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#### Chapters

Chapter 2: Complex Numbers

Chapter 3: Theory of Equations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Two Dimensional Analytical Geometry-II

Chapter 6: Applications of Vector Algebra

Chapter 7: Applications of Differential Calculus

Chapter 8: Differentials and Partial Derivatives

Chapter 9: Applications of Integration

Chapter 10: Ordinary Differential Equations

Chapter 11: Probability Distributions

Chapter 12: Discrete Mathematics

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## Solutions for Chapter 1: Applications of Matrices and Determinants

Below listed, you can find solutions for Chapter 1 of Tamil Nadu Board of Secondary Education Tamil Nadu Board Samacheer Kalvi for Class 12th Mathematics Volume 1 and 2 Answers Guide.

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.1 [Pages 15 - 16]

Find the adjoint of the following:

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

Find the adjoint of the following:

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

Find the adjoint of the following:`1/3[(2, 2, 1),(-2, 1, 2),(1, -2, 2)]`

Find the inverse (if it exists) of the following:

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

Find the inverse (if it exists) of the following:

`[(5, 1, 1),(1, 5, 1),(1, 1, 5)]`

Find the inverse (if it exists) of the following:

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

If `"F"(alpha) = [(cosalpha, 0, sinalpha),(0, 1, 0),(-sinalpha, 0, cosalpha)]`, show that `["F"(alpha)]^-1 = "F"(- alpha)`

If A = `[(5, 3),(-1, -2)]`, show that A^{2} – 3A – 7I_{2} = O_{2}. Hence find A^{–1}

If A = `1/9[(-8, 1, 4),(4, 4, 7),(1, -8, 4)]`, prove that `"A"^-1 = "A"^"T"`

If A = `[(8, -4),(-5, 3)]`, verify that A(adj A) = (adj A)A = |A|I_{2}

If A = `[(3, 2),(7, 5)]` and B = `[(-1, -3),(5, 2)]`, verify that (AB)^{–1} = B^{–}^{1} A^{–}^{1}

If adj(A) = `[(2, -4, 2),(-3, 12, -7),(-2, 0, 2)]`, find A

If adj(A) = `[(0, -2, 0),(6, 2, -6),(-3, 0, 6)]`, find A^{–1}

Find adj(adj(A)) if adj A = `[(1, 0, 1),(0, 2, 0),(-1, 0, 1)]`

A = `[(1, tanx),(-tanx, 1)]`, show that A^{T} A^{–1} = `[(cos 2x, - sin 2x),(sin 2x, cos 2x)]`

Find the matrix A for which A`[(5, 3),(-1, -2)] = [(14, 7),(7, 7)]`

Given A = `[(1, -1),(2, 0)]`, B = `[(3, -2),(1, 1)]` and C = `[(1, 1),(2, 2)]`, find a martix X such that AXB = C

If A = `[(0, 1, 1),(1, 0, 1),(1, 1, 0)]`, show that `"A"^-1 = 1/2("A"^2 - 3"I")`

Decrypt the received encoded message [2 – 3][20 – 4] with the encryption matrix `[(-1, -1),(2, 1)]` and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 – 26 to the letters A – Z respectively, and the number 0 to a blank space

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.2 [Page 27]

Find the rank of the following matrices by minor method:

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

Find the rank of the following matrices by minor method:

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

Find the rank of the following matrices by minor method:

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

Find the rank of the following matrices by minor method:

`[(1, -2, 3),(2, 4, -6),(5, 1, -1)]`

Find the rank of the following matrices by minor method:

`[(0, 1 ,2, 1),(0, 2, 4, 3),(8, 1, 0, 2)]`

Find the rank of the following matrices by row reduction method:

`[(1, 1, 1, 3),(2, -1, 3, 4),(5, -1, 7, 11)]`

Find the rank of the following matrices by row reduction method:

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

Find the rank of the following matrices by row reduction method:

`[(3, -8, 5, 2),(2, -5, 1, 4),(-1, 2, 3, -2)]`

Find the inverse of the following by Gauss-Jordan method:

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

Find the inverse of the following by Gauss-Jordan method:

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

Find the inverse of the following by Gauss-Jordan method:

`[(1, 2, 3),(2, 5, 3),(1, 0, 8)]`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.3 [Page 33]

Solve the following system of linear equations by matrix inversion method:

2x + 5y = – 2, x + 2y = – 3

Solve the following system of linear equations by matrix inversion method:

2x – y = 8, 3x + 2y = – 2

Solve the following system of linear equations by matrix inversion method:

2x + 3y – z = 9, x + y + z = 9, 3x – y – z = – 1

Solve the following system of linear equations by matrix inversion method:

x + y + z – 2 = 0, 6x – 4y + 5z – 31 = 0, 5x + 2y + 2z = 13

If A = `[(-5, 1, 3),(7, 1, -5),(1, -1, 1)]` and B = `[(1, 1, 2),(3, 2, 1),(2, 1, 3)]`, Find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2

A man is appointed in a job with a monthly salary of certain amount and a fixed amount of annual increment. If his salary was ₹ 19,800 per month at the end of the first month after 3 years of service and ₹ 23,400 per month at the end of the first month after 9 years of service, find his starting salary and his annual increment. (Use matrix inversion method to solve the problem.)

Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method

The prices of three commodities A, B and C are ₹ x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B . Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn ₹ 15,000, ₹ 1,000 and ₹ 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.4 [Pages 35 - 36]

Solve the following systems of linear equations by Cramer’s rule:

5x – 2y + 16 = 0, x + 3y – 7 = 0

Solve the following systems of linear equations by Cramer’s rule:

`3/2 + 2y = 12, 2/x + 3y` = 13

Solve the following systems of linear equations by Cramer’s rule:

3x + 3y – z = 11, 2x – y + 2z = 9, 4x + 3y + 2z = 25

Solve the following systems of linear equations by Cramer’s rule:

`3/x - 4/y - 2/z - 1` = 0, `1/x + 2/y + 1/z - 2` = 0, `2/x - 5/y - 4/z + 1` = 0

In a competitive examination, one mark is awarded for every correct answer while `1/4` mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly? (Use Cramer’s rule to solve the problem).

A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution? (Use Cramer’s rule to solve the problem).

A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. However, pump B can pump water in or out at the same rate. If pump B is inadvertently run in reverse, then the tank will be filled in 30 minutes. How long would it take each pump to fill the tank by itself? (Use Cramer’s rule to solve the problem)

A family of 3 people went out for dinner in a restaurant. The cost of two dosai, three idlies and two vadais is ₹ 150. The cost of the two dosai, two idlies and four vadais is ₹ 200. The cost of five dosai, four idlies and two vadais is ₹ 250. The family has ₹ 350 in hand and they ate 3 dosai and six idlies and six vadais. Will they be able to manage to pay the bill within the amount they had?

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.5 [Page 37]

Solve the following systems of linear equations by Gaussian elimination method:

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

Solve the following systems of linear equations by Gaussian elimination method:

2x + 4y + 6z = 22, 3x + 8y + 5z = 27, – x + y + 2z = 2

If ax² + bx + c is divided by x + 3, x – 5, and x – 1, the remainders are 21, 61 and 9 respectively. Find a, b and c. (Use Gaussian elimination method.)

An amount of ₹ 65,000 is invested in three bonds at the rates of 6%, 8% and 9% per annum respectively. The total annual income is ₹ 4,800. The income from the third bond is ₹ 600 more than that from the second bond. Determine the price of each bond. (Use Gaussian elimination method.)

A boy is walking along the path y = ax^{2} + bx + c through the points (– 6, 8), (– 2, – 12), and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.6 [Page 42]

Test for consistency and if possible, solve the following systems of equations by rank method:

x – y + 2z = 2, 2x + y + 4z = 7, 4x – y + z = 4

Test for consistency and if possible, solve the following systems of equations by rank method:

3x + y + z = 2, x – 3y + 2z = 1, 7x – y + 4z = 5

Test for consistency and if possible, solve the following systems of equations by rank method:

2x + 2y + z = 5, x – y + z = 1, 3x + y + 2z = 4

Test for consistency and if possible, solve the following systems of equations by rank method:

2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have no solution

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have unique solution

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have infinitely many solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have no solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have a unique solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have an infinite number of solutions

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.7 [Page 47]

Solve the following system of homogenous equations:

3x + 2y + 7z = 0, 4x – 3y – 2z = 0, 5x + 9y + 23z = 0

Solve the following system of homogenous equations:

2x + 3y – z = 0, x – y – 2z = 0, 3x + y + 3z = 0

Determine the values of λ for which the following system of equations x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has a unique solution

Determine the values of λ for which the following system of equations x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has a non-trivial solution

By using Gaussian elimination method, balance the chemical reaction equation:

\[\ce{C2H + O2 -> H2O + CO2}\]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.8 [Pages 48 - 50]

Choose the correct alternative:

If |adj(adj A)| = |A|^{9}, then the order of the square matrix A is

3

4

2

5

Choose the correct alternative:

If A is a 3 × 3 non-singular matrix such that AA^{T} = A^{T} A and B = A^{-1}A^{T}, then BB^{T} =

A

B

I

_{3}B

^{T}

Choose the correct alternative:

If A = `[(3, 5),(1, 2)]` B = adj A and C = 3A, then `(|"adj B"|)/|"C"|` =

`1/3`

`1/9`

`1/4`

1

Choose the correct alternative:

If A = `[(1, -2),(1, 4)] = [(6, 0),(0, 6)]`, then A =

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

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

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

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

Choose the correct alternative:

If A = `[(7, 3),(4, 2)]` then 9I_{2} – A =

`"A"^-1`

`"A"^-1/2`

`3"A"^-1`

`2"A"^-1`

Choose the correct alternative:

If A = `[(2, 0),(1, 5)]` and B = `[(1, 4),(2, 0)]` then |adj (AB)| =

– 40

– 80

– 60

– 20

Choose the correct alternative:

If + = `[(1, x, 0),(1, 3, 0),(2, 4, -2)]` is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

15

12

14

11

Choose the correct alternative:

If A = `[(3, 1, -1),(2, -2, 0),(1, 2, -1)]` and `"A"^-1 = [("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)]` then the value of a_{23} is

0

– 2

– 3

– 1

Choose the correct alternative:

If A B, and C are invertible matrices of some order, then which one of the following is not true?

adj A = |A|A

^{–}^{1}adj(AB) = (adj A)(adj B)

det A

^{–1}= (det A)^{–}^{1}(ABC)

^{–}^{1}= C^{–}^{1}B^{–}^{1}A^{–}^{1}

Choose the correct alternative:

If `("AB")^-1 = [(12, -17),(-19, 27)]` and `"A"^-1 = [(1, -1),(-2, 3)]` then `"B"^-1` =

`[(2, -5),(- 3, 8)]`

`[(8, 5),(3, 2)]`

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

`[(8, -5),(-3, 2)]`

Choose the correct alternative:

If A^{T}A^{–}^{1} is symmetric, then A^{2} =

A

^{–1}(A

^{T})^{2}A

^{T}(A

^{–}^{1})^{2}

Choose the correct alternative:

If A is a non-singular matrix such that A^{–1} = `[(5, 3),(-2, -1)]`, then (AT)^{–}^{1} =

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

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

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

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

Choose the correct alternative:

If A = `[(3/5, 4/5),(x, 3/5)]` and A^{T} = A^{–1}, then the value of x is

`(-4)/5`

`(-3)/5`

`3/5`

`4/5`

Choose the correct alternative:

If A = `[(1, tan theta/2),(- tan theta/2, 1)]` and AB = I_{2}, then B =

`(cos^2 theta/2)"A"`

`(cos^2 theta/2)"A"^"T"`

`(cos^2theta)"I"`

`(sin^2 theta/2)"A"`

Choose the correct alternative:

If A = `[(costheta, sintheta),(-sintheta, costheta)]` and A(adj A) = `[("k", 0),(0, "k")]`, then k =

0

sin θ

cos θ

1

Choose the correct alternative:

If A = `[(2, 3),(5, -2)]` be such that λA^{–1} = A, then λ is

17

14

19

21

Choose the correct alternative:

If adj A = `[(2, 3),(4, 1)]` and adj B = `[(1, -2),(-3, 1)]` then adj (AB) is

`[(-7, -1),(7, -9)]`

`[(-6, 5),(-2, -10)]`

`[(-7, 7),(-1, -9)]`

`[(-6, -2),(5, -10)]`

Choose the correct alternative:

The rank of the matrix `[(1, 2, 3, 4),(2, 4, 6, 8),(-1, -2, -3, -4)]` is

1

2

4

3

Choose the correct alternative:

If x^{a}y^{b} = e^{m}, x^{c}y^{d} = e^{n}, Δ_{1} = `|("m", "b"),("n", "d")|`, Δ_{2} = `|("a", "m"),("c", "n")|`, Δ_{3} = `|("a", "b"),("c", "d")|`, then the values of x and y are respectively,

`"e"^(Delta_2/Delta_1), "e"^(Delta_3/Delta_1)`

`log(Delta_1/Delta_3), log(Delta_2/Delta_3)`

`log(Delta_2/Delta_1), log(Delta_3/Delta_1)`

`"e"^(Delta_1/Delta_3), "e"^(Delta_2/Delta_3)`

Choose the correct alternative:

Which of the following is/are correct?

(i) Adjoint of a symmetric matrix is also a symmetric matrix.

(ii) Adjoint of a diagonal matrix is also a diagonal matrix.

(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λ^{n} adj (A).

(iv) A(adj A) = (adj A)A = |A|I

Only (i)

(ii) and (iii)

(iii) and (iv)

(i), (ii) and (iv)

Choose the correct alternative:

If ρ(A) ρ([A|B]), then the system AX = B of linear equations is

consistent and has a unique solution

consistent

consistent and has infinitely many solution

inconsistent

Choose the correct alternative:

If 0 ≤ θ ≤ π and the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ) x – y + z = 0, (sin θ) x + y + z = 0 has a non-trivial solution then θ is

`(2pi)/3`

`(3pi)/4`

`(5pi)/6`

`pi/4`

Choose the correct alternative:

The augmented matrix of a system of linear equations is `[(1, 2, 7, 3),(0, 1, 4, 6),(0, 0, lambda - 7, mu + 7)]`. This system has infinitely many solutions if

λ = 7, µ ≠ – 5

λ = – 7, µ = 5

λ ≠ 7, µ ≠ – 5

λ = 7, µ = – 5

Choose the correct alternative:

Let A = `[(2, -1, 1),(-1, 2, -1),(1, -1, 2)]` and 4B = `[(3, 1, -1),(1, 3, x),(-1, 1, 3)]`. If B is the inverse of A, then the value of x is

2

4

3

1

Choose the correct alternative:

If A = `[(3, -3, 4),(2, -3, 4),(0, -1, 1)]`, then adj(adj A) is

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

`[(6, -6, 8),(4, -6, 8),(0, -2, 2)]`

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

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

## Solutions for Chapter 1: Applications of Matrices and Determinants

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 1 - Applications of Matrices and Determinants

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 1 Applications of Matrices and Determinants are Introduction to Applications of Matrices and Determinants, Inverse of a Non-singular Square Matrix, Elementary Transformations of a Matrix, Applications of Matrices: Solving System of Linear Equations, Applications of Matrices: Consistency of System of Linear Equations by Rank Method.

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