##### Question 1:

##### Answer:

= 2 × (-1) - (-5) × 4 = - 2 + 20 = 18.

##### Question 2:

Evaluate

##### Answer:

##### Question 3:

Evaluate

##### Answer:

##### Question 4:

##### Answer:

We have :

##### Question 5:

##### Answer:

We have :

##### Question 6:

Evaluate the determinants :

##### Answer:

##### Question 7:

Evaluate the determinants :

##### Answer:

= 3 (1 × 1 - 3 × (-2)) + 4 (1 × 1 - 2 × (-2))

+ 5 (1 × 3 - 2 × 1)

= 3 (1 + 6) + 4 (1 + 4) + 5 (3 - 2)

= 3 (7) + 4 (5) + 5 (1) = 21 + 20 + 5 = 46.

##### Question 8:

Evaluate the determinants :

##### Answer:

= 0 - (-1 × 0 - (-2) × (-3)) + 2 ((-1) × 3 - (-2) × 0)

= 0 - (- 0 - 6) + 2 (- 3 + 0) = 6 - 6 = 0.

##### Question 9:

Evaluate the determinants :

##### Answer:

= 2 (2 × 0 - (-5) × (-1)) + 3 ((-1) × (-1) - 2 × (-2))

= 2 (0 - 5) + 3 (1 + 4) = - 10 + 15 = 5.

##### Question 10:

##### Answer:

= (- 9 + 12) - (-18 + 15) - 2 (8 - 5)

= 3 + 3 - 6 = 0.

##### Question 11:

Find values of x, if :

##### Answer:

We have :

$\Rightarrow $ 2 - 20 = 2x^{2} - 24

$\Rightarrow $ - 18 = 2x^{2} - 24

$\Rightarrow $ 2x^{2} = 24 - 18

$\Rightarrow $ 2x^{2} = 6

$\Rightarrow $ x^{2} = 3.

##### Question 12:

Find values of x, if :

##### Answer:

We have :

$\Rightarrow $ 2 × 5 - 4 × 3 = 5 × x - 2x × 3

$\Rightarrow $ 10 - 12 = 5x - 6x

$\Rightarrow $ - x = - 2.

Hence, x = 2.

##### Question 13:

- 6
- $\pm $ 6
- - 6
- 0

##### Answer:

$\pm $ 6

##### Question 14:

Using the property of determinants and without expanding prove that :

##### Answer:

##### Question 15:

Using the property of determinants and without expanding prove that :

##### Answer:

##### Question 16:

Using the property of determinants and without expanding prove that :

##### Answer:

##### Question 17:

Using the property of determinants and without expanding prove that :

##### Answer:

##### Question 18:

Using the property of determinants and without expanding prove that :

##### Answer:

##### Question 19:

Using the property of determinants and without expanding prove that :

##### Answer:

##### Question 20:

Using properties of determinants, prove the following

##### Answer:

##### Question 21:

Prove that :

##### Answer:

##### Question 22:

Prove that :

##### Answer:

##### Question 23:

Prove that :

(xy + yz + zx).

##### Answer:

##### Question 24:

Prove that :

##### Answer:

##### Question 25:

Prove that :

##### Answer:

##### Question 26:

Prove that :

##### Answer:

##### Question 27:

Prove that :

##### Answer:

= 2 (x + y + z) [(x + y + z)^{2} + 0]

= 2 (x + y + z)^{3}, which is true

##### Question 28:

Prove that :

##### Answer:

##### Question 29:

##### Answer:

##### Question 30:

Prove that :

##### Answer:

##### Question 31:

Choose the correct answer

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

- k |A|
- k
^{2} - |A|
- 3k |A|

##### Answer:

k^{3} |A|

##### Question 32:

Choose the correct answer Which of the following is correct :

- Determinant is a square matrix.
- Determinant is a number associated to a matrix.
- Determinant is a number associated to a square matrix.
- None of these.

##### Answer:

Determinant is a number associated to a square matrix.

##### Question 33:

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

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

##### Answer:

Area of the triangle

##### Question 34:

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

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

##### Answer:

Area of the triangle

##### Question 35:

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

(-2, -3), (3, 2), (-1, -8).

##### Answer:

Area of the triangle

##### Question 36:

Show that the points :

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

##### Answer:

##### Question 37:

Find the values of ‘k’ if area of triangle is 4 sq. units and vertices are :

(k, 0), (4, 0), (0, 2)

##### Answer:

Area of the triangle

= - (k - 4).

By the question, - (k - 4) = $\pm $ 4.

Taking +ve sign, -(k - 4) = 4 $\Rightarrow $ k = 0.

Taking -ve sign, - (k - 4) = - 4 $\Rightarrow $ k = 8.

Hence, k = 0, 8.

##### Question 38:

Find the values of ‘k’ if area of triangle is 4 sq. units and vertices are :

(-2, 0), (0, 4), (0, k)

##### Answer:

= k - 4

By the question, k - 4 = $\pm $ 4.

Taking +ve sign, k - 4 = 4 $\Rightarrow $ k = 8.

Taking -ve sign, k - 4 = - 4 $\Rightarrow $ k = 0.

Hence, k = 0, 8.

##### Question 39:

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

##### Answer:

Let (x, y) be the third point on the line joining (1, 2) and (3, 6).

The area of the triangle having vertices (x, y), (1, 2) and (3, 6)

##### Question 40:

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

##### Answer:

As in part (i).

##### Question 41:

If area of triangle is 35 sq. units with vertices (2, - 6), (5, 4) and (k, 4), then k is :

- 12
- - 2
- - 12, - 2
- 12, - 2.

##### Answer:

12, - 2

##### Question 42:

Write minors and co-factors of the elements of the following determinants :

##### Answer:

We have :

$\therefore $ Minor ofa_{11} = M_{11} = 3

Co-factor of a_{11} = A_{11} = (- 1)^{1+1} M_{11}

= (- 1)^{2} (3) = 3.

Minor ofa_{12} = M_{12} = 0

Co-factor of a_{12} = (- 1)^{1+2} M12

= (- 1)^{3} (0) = 0.

Minor of a_{21} = M_{21} = - 4

Co-factor of a_{21} = A_{21} = (- 1)^{2+1} M_{21}

= (- 1)^{3} (- 4) = - (- 4) = 4.

Minor of a_{22} = M_{22} = 2.

Co-factor of a_{22} = A_{22} = (- 1)^{2+2} M_{22}

= (- 1)^{4} (2) = (1) (2) = 2.

##### Question 43:

Write minors and co-factors of the elements of the following determinants :

##### Answer:

$\therefore $ M_{11}= d, A_{11} = (- 1)^{1+1} M_{11} = (+ 1) d = d.

M_{12}= b, A_{12} = (- 1)^{1+2} M_{12} = (- 1) b = - b.

M_{21}= c , A_{21} = (- 1)^{2+1} M_{21} = (- 1) c = - c.

M_{22}= a, A_{22} = (- 1)^{2+2} M_{22} = (+ 1) a = a.

##### Question 44:

Write minors and co-factors of the elements of the following determinants :

##### Answer:

##### Question 45:

Write minors and co-factors of the elements of the following determinants :

##### Answer:

##### Question 46:

Using co-factors of elements of second row, evaluate :

##### Answer:

##### Question 47:

Using co-factors of third column, evaluate :

##### Answer:

$\therefore $ $\u2206$= a_{13} A_{13} + a_{23} A_{23} + a_{33} A_{33}

= yz (z - y) + zx (x - z) + xy (y - x)

= yz^{2} - y^{2} z + zx^{2} - z^{2}x + xy^{2} - x^{2}y

= x^{2} (z - y) + x (y^{2} - z^{2}) + yz (z - y)

= x^{2} (z - y) + x (y - z) (y + z) - yz (y - z)

= (y - z) (- x^{2} + xy + xz - yz)

= (y - z) [y (x - z) + x (z - x)]

= (y - z) (z - x) (x - y)

= (x - y) (y - z) (z - x).

##### Question 48:

- a
_{11}A_{11}+ a_{12}A_{32}+ a_{13}A_{33} - a
_{11}A_{11}+ a_{12}A_{21}+ a_{13}A_{31} - a
_{21}A_{11}+ a_{22}A_{12}+ a_{23}A_{13} - a
_{11}A_{11}+ a_{21}A_{21}+ a_{31}A_{31}

##### Answer:

a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}

##### Question 49:

Find the adjoint of the matrices

##### Answer:

##### Question 50:

Find the adjoint of the matrices

##### Answer:

##### Question 51:

Verify A (Adj A) = (Adj A) A = |A| I

##### Answer:

##### Question 52:

Verify A (Adj A) = (Adj A) A = |A| I

##### Answer:

##### Question 53:

Find the inverse of each of the matrices (if it exists) given in

##### Answer:

##### Question 54:

Find the inverse of each of the matrices (if it exists) given in

##### Answer:

##### Question 55:

Find the inverse of each of the matrices (if it exists) given in

##### Answer:

##### Question 56:

Find the inverse of each of the matrices (if it exists) given in

##### Answer:

##### Question 57:

Find the inverse of each of the matrices (if it exists) given in

##### Answer:

##### Question 58:

Find the inverse of each of the matrices (if it exists) given in

##### Answer:

##### Question 59:

Find the inverse of each of the matrices (if it exists) given in

##### Answer:

##### Question 60:

##### Answer:

##### Question 61:

##### Answer:

We have :

##### Question 62:

##### Answer:

##### Question 63:

##### Answer:

We have :

##### Question 64:

##### Answer:

##### Question 65:

Let A be non-singular square matrix of order 3 × 3. Then |adj A| is equal to :

- |A|
- |A|
^{2 } - |A|
^{3 } - 3|A|

##### Answer:

|A|^{2 }

##### Question 66:

If A is an invertible matrix of order 2, then det. (A–^{1}) is equal to :

- det (A)
- 1
- 0

##### Answer:

##### Question 67:

Classify the following system of equations as consistent or inconsistent :

x + 2y = 2

2x + 3y = 3.

##### Answer:

The given equations are :

x + 2y=2

2x + 3y=3.

= 3 – 4 = – 1 ≠ 0.

Hence, the given system of equations is consistent.

##### Question 68:

Classify the following system of equations as consistent or inconsistent :

2x – y = 5

x + y = 4.

##### Answer:

The given equations are :

2x – y=5

x + y=4.

= 2 + 1 = 3 ≠ 0.

Hence, the given system of equations is consistent.

##### Question 69:

Classify the following system of equations as consistent or inconsistent :

x + 3y = 5

2x + 6y = 8.

##### Answer:

The given equations are :

x + 3y = 5

2x + 6y = 8.

Hence, the given system of equations is inconsistent.

##### Question 70:

Classify the following system of equations as consistent or inconsistent :

x + y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4.

##### Answer:

The given equations are :

x + y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

= 1.(6 – 2) – 1. (4 – 2) + 1. (2 – 3)

= 4 – 2 – 1= 1 ≠ 0.

Hence, the given system of equations is consistent.

##### Question 71:

Classify the following system of equations as consistent or inconsistent :

3x – y – 2z = 2

2y – z = – 1

3x – 5y = 3.

##### Answer:

The given equations are :

3x – y – 2z = 2

2y – z = – 1

3x – 5y = 3.

##### Question 72:

Classify the following system of equations as consistent or inconsistent :

5x – y + 4z = 5

2x + 3y + 5z = 2

5x – 2y + 6z = – 1.

##### Answer:

The given equations are :

5x – y + 4z = 5

2x + 3y + 5z = 2

5x – 2y + 6z = – 1.

= 5 (18 + 10) + 1. (12 – 25) + 4 (– 4 – 15)

= 140 – 13 – 76 = 51 ≠ 0.

Hence, the given system of equations is consistent.

##### Question 73:

Solve system of linear equations, using matrix method

5x + 2y = 4

7x + 3y = 5.

##### Answer:

The given system of equations is :

5x + 2y = 4

7x + 3y = 5.

These can be written as AX = B

$\Rightarrow $ X = A^{–1} B ...(1),

##### Question 74:

Solve system of linear equations, using matrix method

2x – y = – 2

3x + 4y = 3.

##### Answer:

The given system of equations is :

2x – y = – 2

3x + 4y = 3.

These can be written as A X = B

$\Rightarrow $ X = A^{–1} B ...(1),

##### Question 75:

Solve system of linear equations, using matrix method

4x – 3y = 3

3x – 5y = 7.

##### Answer:

The given system of equations is :

4x – 3y = 3

3x – 5y = 7.

These can be written as AX = B

$\Rightarrow $ X = A^{–1} B ...(1),

##### Question 76:

Solve system of linear equations, using matrix method

5x + 2y = 3

3x + 2y = 5.

##### Answer:

The given system of equations is :

5x + 2y = 3

3x + 2y = 5.

These can be written as AX = B

$\Rightarrow $ X = A^{–1} B ...(1),

##### Question 77:

Solve system of linear equations, using matrix method

2x + y + z = 1

3y – 5z = 9.

##### Answer:

The given system of equations is :

2x + y + z = 1

##### Question 78:

Solve system of linear equations, using matrix method

x – y + z = 4

2x + y – 3z = 0

x + y + z = 2.

##### Answer:

The given system of equations is :

x – y + z = 4

2x + y – 3z= 0

x + y + z = 2.

These can be written as AX = B

$\Rightarrow $ X = A^{–1} B ...(1),

##### Question 79:

Solve system of linear equations, using matrix method

2x + 3y + 3z = 5

x – 2y + z = – 4

3x – y – 2z = 3.

##### Answer:

The given system of equations is :

2x + 3y + 3z = 5

x – 2y + z = – 4

3x – y – 2z= 3.

These can be written as AX = B

$\Rightarrow $ X = A^{–1} B ...(1),

##### Question 80:

Solve system of linear equations, using matrix method

x – y + 2z = 7

3x + 4y – 5z = – 5

2x – y + 3z = 12.

##### Answer:

The given system of equations is :

x – y + 2z = 7

3x + 4y – 5z = – 5

2x – y + 3z= 12.

These can be written as AX = B

$\Rightarrow $ X = A^{–1} B ...(1),

##### Question 81:

Using A^{–1}, solve the system of equations :

2x – 3y + 5z=11

3x + 2y – 4z=– 5

x + y – 2z = – 3.

##### Answer:

##### Question 82:

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ₹ 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is ₹ 90.

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

##### Answer:

Let the cost of onion, wheat and rice per kg be ₹ x, ₹ y and ₹ z respectively.

By the question,

4x + 3y + 2z = 60 ...(1)

2x + 4y + 6z=90

i.e. x + 2y + 3z = 45 ...(2)

6x + 2y + 3z = 70 ...(3)

These equations can be written as A X = B

$\Rightarrow $ X = A^{–1} B ...(4)

$\Rightarrow $ x = 5, y = 8, z = 8.

Hence, the cost of onion, wheat and rice per kg. is ₹ 5, ₹ 8 and ₹ 8 respectively.

##### Question 83:

- 6
- ± 6
- – 6
- 6, 6

##### Answer:

6

##### Question 84:

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

- k | A |
- k
^{2}| A | - k
^{3}| A | - 3k | A |

##### Answer:

k^{3} | A |

##### Question 85:

Which of the following is correct?

- Determinant is a square
- Determinant is a number associated to a matrix
- Determinant is a number associated to a square matrix
- None of these

##### Answer:

Determinant is a number associated to a square matrix

##### Question 86:

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

- 12
- – 2
- – 12, – 2
- 12, – 2

##### Answer:

12, – 2

##### Question 87:

- a
_{11}A_{31}+ a_{12}A_{32}+ a_{13}A_{33} - a
_{11}A_{11}+ a_{12}A_{21}+ a_{13}A_{33} - a
_{21}A_{11}+ a_{22}A_{12}+ a_{23}A_{13} - a
_{11}A_{11}+ a_{21}A_{21}+ a_{31}A_{31}

##### Answer:

a_{11}A_{11} + a_{12}A_{21} + a_{13}A_{33}

##### Question 88:

Let A be a non-singular matrix of order 3 × 3. Then | adj. A | is equal to :

- | A |
- | A |
^{2} - | A |
^{3} - 3 | A |

##### Answer:

| A |^{2}

##### Question 89:

If A is an invertible matrix of order 2, then det (A^{–1}) is equal to :

- det (A)
- 1
- 0

##### Answer:

det (A)

##### Question 90:

If a, b, c are in A.P., then determinant :

- 0
- 1
- x
- 2x

##### Answer:

0

##### Question 91:

If x, y, z are non-zero real numbers, then the inverse of matrix A =

is :

##### Answer:

##### Question 92:

- Det (A) = 0
- Det (A)$\in $ (2,$\infty $ )
- Det (A) $\in $ (2, 4)
- Det (A) $\in $ [2, 4].

##### Answer:

Det (A)$\in $[2, 4].