} c_{2} and c_{3} c_{3} c_{2}, we get \\ |\begin{array}{l} a 1 0 \\ b 1 + b b \\ 0 1 c \end{array}| \\ Expanding along R_{1}, we obtain \\ = a |\begin{array}{l} 1 + b b \\ 1 c \end{array}| 1 |\begin{array}{l} b b \\ 0 c \end{array}| + 0 \\ = a (c + bc + b) (bc) \\ = ac + abc + ab + bc \\ = abc + bc + ca + ab \\ = abc [a + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}] \end{array}$

Q.3

$\begin{array}{l} If A = |\begin{array}{l} 1 -1 2 \\ 0 2 -3 \\ 3 -2 4 \end{array}| and B |\begin{array}{l} - 2 0 1 \\ 9 2 -3 \\ 6 1 -2 \end{array}| then find AB . \\ Using AB, solve the folowing system of equations : \\ x y +2 z =1 \\ 2 y 3 z =1 \\ 3 x 2 y +4 z = 2 \end{array}$

Ans-

$\begin{array}{l} Here, AB = |\begin{array}{l} 1 -1 2 \\ 0 2 -3 \\ 3 -2 4 \end{array}| |\begin{array}{l} - 2 0 1 \\ 9 2 -3 \\ 6 1 -2 \end{array}| \\ = |\begin{array}{l} 2 9 +1 2 0 2 + 2 1+3 4 \\ 0 +18 18 0+4 3 0 6 + 6 \\ 6 18 + 24 0 4 +43 +6 8 \end{array}| \\ = |\begin{array}{l} 1 0 0 \\ 0 1 0 \\ 0 0 1 \end{array}| \\ Then AB = I \\ Thus A^{1} = B \\ {|\begin{array}{l} 1 -1 2 \\ 0 2 -3 \\ 3 -2 4 \end{array}|}^{1} = |\begin{array}{l} - 2 0 1 \\ 9 2 -3 \\ 6 1 -2 \end{array}| \\ Now given equations can be written as A_{1} X = B_{1} \\ where, A_{1} = |\begin{array}{l} 1 -1 2 \\ 0 2 -3 \\ 3 -2 4 \end{array}| X = [\begin{array}{l} x \\ y \\ z \end{array}] and B_{1} €‹ = [\begin{array}{l} 1 \\ 1 \\ 2 \end{array}] \\ Now A_{1} = A \\ ˆ´ {A_{1}}^{1} = A^{1} \\ ‡’ {A_{1}}^{1} = |\begin{array}{l} - 2 0 1 \\ 9 2 -3 \\ 6 1 -2 \end{array}| \\ as X = {A_{1}}^{1} B_{1} \\ ˆ´ X = |\begin{array}{l} - 2 0 1 \\ 9 2 -3 \\ 6 1 -2 \end{array}| [\begin{array}{l} 1 \\ 1 \\ 2 \end{array}] \\ X = |\begin{array}{l} - 2 + 0 + 1 \\ 9 +2 -6 \\ 6 +1 -4 \end{array}| [\begin{array}{l} 0 \\ 5 \\ 3 \end{array}] \\ Hence, x = 0, y = 5 and z = 3 \end{array}$

Q.4-

$\begin{array}{l} Show that |\begin{array}{l} a b c \\ a + 2 x b +2 y c +2 z \\ x y z \end{array}| = 0 by \\ using properties of determinants . \end{array}$

Ans-

$\begin{array}{l} |\begin{array}{l} a b c \\ a + 2 x b +2 y c +2 z \\ x y z \end{array}| = |\begin{array}{l} a b c \\ a b c \\ x y z \end{array}| + |\begin{array}{l} a b c \\ 2 x 2 y 2 z \\ x y z \end{array}| \\ (\begin{array}{l} All elements of R_{2} are expressed as sum of \\ two terms . Therefore, the determinant can \\ be expressed as sum of two determinant . \end{array}) \\ = 0 + 2 |\begin{array}{l} a b c \\ a b c \\ x y z \end{array}| (2 is taken out common from R_{2} .) \\ (\begin{array}{l} If any two rows or columns of a determinant are \\ identical then the value of determinant is zero . \end{array}) \\ = 0 + 0 = 0 \end{array}$

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CBSE Class 12 Maths Important Questions

Chapter 1- Relations and Functions

Chapter 2- Inverse Trigonometric Functions

Chapter 3- Matrices

Chapter 5- Continuity and Differentiability

Chapter 6- Application of Derivatives

Chapter 7- Integrals

Chapter 8- Application of Integrals

Chapter 9- Differential Equations

Chapter 10- Vector Algebra

Chapter 11- Three Dimensional Geometry

Chapter 12- Linear Programming

Chapter 13- Probability

FAQs (Frequently Asked Questions)

Chapter 4 Determinants of Class 12 Mathematics explains the fundamental concepts and applications of determinants. Learning Determinants is important because they have numerous applications. A Determinant is a function that can be used to connect any square matrix to a real or complex integer. This principle is used to determine whether an equation system has a unique solution. In addition, Determinants have numerous applications in engineering, Science, Economics and Social Science because it is useful for understanding ‘cross- product’ and to find a solution of a system of equations.

Reflection, all zero, proportionality, switching, factor, scaler, multiple, sum, triangle, cofactor matrix and invariance are the properties of a Determinant. There are various applications of Determinants in all of these properties, such as reflection, which depicts that it does not change with changes in rows and columns, and zero, which depicts that if rows and columns are similar to others, then the determinant will also be zero.

Determinants are very important in algebra. To solve linear equations, you must first understand determinants. This is a significant concept that is also useful in calculating area, volume and other integrals. Determinants can also be used to calculate the values in square matrices.

Important Questions Class 12 Maths Chapter 4