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NCERT solutions Mathematics Class 12 Part 1 chapter 4 Determinants

Chapters

NCERT Solutions for Mathematics Class 12 Part 2

NCERT Mathematics Class 12 Part 1

Mathematics Textbook for Class 12 Part 1

Chapter 4 - Determinants

Pages 108 - 109

Evaluate the determinants in Exercises 1 and 2. 

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

Q 1 | Page 108

Evaluate the determinants in Exercises 1 and 2. 

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

Q 2.1 | Page 108

Evaluate the determinants in Exercises 1 and 2.

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

Q 2.2 | Page 108

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

Q 3 | Page 108

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

Q 4 | Page 108

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

Q 5.1 | Page 108

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

Q 5.2 | Page 108

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

Q 5.3 | Page 108

Evaluate the determinants

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

Q 5.4 | Page 108

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

Q 6 | Page 109

Find values of x, if 2451=2x46x

Q 7.1 | Page 109

Find values of x, if 2345=x32x5

Q 7.2 | Page 109

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

(A) 6

(B) ±6

(C) −6

(D) 0

Q 8 | Page 109

Pages 119 - 121

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

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

Q 1 | 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 2 | Page 119

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

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

Q 3 | 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 4 | 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 5 | Page 119

By using properties of determinants, show that:

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

Q 6 | 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 7 | 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 | 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 8.2 | 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 9 | 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 10.1 | 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 10.2 | 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 11.1 | 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 11.2 | Page 120

By using properties of determinants, show that:

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

Q 12 | 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 13 | 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 14 | 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 15 | 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

Q 16 | Page 121

Pages 122 - 123

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

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

Q 1.1 | 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 1.2 | 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 1.3 | Page 122

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

Q 2 | Page 123

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

 

Q 3.1 | Page 123

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

Q 3.2 | Page 123

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

Q 4.1 | Page 123

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

Q 4.2 | 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

Q 5 | Page 123

Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 1.1 | Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 1.2 | Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 2.1 | Page 126

Write Minors and Cofactors of the elements of following determinants:

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

Q 2.2 | Page 126

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

Q 3 | Page 126

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

Q 4 | 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

 

Q 5 | Page 126

Pages 131 - 132

Find adjoint of each of the matrices.'

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

Q 1 | Page 131

Find adjoint of each of the matrices.

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

Q 2 | Page 131

Verify A (adj A) = (adj A) A = |A|I

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

Q 3 | Page 131

Verify A (adj A) = (adj AA = |A|I

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

Q 4 | Page 131

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

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

Q 5 | Page 132

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

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

Q 6 | Page 132

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

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

Q 7 | Page 132

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

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

Q 8 | Page 132

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

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

Q 9 | Page 132

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

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

Q 10 | 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 11 | Page 132

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

Q 12 | Page 132

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

Q 13 | Page 132

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

Q 14 | 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 15 | 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 16 | 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 17 | 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

 

Q 18 | Page 132

Pages 136 - 137

Examine the consistency of the system of equations.

+ 2= 2

2x + 3= 3

Q 1 | Page 136

Examine the consistency of the system of equations.

2− y = 5

x + = 4

Q 2 | Page 136

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Q 3 | Page 136

Examine the consistency of the system of equations.

x + y z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

 

Q 4 | Page 136

Examine the consistency of the system of equations.

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

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

Solve system of linear equations, using matrix method.

5x + 2y = 4

7x + 3y = 5

Q 7 | Page 136

Solve system of linear equations, using matrix method.

2x – y = –2

3x + 4y = 3

Q 8 | Page 136

Solve system of linear equations, using matrix method.

4x – 3y = 3

3x – 5y = 7

Q 9 | Page 136

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Q 10 | Page 136

Solve system of linear equations, using matrix method.

2x + y + z = 1

x – 2y – z = 3/2

3y – 5z = 9

Q 11 | Page 136

Solve system of linear equations, using matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2

Q 12 | Page 136

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

Q 13 | Page 136

Solve system of linear equations, using matrix method.

x − y + 2z = 7

3x + 4y − 5z = −5

2x − y + 3z = 12

Q 14 | Page 136

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 15 | 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.

Q 16 | Page 137

Pages 141 - 143

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 1 | 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 2 | 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 3 | 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 4 | Page 141

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

Q 5 | 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 6 | 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 7 | Page 141

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

Q 8 | Page 142

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

Q 9 | Page 142

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

Q 10 | 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 11 | 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 12 | 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 13 | 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 14 | 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 15 | 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 16 | Page 142

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

Q 17 | 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)]`

Q 18 | 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]

Q 19 | Page 143

NCERT Mathematics Class 12 Part 1

Mathematics Textbook for Class 12 Part 1
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