#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

## Chapter 8: Solution of Simultaneous Linear Equations

#### Chapter 8: Solution of Simultaneous Linear Equations Exercise 8.1, 8.10 solutions [Pages 14 - 18]

Solve the following system of equations by matrix method:

5x + 7y + 2 = 0

4x + 6y + 3 = 0

Solve the following system of equations by matrix method:

5x + 2y = 3

3x + 2y = 5

Solve the following system of equations by matrix method:

3x + 4y − 5 = 0

x − y + 3 = 0

Solve the following system of equations by matrix method:

3x + y = 19

3x − y = 23

Solve the following system of equations by matrix method:

3x + 7y = 4

x + 2y = −1

Solve the following system of equations by matrix method:

3x + y = 7

5x + 3y = 12

Solve the following system of equations by matrix method:

x + y − z = 3

2x + 3y + z = 10

3x − y − 7z = 1

Solve the following system of equations by matrix method:

x + y + z = 3

2x − y + z = − 1

2x + y − 3z = − 9

Solve the following system of equations by matrix method:

6x − 12y + 25z = 4

4x + 15y − 20z = 3

2x + 18y + 15z = 10

Solve the following system of equations by matrix method:

3x + 4y + 7z = 14

2x − y + 3z = 4

x + 2y − 3z = 0

Solve the following system of equations by matrix method:

\[\frac{2}{x} - \frac{3}{y} + \frac{3}{z} = 10\]

\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 10\]

\[\frac{3}{x} - \frac{1}{y} + \frac{2}{z} = 13\]

Solve the following system of equations by matrix method:

5x + 3y + z = 16

2x + y + 3z = 19

x + 2y + 4z = 25

Solve the following system of equations by matrix method:

3x + 4y + 2z = 8

2y − 3z = 3

x − 2y + 6z = −2

Solve the following system of equations by matrix method:

2x + y + z = 2

x + 3y − z = 5

3x + y − 2z = 6

Solve the following system of equations by matrix method:

2x + 6y = 2

3x − z = −8

2x − y + z = −3

Solve the following system of equations by matrix method:

x − y + z = 2

2x − y = 0

2y − z = 1

Solve the following system of equations by matrix method:

8x + 4y + 3z = 18

2x + y +z = 5

x + 2y + z = 5

Solve the following system of equations by matrix method:

x + y + z = 6

x + 2z = 7

3x + y + z = 12

Solve the following system of equations by matrix method:

Solve the following system of equations by matrix method:

x − y + 2z = 7

3x + 4y − 5z = −5

2x − y + 3z = 12

Show that the following systems of linear equations is consistent and also find their solutions:

6x + 4y = 2

9x + 6y = 3

Show that the following systems of linear equations is consistent and also find their solutions:

2x + 3y = 5

6x + 9y = 15

Show that the following systems of linear equations is consistent and also find their solutions:

5x + 3y + 7z = 4

3x + 26y + 2z = 9

7x + 2y + 10z = 5

Show that the following systems of linear equations is consistent and also find their solutions:

x − y + z = 3

2x + y − z = 2

−x −2y + 2z = 1

Show that the following systems of linear equations is consistent and also find their solutions:

x + y + z = 6

x + 2y + 3z = 14

x + 4y + 7z = 30

Show that the following systems of linear equations is consistent and also find their solutions:

2x + 2y − 2z = 1

4x + 4y − z = 2

6x + 6y + 2z = 3

Show that each one of the following systems of linear equation is inconsistent:

2x + 5y = 7

6x + 15y = 13

Show that each one of the following systems of linear equation is inconsistent:

2x + 3y = 5

6x + 9y = 10

Show that each one of the following systems of linear equation is inconsistent:

4x − 2y = 3

6x − 3y = 5

Show that each one of the following systems of linear equation is inconsistent:

4x − 5y − 2z = 2

5x − 4y + 2z = −2

2x + 2y + 8z = −1

Show that each one of the following systems of linear equation is inconsistent:

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

Show that each one of the following systems of linear equation is inconsistent:

x + y − 2z = 5

x − 2y + z = −2

−2x + y + z = 4

If \[A = \begin{bmatrix}2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2\end{bmatrix}\], find A^{−1} and hence solve the system of linear equations 2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y + 2z = −3

Find A^{−1}, if \[A = \begin{bmatrix}1 & 2 & 5 \\ 1 & - 1 & - 1 \\ 2 & 3 & - 1\end{bmatrix}\] . Hence solve the following system of linear equations:x + 2y + 5z = 10, x − y − z = −2, 2x + 3y − z = −11

^{−1}. Using A

^{−1}, solve the system of linear equations x − 2y = 10, 2x + y + 3z = 8, −2y + z = 7.

^{−1}and hence solve the following system of equations:

If \[A = \begin{bmatrix}1 & 2 & 0 \\ - 2 & - 1 & - 2 \\ 0 & - 1 & 1\end{bmatrix}\] , find A^{−1}. Using A^{−1}, solve the system of linear equations x − 2y = 10, 2x − y − z = 8, −2y + z = 7

Given \[A = \begin{bmatrix}2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5\end{bmatrix}, B = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}\] , find BA and use this to solve the system of equations y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17

If \[A = \begin{bmatrix}2 & 3 & 1 \\ 1 & 2 & 2 \\ 3 & 1 & - 1\end{bmatrix}\] , find A^{–1} and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.

Use product \[\begin{bmatrix}1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4\end{bmatrix}\begin{bmatrix}- 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2\end{bmatrix}\] to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

An amount of Rs 10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and second investment is Rs 190 short of the income from the third. Find the investment in each using matrix method.

A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.

The prices of three commodities P, Q and R are Rs x, y and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. Cpurchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earn Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. *x*, *y* and *z* respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of *x*, *y* and *z*. What values are described in this equations?

Two factories decided to award their employees for three values of (a) adaptable tonew techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ *x*, ₹ *y* and ₹ *z* per person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then

i) represent the above situation by matrix equation and form linear equation using matrix multiplication.

ii) Solve these equation by matrix method.

iii) Which values are reflected in the questions?

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹1,600. School B wants to spend ₹2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹2,200. School Q wants to spend ₹3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹1,200, using matrices, find the award money for each value.

Apart from these three values, suggest one more value which should be considered for award.

A total amount of ₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and \[8\frac{1}{2}\] % respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.

A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.

#### Chapter 8: Solution of Simultaneous Linear Equations Exercise 8.2 solutions [Pages 20 - 21]

2*x* − *y* + *z* = 0

3*x* + 2*y* − *z* = 0*x* + 4*y* + 3*z* = 0

2*x* − *y* + 2*z* = 0

5*x* + 3*y* − *z* = 0*x* + 5*y* − 5*z* = 0

3*x* − *y* + 2*z* = 0

4*x* + 3*y* + 3*z* = 0

5*x* + 7*y* + 4*z* = 0

*x* + *y* − 6*z* = 0*x* − *y* + 2*z* = 0

−3*x* + *y* + 2*z* = 0

*x* + *y* + *z* = 0*x* − *y* − 5*z* = 0*x* + 2*y* + 4*z* = 0

x + y − z = 0

x − 2y + z = 0

3x + 6y − 5z = 0

3x + y − 2z = 0

x + y + z = 0

x − 2y + z = 0

2x + 3y − z = 0

x − y − 2z = 0

3x + y + 3z = 0

#### Chapter 8: Solution of Simultaneous Linear Equations Exercise 8.2 solutions [Page 21]

If \[\begin{bmatrix}1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}\], find x, y and z.

If \[\begin{bmatrix}1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ - 1 \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}\] , find x, y and z.

Solve the following for x and y: \[\begin{bmatrix}3 & - 4 \\ 9 & 2\end{bmatrix}\binom{x}{y} = \binom{10}{ 2}\]

#### Chapter 8: Solution of Simultaneous Linear Equations Exercise 8.2 solutions [Pages 21 - 23]

The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has

a unique solution

no solution

an infinite number of solutions

zero solution as the only solution

The number of solutions of the system of equations

2x + y − z = 7

x − 3y + 2z = 1

x + 4y − 3z = 5

is

3

2

1

0

Let \[X = \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}, A = \begin{bmatrix}1 & - 1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}3 \\ 1 \\ 4\end{bmatrix}\] . If AX = B, then X is equal to

\[\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}\]

\[\begin{bmatrix}- 1 \\ - 2 \\ - 3\end{bmatrix}\]

\[\begin{bmatrix}- 1 \\ - 2 \\ - 3\end{bmatrix}\]

\[\begin{bmatrix}- 1 \\ 2 \\ 3\end{bmatrix}\]

\[\begin{bmatrix}0 \\ 2 \\ 1\end{bmatrix}\]

The number of solutions of the system of equations:

2x + y − z = 7

x − 3y + 2z = 1

x + 4y − 3z = 5

3

2

1

0

The system of linear equations:

x + y + z = 2

2x + y − z = 3

3x + 2y + kz = 4 has a unique solution if

k ≠ 0

−1 < k < 1

−2 < k < 2

k = 0

Consider the system of equations:

a_{1}x + b_{1}y + c_{1}z = 0

a_{2}x + b_{2}y + c_{2}z = 0

a_{3}x + b_{3}y + c_{3}z = 0,

if \[\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}\]= 0, then the system has

more than two solutions

one trivial and one non-trivial solutions

no solution

only trivial solution (0, 0, 0)

Let a, b, c be positive real numbers. The following system of equations in x, y and z

(a) no solution

(b) unique solution

(c) infinitely many solutions

(d) finitely many solutions

no solution

unique solution

infinitely many solutions

finitely many solutions

For the system of equations:

x + 2y + 3z = 1

2x + y + 3z = 2

5x + 5y + 9z = 4

there is only one solution

there exists infinitely many solution

there is no solution

none of these

The existence of the unique solution of the system of equations:

x + y + z = λ

5x − y + µz = 10

2x + 3y − z = 6

depends on

µ only

λ only

λ and µ both

neither λ nor µ

The system of equations:

x + y + z = 5

x + 2y + 3z = 9

x + 3y + λz = µ

has a unique solution, if

(a) λ = 5, µ = 13

(b) λ ≠ 5

(c) λ = 5, µ ≠ 13

(d) µ ≠ 13

λ = 5, µ = 13

λ ≠ 5

λ = 5, µ ≠ 13

µ ≠ 13

## Chapter 8: Solution of Simultaneous Linear Equations

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

#### Textbook solutions for Class 12

## RD Sharma solutions for Class 12 Mathematics chapter 8 - Solution of Simultaneous Linear Equations

RD Sharma solutions for Class 12 Maths chapter 8 (Solution of Simultaneous Linear Equations) 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 Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 8 Solution of Simultaneous Linear Equations are Minors and Co-factors, Area of a Triangle, Introduction of Determinant, Determinants of Matrix of Order One and Two, Determinant of a Square Matrix, Properties of Determinants, Adjoint and Inverse of a Matrix, Elementary Transformations, Applications of Determinants and Matrices, Determinant of a Matrix of Order 3 × 3, Rule A=KB.

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