NCERT solutions for Class 12 Maths chapter 4 - Determinants [Latest edition]

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Solutions for Chapter 4: Determinants

Below listed, you can find solutions for Chapter 4 of CBSE, Karnataka Board PUC NCERT for Class 12 Maths.


Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6Exercise 4.7
Exercise 4.1 [Pages 108 - 109]

NCERT solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.1 [Pages 108 - 109]

Exercise 4.1 | Q 1 | Page 108

Evaluate the determinants in Exercises 1 and 2. 

`|(2,4),(-5, -1)|`

Exercise 4.1 | Q 2.1 | Page 108

Evaluate the determinants in Exercises 1 and 2. 

`|(cos theta, -sin theta),(sin theta, cos theta)|`

Exercise 4.1 | Q 2.2 | Page 108

Evaluate the determinants in Exercises 1 and 2.

`|(x^2-x-1, x -1),(x+1, x+1)|`

Exercise 4.1 | Q 3 | Page 108

if A = `[(1,2),(4,2)]` then show that |2A| = 4|A|

Exercise 4.1 | Q 4 | Page 108

if A=`[(1,0,1),(0,1,2),(0,0,4)]` then show that `|3A| = 27|A|`

Exercise 4.1 | Q 5.1 | Page 108

Evaluate the determinants `|(3,-1,-2),(0,0,-1),(3,-5,0)|`

Exercise 4.1 | Q 5.2 | Page 108

Evaluate the determinants `|(0,1,2),(-1,0,-3),(-2,3,0)|`

Exercise 4.1 | Q 5.3 | Page 108

Evaluate the determinants `|(3,-4,5),(1,1,-2),(2,3,1)|`

Exercise 4.1 | Q 5.4 | Page 108

Evaluate the determinants

`|(2,-1,-2),(0,2,-1),(3,-5,0)|`

Exercise 4.1 | Q 6 | Page 109

if A  = `[(1,1,-2),(2,1,-3),(5,4,-9)]`, Find |A|

Exercise 4.1 | Q 7.1 | Page 109

Find values of x, if 2451=2x46x

Exercise 4.1 | Q 7.2 | Page 109

Find values of x, if `|[2,3],[4,5]|=|[x,3],[2x,5]|`

Exercise 4.1 | Q 8 | Page 109

if `|(x, 2),(18, x)| = |(6,2),(18,6)|`, then x is equal to 

(A) 6

(B) ±6

(C) −6

(D) 0

Exercise 4.2 [Pages 119 - 121]

NCERT solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.2 [Pages 119 - 121]

Exercise 4.2 | Q 1 | Page 119

Using the property of determinants and without expanding, prove that:

`|(x, a, x+a),(y,b,y+b),(z,c, z+ c)| = 0`

Exercise 4.2 | Q 2 | Page 119

Using the property of determinants and without expanding, prove that:

`|(a-b,b-c,c-a),(b-c,c-a,a-b),(a-a,a-b,b-c)| = 0`

Exercise 4.2 | Q 3 | Page 119

Using the property of determinants and without expanding, prove that:

`|(2,7,65),(3,8,75),(5,9,86)| = 0`

Exercise 4.2 | Q 4 | Page 119

Using the property of determinants and without expanding, prove that:

`|(1, bc, a(b+c)),(1, ca, b(c+a)),(1, ab, c(a+b))| = 0`

Exercise 4.2 | Q 5 | Page 119

Using the property of determinants and without expanding, prove that:

`|(b+c, q+r, y+z),(c+a, r+p, z +x),(a+b, p+q, x + y )| = 2|(a,p,x),(b,q,y),(c, r,z)|`

Exercise 4.2 | Q 6 | Page 120

By using properties of determinants, show that:

`|(0,a, -b),(-a,0, -c),(b, c,0)| = 0`

Exercise 4.2 | Q 7 | Page 120

By using properties of determinants, show that:

`|(-a^2, ab, ac),(ba, -b^2, bc),(ca,cb, -c^2)| = 4a^2b^2c^2`

Exercise 4.2 | Q 8 | Page 120

By using properties of determinants, show that:

`|(1,a,a^2),(1,b,b^2),(1,c,c^2)| = (a - b)(b-c)(c-a)`

Exercise 4.2 | Q 8.2 | Page 120

By using properties of determinants, show that:

`|(1,1,1),(a,b,c),(a^3, b^3,c^3)|` = (a-b)(b-c)(c-a)(a+b+c)

Exercise 4.2 | Q 9 | Page 120

By using properties of determinants, show that:

`|(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)| = (x-y)(y-z)(z-x)(xy+yz+zx)`

Exercise 4.2 | Q 10.1 | Page 120

By using properties of determinants, show that:

`|(x+4,2x,2x),(2x,x+4,2x),(2x , 2x, x+4)| = (5x + 4)(4-x)^2`

Exercise 4.2 | Q 10.2 | Page 120

By using properties of determinants, show that:

`|(y+k,y, y),(y, y+k, y),(y, y, y+k)| = k^2(3y + k)`

Exercise 4.2 | Q 11.1 | Page 120

By using properties of determinants, show that:

`|(a-b-c, 2a,2a),(2b, b-c-a,2b),(2c,2c, c-a-b)| = (a + b + c)^2`

Exercise 4.2 | Q 11.2 | Page 120

By using properties of determinants, show that:

`|(x+y+2z, x, y),(z, y+z+2z,y),(z,x,z+x+2y)| = 2(x+y+z)^3`

Exercise 4.2 | Q 12 | Page 121

By using properties of determinants, show that:

`|(1,x,x^2),(x^2,1,x),(x,x^2,1)| = (1-x^3)^2`

Exercise 4.2 | Q 13 | Page 121

By using properties of determinants, show that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1+a^2+b^2)`

Exercise 4.2 | Q 14 | Page 121

By using properties of determinants, show that:

`|(a^2+1, ab, ac),(ab, b^2+1, bc),(ca, cb, c^2+1)| = 1+a^2+b^2+c^2`

Exercise 4.2 | Q 15 | Page 121

Let A be a square matrix of order 3 × 3, then | kA| is equal to

(A) k|A|

(B) k2 | A |

(C) k3 | A |

(D) 3k | A |

Exercise 4.2 | Q 16 | Page 121

Which of the following is correct?

A. Determinant is a square matrix.

B. Determinant is a number associated to a matrix.

C. Determinant is a number associated to a square matrix.

D. None of these

Exercise 4.3 [Pages 122 - 123]

NCERT solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.3 [Pages 122 - 123]

Exercise 4.3 | Q 1.1 | Page 122

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

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

Exercise 4.3 | Q 1.2 | Page 122

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

(2, 7), (1, 1), (10, 8)

Exercise 4.3 | Q 1.3 | Page 122

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

(−2, −3), (3, 2), (−1, −8)

Exercise 4.3 | Q 2 | Page 123

Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

Exercise 4.3 | Q 3.1 | Page 123

Find values of k if area of triangle is 4 square units and vertices are (k, 0), (4, 0), (0, 2)

 

Exercise 4.3 | Q 3.2 | Page 123

Find values of k if area of triangle is 4 square units and vertices are (−2, 0), (0, 4), (0, k)

Exercise 4.3 | Q 4.1 | Page 123

Find equation of line joining (1, 2) and (3, 6) using determinants

Exercise 4.3 | Q 4.2 | Page 123

Find equation of line joining (3, 1) and (9, 3) using determinants

Exercise 4.3 | Q 5 | Page 123

If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is

A. 12

B. −2

C. −12, −2

D. 12, −2

Exercise 4.4 [Page 126]

NCERT solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.4 [Page 126]

Exercise 4.4 | Q 1.1 | Page 126

Write Minors and Cofactors of the elements of following determinants:

`|(2,-4),(0,3)|`

Exercise 4.4 | Q 1.2 | Page 126

Write Minors and Cofactors of the elements of following determinants:

`|(a,c),(b,d)|`

Exercise 4.4 | Q 2.1 | Page 126

Write Minors and Cofactors of the elements of following determinants:

`|(1,0,0),(0,1,0),(0,0,1)|`

Exercise 4.4 | Q 2.2 | Page 126

Write Minors and Cofactors of the elements of following determinants:

`|(1,0,4),(3,5,-1),(0,1,2)|`

Exercise 4.4 | Q 3 | Page 126

Using Cofactors of elements of second row, evaluate `triangle = |(5,3,8),(2,0,1),(1,2, 3)|`

Exercise 4.4 | Q 4 | Page 126

Using Cofactors of elements of third column, evaluate `triangle = |(1,x,yz),(1,y,zx),(1,z,xy)|`

Exercise 4.4 | Q 5 | Page 126

if `triangle = |(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|` and Aij is Cofactors of aij, then value of Δ is given by

(A) a11 A31+ a12 A32 + a13 A33

(B) a11 A11+ a12 A21 + a13 A31

(C) a21 A11+ a22 A12 + a23 A13

(D) a11 A11+ a21 A21 + a31 A31

 

Exercise 4.5 [Pages 131 - 132]

NCERT solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.5 [Pages 131 - 132]

Exercise 4.5 | Q 1 | Page 131

Find adjoint of each of the matrices.'

`[(1,2),(3,4)]`

Exercise 4.5 | Q 2 | Page 131

Find adjoint of each of the matrices.

`[(1,-1,2),(2,3,5),(-2,0,1)]`

Exercise 4.5 | Q 3 | Page 131

Verify A (adj A) = (adj A) A = |A|I

`[(2,3),(-4,-6)]`

Exercise 4.5 | Q 4 | Page 131

Verify A (adj A) = (adj AA = |A|I

`[(1,-1,2),(3,0,-2),(1,0,3)]`

Exercise 4.5 | Q 5 | Page 132

Find the inverse of each of the matrices (if it exists).

`[(2,-2),(4,3)]`

Exercise 4.5 | Q 6 | Page 132

Find the inverse of each of the matrices (if it exists).

`[(-1,5),(-3,2)]`

Exercise 4.5 | Q 7 | Page 132

Find the inverse of each of the matrices (if it exists).

`[(1,2,3),(0,2,4),(0,0,5)]`

Exercise 4.5 | Q 8 | Page 132

Find the inverse of each of the matrices (if it exists).

`[(1,0,0),(3,3,0),(5,2,-1)]`

Exercise 4.5 | Q 9 | Page 132

Find the inverse of each of the matrices (if it exists).

`[(2,1,3),(4,-1,0),(-7,2,1)]`

Exercise 4.5 | Q 10 | Page 132

Find the inverse of each of the matrices (if it exists).

`[(1,-1,2),(0,2,-3),(3,-2,4)]`

Exercise 4.5 | Q 11 | Page 132

Find the inverse of each of the matrices (if it exists).

`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`

Exercise 4.5 | Q 12 | Page 132

Let `A =[(3,7),(2,5)] and B = [(6,8),(7,9)]`. Verify that `(AB)^(-1) = B^(-1)A^(-1)`

Exercise 4.5 | Q 13 | Page 132

if A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = O. Hence find A–1.

Exercise 4.5 | Q 14 | Page 132

For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A2 + aA + bI O.

Exercise 4.5 | Q 15 | Page 132

For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A3 − 6A2 + 5A + 11 I = O. Hence, find A−1.

Exercise 4.5 | Q 16 | Page 132

if A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A3 − 6A2 + 9A − 4I = O and hence find A−1

Exercise 4.5 | Q 17 | Page 132

Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to

(A) |A |

(B) | A|2

(C) | A|3

(D) 3|A|

Exercise 4.5 | Q 18 | Page 132

If A is an invertible matrix of order 2, then det (A−1) is equal to

(A) det (A)

(B) 1/det (A)

(C) 1

(D) 0

 

Exercise 4.6 [Pages 136 - 137]

NCERT solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.6 [Pages 136 - 137]

Exercise 4.6 | Q 1 | Page 136

Examine the consistency of the system of equations.

+ 2= 2

2x + 3= 3

Exercise 4.6 | Q 2 | Page 136

Examine the consistency of the system of equations.

2− y = 5

x + = 4

Exercise 4.6 | Q 3 | Page 136

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Exercise 4.6 | Q 4 | Page 136

Examine the consistency of the system of equations.

x + y z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

 

Exercise 4.6 | Q 6 | Page 136

Examine the consistency of the system of equations.

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

Exercise 4.6 | Q 6 | Page 136

Examine the consistency of the system of equations.

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

Exercise 4.6 | Q 7 | Page 136

Solve system of linear equations, using matrix method.

5x + 2y = 4

7x + 3y = 5

Exercise 4.6 | Q 8 | Page 136

Solve system of linear equations, using matrix method.

2x – y = –2

3x + 4y = 3

Exercise 4.6 | Q 9 | Page 136

Solve system of linear equations, using matrix method.

4x – 3y = 3

3x – 5y = 7

Exercise 4.6 | Q 10 | Page 136

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Exercise 4.6 | Q 11 | Page 136

Solve system of linear equations, using matrix method.

2x + y + z = 1

x – 2y – z = 3/2

3y – 5z = 9

Exercise 4.6 | Q 12 | Page 136

Solve system of linear equations, using matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2

Exercise 4.6 | Q 13 | Page 136

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

Exercise 4.6 | Q 14 | Page 136

Solve system of linear equations, using matrix method.

x − y + 2z = 7

3x + 4y − 5z = −5

2x − y + 3z = 12

Exercise 4.6 | Q 15 | Page 137

If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A−1. Using A−1 solve the system of equations

2x – 3y + 5z = 11
3x + 2y – 4z = – 5
x + y – 2z = – 3

Exercise 4.6 | Q 16 | Page 137

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Exercise 4.7 [Pages 141 - 143]

NCERT solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.7 [Pages 141 - 143]

Exercise 4.7 | Q 1 | Page 141

Prove that the determinant `|(x,sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1, x)|` is independent of θ.

Exercise 4.7 | Q 2 | Page 141

Without expanding the determinant, prove that

`|(a, a^2,bc),(b,b^2, ca),(c, c^2,ab)| = |(1, a^2, a^3),(1, b^2, b^3),(1, c^2, c^3)|`

Exercise 4.7 | Q 3 | Page 141

Evaluate `|(cos alpha cos beta, cos alpha sin beta, -sin alpha),(-sin beta, cos beta, 0),(sin alpha cos beta, sin alpha sin beta,cos alpha )|`

Exercise 4.7 | Q 4 | Page 141

If ab and are real numbers, and triangle =`|(b+c, c+a, a+b),(c+a,a+b, b+c),(a+b, b+c, c+a)|` = 0 Show that either a + b + c = 0 or a = b = c.

Exercise 4.7 | Q 5 | Page 141

Solve the equations `|(x+a,x,x),(a,x+a,x),(x,x,x+a)| = 0, a != 0`

Exercise 4.7 | Q 6 | Page 141

Prove that `|(a^2, bc, ac+c^2),(a^2+ab, b^2, ac),(ab, b^2+bc, c^2)| = 4a^2b^2c^2`

Exercise 4.7 | Q 7 | Page 141

if A^(-1) =`[(3,-1,1),(-15,6,-5),(5,-2,2)] and B = [(1,2,-2),(-1,3,0),(0,-2,1)] " find " (AB)^(-1)`

Exercise 4.7 | Q 8 | Page 142

Let A = `[(1,-2,1),(-2,3,1),(1,1,5)]` verify that

1) [adj A]–1 = adj (A–1)

2) (A–1)–1 = A

Exercise 4.7 | Q 9 | Page 142

Evaluate `|(x, y, x+y),(y, x+y, x),(x+y, x, y)|`

Exercise 4.7 | Q 10 | Page 142

Evaluate `|(1,x,y),(1,x+y,y),(1,x,x+y)|`

Exercise 4.7 | Q 11 | Page 142

Using properties of determinants, prove that:

`|(alpha, alpha^2,beta+gamma),(beta, beta^2, gamma+alpha),(gamma, gamma^2, alpha+beta)|` =  (β – γ) (γ – α) (α – β) (α + β + γ)

Exercise 4.7 | Q 12 | Page 142

Using properties of determinants, prove that:

`|(x, x^2, 1+px^3),(y, y^2, 1+py^3),(z, z^2, 1+pz^2)|` = (1 + pxyz) (x – y) (y – z) (z – x), where p is any scalar.

Exercise 4.7 | Q 13 | Page 142

Using properties of determinants, prove that:

`|(3a, -a+b, -a+c),(-b+a, 3b, -b+c),(-c+a, -c+b, 3c)|`= 3(a + b + c) (ab + bc + ca)

Exercise 4.7 | Q 14 | Page 142

Using properties of determinants, prove that:

`|(1, 1+p, 1+p+q),(2, 3+2p, 4+3p+2q),(3,6+3p,10+6p+3q)| =  1`                 

Exercise 4.7 | Q 15 | Page 142

Using properties of determinants, prove that

`|(sin alpha, cos alpha, cos(alpha+ delta)),(sin beta, cos beta, cos (beta + delta)),(sin gamma, cos gamma, cos (gamma+ delta))| = 0`

Exercise 4.7 | Q 16 | Page 142

Solve the system of the following equations

`2/x+3/y+10/z = 4`

`4/x-6/y + 5/z = 1`

`6/x + 9/y - 20/x = 2`

Exercise 4.7 | Q 17 | Page 143

Choose the correct answer.

If abc, are in A.P., then the determinant

`|(x+2, x+3,x +2a),(x+3,x+4,x+2b),(x+4,x+5,x+2c)|`

A. 0

B. 1

C. x

D. 2x

Exercise 4.7 | Q 18 | Page 143

Choose the correct answer.

If xyz are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is

A) `[(x^(-1),0,0),(0, y^(-1),0),(0,0,z^(-1))]`

B) `xyz[(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1))]`

c) `1/xyz[(x,0,0),(0,y,0),(0,0,z)]`

D) `1/xyz [(1,0,0),(0,1,0),(0,0,1)]`

Exercise 4.7 | Q 19 | Page 143

Choose the correct answer.

Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4]

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Solutions for Chapter 4: Determinants

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6Exercise 4.7

NCERT solutions for Class 12 Maths chapter 4 - Determinants

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Class 12 Maths CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC 4 (Determinants) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Class 12 Maths chapter 4 Determinants are Applications of Determinants and Matrices, Elementary Transformations, Inverse of a Square Matrix by the Adjoint Method, Properties of Determinants, Determinant of a Square Matrix, Determinants of Matrix of Order One and Two, Determinant of a Matrix of Order 3 × 3, Rule A=KB, Introduction of Determinant, Minors and Co-factors, Area of a Triangle.

Using NCERT Class 12 Maths solutions Determinants exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Class 12 Maths students prefer NCERT Textbook Solutions to score more in exams.

Get the free view of Chapter 4, Determinants Class 12 Maths additional questions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC, and you can use Shaalaa.com to keep it handy for your exam preparation.

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