#### Chapters

Chapter 1.2: Matrices

Chapter 1.3: Differentiation

Chapter 1.4: Applications of Derivatives

Chapter 1.5: Integration

Chapter 1.6: Definite Integration

Chapter 1.7: Application of Definite Integration

Chapter 1.8: Differential Equation and Applications

Chapter 2.1: Commission, Brokerage and Discount

Chapter 2.2: Insurance and Annuity

Chapter 2.3: Linear Regression

Chapter 2.4: Time Series

Chapter 2.5: Index Numbers

Chapter 2.6: Linear Programming

Chapter 2.7: Assignment Problem and Sequencing

Chapter 2.8: Probability Distributions

## Chapter 1: Matrices

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.1

#### MCQ [1 Mark]

**Choose the correct alternative:**

If A = `[(6, 0),("p", "q")]` is a scalar matrix, then the values of p and q are ______ respectively.

6 and 0

0 and 6

5 and 6

0 and 1

**Choose the correct alternative:**

If B = `[(6, 3),(-2, "k")]` is singular matrix, then the value of k is ______

– 1

2

0

1

**Choose the correct alternative:**

If A = `[(1, 3/5, x),(y, -5, -7),(-4, -7, 0)]` is a symmetric matrix, then the values of x and y are ______ respectively.

`3/5` and 4

`5/3` and – 4

`3/5` and – 4

– 4 and `3/5`

**Choose the correct alternative:**

`[(3, 2, 1)][(2),(-2),(-1)]` = ______

[2]

1

[1]

2

**Choose the correct alternative:**

If A = `[(2, 0),(0, 2)]`, then A^{2} – 3I = ______

O

I

A

3A

**Choose the correct alternative:**

For any square matrix B, matrix B + B^{T} is ______

Null matrix

Identity matrix

Symmetric matrix

Skew symmetric matrix

**Choose the correct alternative:**

If A and B are two square matrices of order 3, then (AB)^{T} = ______

B

^{T}A^{T}A

^{T}B^{T}(BA)

^{T}I

**Choose the correct alternative:**

If A = `[(1, 2),(2, -1)]`, then adj (A) = ______

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

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

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

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

**Choose the correct alternative:**

If A is a non singular matrix of order 3, then |adj (A)| = ______

|A|

^{2}|A|

^{3}0

1

**Choose the correct alternative:**

If A^{2} + 5A + 3I = 0, |A| ≠ 0, then A–1 = ______

`(-1)/3 ("A" + 5"I")`

`(-1)/5 ("A" + 3"I")`

(A + 15I)

`(-1)/3 ("I" + 5"A")`

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.2

#### [1 mark]

**State whether the following statement is True or False:**

If A is non singular, then |A| = 0

True

False

**State whether the following statement is True or False:**

Inverse of `[(2, 0),(0, 3)]` is `[(1/2, 0),(0, 1/3)]`

True

False

**State whether the following statement is True or False:**

If `[(3, 0),(0, 2)][(x),(y)] = [(3),(2)]`, then x = 1 and y = – 1

True

False

**State whether the following statement is True or False:**

Every square matrix of order n can be expressed as sum of symmetric and skew symmetric matrix

True

False

**State whether the following statement is True or False:**

If A = `[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`, then A^{T} = A

True

False

**State whether the following statement is True or False:**

Matrix `[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]` is a singular

True

False

**State whether the following statement is True or False:**

`[(2, 0, 0),(3, -1, 0),(-7, 3, 1)]` is a skew symmetric matrix

True

False

**State whether the following statement is True or False:**

If O(A) = m × n and O(B) = n × p with m ≠ p, then BA exists but AB does not exist.

True

False

**State whether the following statement is True or False:**

After applying elementary transformation R_{1} – 3R_{2} on matrix `[(3, -2),(1, 4)]` we get `[(0, -12),(1, 4)]`

True

False

**State whether the following statement is True or False:**

If A and B are two square matrices such that AB = BA, then (A – B)^{2} = A^{2} – 2AB + B^{2}^{ }

True

False

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.3

#### Fill in the blanks [1 Mark]

If A = [a_{ij}]_{2×2}, where a_{ij} = i – j, then A = ______

In a Skew symmetric matrix, all diagonal elements are ______

If A = `[(1, -2),(5, 3)]`, B = `[(1, -3),(4, -7)]`, then A – 3B = ______

If A = `[(4, 3, 2),(-1, 2, 0)]`, B = `[(1, 2),(-1, 0),(1, -2)]`, then |AB| = ______

The value of Cofactor of element a_{21} in matrix A = `[(1, 2),(5, -8)]` is ______

The value of Minor of element b_{22} in matrix B = `[(2, -2),(4, 5)]` is ______

If matrix form of given equations 3x – y = 1 and y + 4x = 6 is AX = B, then A = ______

The suitable elementary row transformation which will reduce the matrix `[(1, 0),(2, 1)]` into identity matrix is ______

If A = `[(2, 0, 0),(0, 1, 0),(0, 0, 1)]`, then |adj (A)| = ______

If A is a square matrix of order 2 such that A(adj A) = `[(7, 0),(0, 7)]`, then |A| = ______

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.4

#### Solve the following [3 Marks]

If A = `[(1, 2),(-1, -2)], "B" = [(2, "a"),(-1, "b")]` and (A + B)^{2} A^{2} + B^{2}, find the values of a and b.

Find matrices A and B, if `2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]`

If A = `[(3, 1),(-1, 2)]`, then prove that A^{2} – 5A + 7I = O, where I is unit matrix of order 2

Find k, if A = `[(3, -2),(4, -2)]` and A^{2} = kA – 2I, where I is identity matrix of order 2

If A = `[(2, 1),(0, 3),(1, -1)]` and B = `[(0, 3, 5),(1, -7, 2)]`, then verify (BA)^{T} = A^{T}B^{T}

Find the inverse of matrix B = `[(3,1, 5),(2, 7, 8),(1, 2, 5)]` by using adjoint method

If A = `[(3, 1),(1, 5)]` nd B = `[(1, 2),(5, -2)]`, then verify |AB| = |A||B|

If A = `[(1, 3, 3),(3, 1, 3),(3, 3, 1)]`, then show that A^{2} – 5A is a scalar matrix

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.5

#### [4 Marks]

Find the inverse of matrix A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)]` by using elementary row transformations

**Solve the following equations by method of inversion : **

4x – 3y – 2 = 0, 3x – 4 + 6 = 0

**Solve the following equations by method of reduction :**

x – 3y + z = 2 , 3x + y + z = 1 and 5x + y + 3z = 3

The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding the first and third number we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.

The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 Matrices Q.6

#### Activities

Complete the following activity to verify A. adj (A) = det (A) I.

Given A = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` then

|A| = 2(____) – 0(____) + ( ) (____)

= 6 – 0 – 5

= ______ ≠ 0

Cofactors of all elements of matrix A are

A_{11} = `(-1)^2 |("( )", "( )"),("( )", "( )")|` = (______),

A_{12} = `(-1)^3 |(5, "( )"),("( )", 3)|` = – 15,

A_{13} = `(-1)^4 |(5, "( )"),("( )", 1)|` = 5,

A_{21} = _______, A_{22} = _______, A_{23 }= _______,

A_{31} = `(-1)^4 |("( )", "( )"),("( )", "( )")|` = (______),

A_{32} = `(-1)^5 |(2, "( )"),("( )", 0)|` = ( ),

A_{33} = `(-1)^6 |(2, "( )"),("( )", 1)|` = 2,.

Cofactors of matrix A = `[(3, "____", "____"),("____", "____",-2),(1, "____", "____")]`

adj (A) = `[("____", "____", "____"),("____", "____","____"),("____","____","____")]`

A.adj (A) = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)] [("( )", -1, 1), (-15, "( )", -5),("( )", -2, "( )")] = [(1, 0, "( )"),("( )", "( )", "( )"),(0, "( )", "( )")]` = |A|I

Complete the following activity to find inverse of matrix using elementary column transformations and hence verify.

`[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` B^{−1} = `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

C_{1} → C_{1} + C_{3}

`[("( )", 0, -1),("( )", 1, 0),("( )", 1, 3)]` B^{−1} = `[("( )", 0, 0),("( )", 1, 0),("( )", 0, 1)]`

C_{3} → C_{3} + C_{1}

`[(1, 0, 0),("( )", 1, "( )"),(3, 1, "( )")]` B^{−1} = `[(1, 0, "( )"),(0, 1, 0),("( )", 0, "( )")]`

C_{1} → C_{1} – 5C_{2}, C_{3} → C_{3} – 5C_{2}

`[(1, "( )", 0),(0, 1, 0),("( )", 1, "( )")]` B^{−1} = `[(1, 0, "( )"),("( )", 1, -5),(1, "( )", 2)]`

C_{1} → C_{1} – 2C_{3}, C_{2} → C_{2} – C_{3 }

`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]` B^{−1} = `[(3, -1, "( )"),("( )", 6, -5),(5, "( )", "( )")]`

B^{−1} = `[("( )", "( )", "( )"),("( )", "( )", "( )"),("( )", "( )", "( )")]`

`[(2, "( )", -1),("( )", 1, 0),(0, 1, "( )")] [(3, "( )", "( )"),("( )", 6, "( )"),("( )", -2, "( )")] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

**Complete the following activity.**

The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.

**Solution:** Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + ( )y + 2( ) = ( )

x + 2y + ( )( ) = ( )

( )x + 2y + 3z = ( )

Matrix form of above equations is,

`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`

R_{1} ↔ R_{2}

`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`

R_{2} – 4R_{1}, R_{3} – 6R_{1}

`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`

`(-1)/5 "R"_2, (-1)/5 "R"`,

`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`

R_{3} – 2R_{2}

`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`

By pre multiplying we get,

x + 2y + ( )z = ( ) .....(i)

y + 2z = 24 ......(ii)

–z = ( ) ......(iii)

From (iii), we get, z = ( )

From (ii), we get, y = ( )

From (i), we get, x = ( )

Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.

## Chapter 1: Matrices

## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 - Matrices

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 Matrices are Determinant of a Matrix, Types of Matrices, Algebra of Matrices, Properties of Matrices, Elementary Transformations, Inverse of Matrix, Application of Matrices.

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