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

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## Chapter 4: Determinants

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

Q 1 | Page 108

Evaluate the determinants in Exercises 1 and 2.

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

Q 2.1 | Page 108

Evaluate the determinants in Exercises 1 and 2.

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

Q 2.2 | Page 108

Evaluate the determinants in Exercises 1 and 2.

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

Q 3 | Page 108

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

Q 4 | Page 108

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

Q 5.1 | Page 108

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

Q 5.2 | Page 108

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

Q 5.3 | Page 108

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

Q 5.4 | Page 108

Evaluate the determinants

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

Q 6 | Page 109

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

Q 7.1 | Page 109

Find values of x, if 2451=2x46x

Q 7.2 | Page 109

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

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

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

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

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

Q 3 | Page 119

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

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

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

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)|

Q 6 | Page 120

By using properties of determinants, show that:

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

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

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)

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)

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)

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

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)

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

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

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

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)

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

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 |

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

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

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)

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)

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)

Q 2 | Page 123

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

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)

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)

Q 4.1 | Page 123

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

Q 4.2 | Page 123

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

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

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

Q 1.1 | Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 1.2 | Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 2.1 | Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 2.2 | Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 3 | Page 126

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

Q 4 | Page 126

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

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

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

Q 1 | Page 131

Find adjoint of each of the matrices.'

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

Q 2 | Page 131

Find adjoint of each of the matrices.

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

Q 3 | Page 131

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

Q 4 | Page 131

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

Q 5 | Page 132

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

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

Q 6 | Page 132

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

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

Q 7 | Page 132

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

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

Q 8 | Page 132

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

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

Q 9 | Page 132

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

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

Q 10 | Page 132

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

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

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)]

Q 12 | Page 132

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

Q 13 | Page 132

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

Q 14 | Page 132

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

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.

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

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|

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

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

Q 1 | Page 136

Examine the consistency of the system of equations.

+ 2= 2

2x + 3= 3

Q 2 | Page 136

Examine the consistency of the system of equations.

2− y = 5

x + = 4

Q 3 | Page 136

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Q 4 | Page 136

Examine the consistency of the system of equations.

x + y z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Q 6 | Page 136

Examine the consistency of the system of equations.

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

Q 6 | Page 136

Examine the consistency of the system of equations.

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

Q 7 | Page 136

Solve system of linear equations, using matrix method.

5x + 2y = 4

7x + 3y = 5

Q 8 | Page 136

Solve system of linear equations, using matrix method.

2x – y = –2

3x + 4y = 3

Q 9 | Page 136

Solve system of linear equations, using matrix method.

4x – 3y = 3

3x – 5y = 7

Q 10 | Page 136

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Q 11 | Page 136

Solve system of linear equations, using matrix method.

2x + y + z = 1

x – 2y – z = 3/2

3y – 5z = 9

Q 12 | Page 136

Solve system of linear equations, using matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2

Q 13 | Page 136

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

Q 14 | Page 136

Solve system of linear equations, using matrix method.

x − y + 2z = 7

3x + 4y − 5z = −5

2x − y + 3z = 12

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

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.

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

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.is independent of θ.

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)|

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 )|

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.

Q 5 | Page 141

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

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

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)

Q 8 | Page 142

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

2) (A–1)–1 = A

Q 9 | Page 142

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

Q 10 | Page 142

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

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)

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.

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)

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

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

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

Q 17 | Page 143

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

Q 18 | Page 143

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)]

Q 19 | Page 143

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]

## NCERT solutions for Class 12 Maths chapter 4 - Determinants

NCERT solutions for Class 12 Maths chapter 4 (Determinants) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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

Concepts covered in Class 12 Maths chapter 4 Determinants are Applications of Determinants and Matrices, Elementary Transformations, Adjoint and Inverse of a Matrix, 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, Area of a Triangle, Minors and Co-factors.

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

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