NCERT Solutions For Class 12 Maths Chapter 4 Determinants

Mathematics strengthens logical reasoning and analytical skills, and Chapter 4: Determinants is one of the most important chapters in Class 12 Maths. This chapter focuses on the concept of determinants, their properties, evaluation of determinants up to order three, minors and cofactors, adjoint and inverse of a matrix, and solving systems of linear equations using determinants.

NCERT Solutions For Class 12 Maths Chapter 4 Determinants

Determinants

Difficult

Q.

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)

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Determinants

Medium

Q.

Examine the consistency of the system of equations.

 x + 3y = 5
2x + 6y = 8

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Determinants

Medium

Q.

 $\begin{array}{l}\mathrm{Prove}\quad \mathrm{that}\quad \mathrm{the}\quad \mathrm{determinant}\mathrm{\quad }\left|\begin{array}{ccc}\mathrm{x}& \mathrm{sin\theta }& \mathrm{cos\theta }\\ -\mathrm{sin\theta }& -\mathrm{x}& 1\\ \mathrm{cos\theta }& 1& \mathrm{x}\end{array}\right|\mathrm{\quad }\mathrm{is}\\ \mathrm{independent}\quad \mathrm{of}\quad \quad \mathrm{\theta }\mathrm{.}\end{array}$ 

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Determinants

Difficult

Q.

$\begin{array}{l}\mathbf{If}\hspace{0.17em}\hspace{0.17em}\mathbf{A}=\left|\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right|,\mathrm{\hspace{0.17em}}\mathrm{find} {{\mathrm{A}}^{-}}^1.\mathrm{\hspace{0.17em}}{\mathrm{UsingA}}^{-1}\mathrm{solve} \mathrm{the} \mathrm{system} \mathrm{of}\\ \mathrm{equations}:\\ 2\mathrm{x}-3\mathrm{y}+5\mathrm{z}=11\\ 3\mathrm{x}+2\mathrm{y}-4\mathrm{z}=-5\\ \mathrm{x}+\mathrm{y}-2\mathrm{z}=-3\end{array}$

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Determinants

Difficult

Q.

Solve system of linear equations, using matrix method,

x – y + 2z = 7
3x + 4y – 5z = – 5
2x – y + 3z = 12

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Determinants

Difficult

Q.

Solve system of linear equations, using matrix method,

2x + 3y + 3z = 5
x – 2y + z = – 4
3x – y – 2z = 3

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Determinants

Difficult

Q.

Solve system of linear equations, using matrix method,

x – y + z = 4
2x + y – 3z = 0
x + y + z = 2

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Determinants

Medium

Q.

Examine the consistency of the system of equations.

5x – y + 4z = 5
2x + 3y + 5z = 2
5x –2y + 6z = –1

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Determinants

Medium

Q.

Examine the consistency of the system of equations.

3x – y – 2z = 2
2y – z = – 1
3x –5y = 3

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Determinants

Difficult

Q.

$\begin{array}{l}\mathrm{If} \mathrm{A}=\hspace{0.17em}\hspace{0.17em}\left|\begin{array}{ccc}\hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}\mathbf{2}& -\mathbf{1}& 1\\ -\mathbf{1}& \hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}\mathbf{2}& -\mathbf{1}\mathbf{ }\mathbf{ }\\ \hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}\mathbf{1}& -\mathbf{1}& \hspace{0.17em}\hspace{0.17em}\mathbf{2}\end{array}\right|, \mathrm{verify} \mathrm{that}\\ {\mathrm{A}}^3-6{\mathrm{A}}^2+9\mathrm{A}-4\mathrm{I}=\mathrm{O} \mathrm{and} \mathrm{hence} \mathrm{find} {\mathrm{A}}^{-1}.\end{array}$

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Determinants

Difficult

Q.

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)

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Determinants

Difficult

Q.

$\begin{array}{l}\mathrm{For} \mathrm{the} \mathrm{matrix} \mathrm{A}=[\hspace{0.17em}\hspace{0.17em}\begin{array}{ccc}1& 1& 1\\ 1& 2& -\mathbf{3}\\ 2& -\mathbf{1}& \hspace{0.17em}\hspace{0.17em}\mathbf{3}\end{array}], \mathrm{show} \mathrm{that}\\ {\mathrm{A}}^3-6{\mathrm{A}}^2+5\mathrm{A}+11\mathrm{I}=\mathrm{O}. \mathrm{Hence} \mathrm{find} {\mathrm{A}}^{-1}.\end{array}$

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Determinants

Difficult

Q.

 $\begin{array}{l}\mathrm{For}\quad \mathrm{matrix}\hspace{0.17em}\hspace{0.17em}\mathrm{A}=\left[\begin{array}{l}3\quad \quad 2\\ 1\quad \quad \quad 1\end{array}\right],\quad \mathrm{find}\quad \mathrm{the}\quad \mathrm{number}\quad \mathrm{a}\quad \mathrm{and}\quad \mathrm{b}\quad \mathrm{such}\quad \mathrm{that}\\ {\mathrm{A}}^{2}+\mathrm{aA}+\mathrm{bI}=\mathrm{O}.\end{array}$ 

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Determinants

Difficult

Q.

  $$\mathrm{If\ } A = \begin{bmatrix}3 & 1 \\-1 & 2\end{bmatrix},\ \mathrm{show\ that\ } A^2 - 5A + 7I = O.\ \mathrm{Hence,\ find\ } A^{-1}.$$​​

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Determinants

Medium

Q.

  $$\mathrm{Let\ } A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \mathrm{\ and\ } B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}. \ \mathrm{Verify\ that\ } (AB)^{-1} = B^{-1} A^{-1}.$$​​

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Determinants

Difficult

Q.

Verify A(adj A) = (adj A)A = |A|I    

$\left|\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \mathrm{\hspace{0.17em}} 3\\ -4 -6\end{array}\right|$

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Determinants

Difficult

Q.

 $$\begin{array}{l}\mathrm{Using}\space \mathrm{Co}\space \mathrm{factors}\space \mathrm{of}\space \mathrm{elements}\space \mathrm{of}\space \mathrm{third}\space \mathrm{column},\space \mathrm{evaluate}\\ \mathrm{\Delta }=\left|\begin{array}{l}1\space \; \space \space \mathrm{x}\space \space \mathrm{yz}\\ 1\space \space \space \mathrm{y}\space \space \mathrm{zx}\\ 1\space \space \mathrm{z}\space \; \space \mathrm{xy}\end{array}\right|.\end{array}$$ 

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Determinants

Difficult

Q.

  Using cofactors of elements of second row, evaluate

 $$\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$$​​

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Determinants

Difficult

Q.

(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.

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Determinants

Medium

Q.

 $$\begin{array}{l}\mathrm{Evaluate}:\\ \mathrm{\space }\left|\begin{array}{ccc}\mathrm{cos\alpha cos\beta }& \mathrm{cos\alpha sin\beta }& -\mathrm{sin\alpha }\\ -\mathrm{sin\beta }& \mathrm{cos\beta }& \space 0\\ \mathrm{sin\alpha cos\beta }& \mathrm{sin\alpha sin\beta }& \mathrm{cos\alpha }\end{array}\right|.\end{array}$$ 

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These NCERT Solutions for Class 12 Maths Chapter 4 Determinants are prepared strictly according to the latest CBSE syllabus and NCERT guidelines. All questions are solved in a step-by-step manner using simple explanations to help students understand concepts clearly and perform well in board examinations as well as competitive exams like JEE.

The chapter plays a crucial role in building the foundation for higher-level mathematics and is frequently asked in exams due to its conceptual depth and scoring potential.

Q1. Using cofactors of elements of the second row, evaluate:

Δ =
| 5   3   8 |
| 2   0   1 |
| 1   2   3 |

Solution:

Expanding along the second row:

Δ = 2C₂₁ + 0C₂₂ + 1C₂₃

C₂₁ = − | 3   8 ; 2   3 | = −(9 − 16) = 7

C₂₃ = | 5   3 ; 1   2 | = (10 − 3) = 7

Δ = 2(7) + 1(7) = 21

Q2. For the matrix

A =
3   2
1   1

find numbers a and b such that:

A² + aA + bI = O

Solution:

First find A²:

A² =
(3   2)
(1   1)
×
(3   2)
(1   1)

A² =
11   8
4   3

Now substitute in the given equation:

(11   8; 4   3) + a(3   2; 1   1) + b(1   0; 0   1) = (0   0; 0   0)

Comparing elements:

11 + 3a + b = 0

8 + 2a = 0

4 + a = 0

3 + a + b = 0

From 4 + a = 0 ⇒ a = −4

Substitute in 8 + 2a = 0 ⇒ correct

11 − 12 + b = 0 ⇒ b = 1

a = −4, b = 1

Q3. Find the area of the triangle with vertices:

(i) (1, 0), (6, 0), (4, 3)

Solution:

Area = ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |

= ½ | 1(0 − 3) + 6(3 − 0) + 4(0 − 0) |

= ½ | −3 + 18 |
= 7.5 sq units

Q4. For the matrix

A =
1   1   1
1   2   −3
2   −1   3

show that A is invertible and find A⁻¹.

Solution:

First find |A|:

|A| = 1(6 − 3) − 1(3 − (−6)) + 1(−1 − 4)

|A| = 3 − 9 − 5 = −11 ≠ 0

Since |A| ≠ 0, matrix A is invertible.

A⁻¹ = (1 / |A|) × adj(A)

A⁻¹ =
−1/11 × adj(A)

Hence, inverse of A exists.

FAQs: NCERT Solutions for Class 12 Maths Chapter 4 Determinants

1. Is Chapter 4 Determinants important for the Class 12 board exam?

Yes, Determinants is a high-weightage chapter in the Class 12 Maths board exam. Questions are commonly asked from properties of determinants, inverse of matrices, and solving linear equations.

2. Are NCERT Solutions enough to score well in Determinants?

Yes. Practicing NCERT textbook questions and examples thoroughly is sufficient to score well in this chapter, as CBSE largely follows NCERT-based questions.

3. Does Chapter 4 Determinants come in JEE exams?

Yes. Determinants are an important topic in JEE Main and JEE Advanced, especially in questions related to matrices and systems of linear equations.

4. Which topics in Determinants should be practised more?

Students should focus more on:

• Properties of determinants

• Minors and cofactors

• Adjoint and inverse of a matrix

• Solving linear equations using determinants

5. Are these solutions based on the latest CBSE syllabus?

Yes, these NCERT Solutions for Class 12 Maths Chapter 4 are fully aligned with the latest CBSE curriculum and NCERT textbook.

6. Can these solutions be used for last-minute revision?

Absolutely. The clear, step-wise format makes these solutions ideal for quick revision before exams.