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

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

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) Chapter 8: Solution of Simultaneous Linear Equations

Ex. 8.1Ex. 8.10Ex. 8.2Others

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

Ex. 8.1 | Q 1.1 | Page 14

Solve the following system of equations by matrix method:
5x + 7y + 2 = 0
4x + 6y + 3 = 0

Ex. 8.1 | Q 1.2 | Page 14

Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5

Ex. 8.1 | Q 1.3 | Page 14

Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 0

Ex. 8.1 | Q 1.4 | Page 14

Solve the following system of equations by matrix method:
3x + y = 19
3x − y = 23

Ex. 8.1 | Q 1.5 | Page 14

Solve the following system of equations by matrix method:
3x + 7y = 4
x + 2y = −1

Ex. 8.1 | Q 1.6 | Page 14

Solve the following system of equations by matrix method:
3x + y = 7
5x + 3y = 12

Ex. 8.1 | Q 2.01 | Page 14

Solve the following system of equations by matrix method:
x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1

Ex. 8.1 | Q 2.02 | Page 14

Solve the following system of equations by matrix method:
x + y + z = 3
2x − y + z = − 1
2x + y − 3z = − 9

Ex. 8.1 | Q 2.03 | Page 14

Solve the following system of equations by matrix method:
6x − 12y + 25z = 4
4x + 15y − 20z = 3
2x + 18y + 15z = 10

Ex. 8.1 | Q 2.04 | Page 14

Solve the following system of equations by matrix method:
3x + 4y + 7z = 14
2x − y + 3z = 4
x + 2y − 3z = 0

Ex. 8.1 | Q 2.05 | Page 14

Solve the following system of equations by matrix method:
$\frac{2}{x} - \frac{3}{y} + \frac{3}{z} = 10$
$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 10$
$\frac{3}{x} - \frac{1}{y} + \frac{2}{z} = 13$

Ex. 8.1 | Q 2.06 | Page 14

Solve the following system of equations by matrix method:
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25

Ex. 8.1 | Q 2.07 | Page 14

Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2

Ex. 8.1 | Q 2.08 | Page 14

Solve the following system of equations by matrix method:
2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6

Ex. 8.1 | Q 2.09 | Page 14

Solve the following system of equations by matrix method:
2x + 6y = 2
3x − z = −8
2x − y + z = −3

Ex. 8.1 | Q 2.1 | Page 14

Solve the following system of equations by matrix method:
x − y + z = 2
2x − y = 0
2y − z = 1

Ex. 8.1 | Q 2.11 | Page 14

Solve the following system of equations by matrix method:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5

Ex. 8.1 | Q 2.12 | Page 14

Solve the following system of equations by matrix method:
x + y + z = 6
x + 2z = 7
3x + y + z = 12

Ex. 8.1 | Q 2.13 | Page 14

Solve the following system of equations by matrix method:

$\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4, \frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1, \frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2; x, y, z \neq 0$

Ex. 8.1 | Q 2.14 | Page 14

Solve the following system of equations by matrix method:
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12

Ex. 8.1 | Q 3.1 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
6x + 4y = 2
9x + 6y = 3

Ex. 8.1 | Q 3.2 | Page 15

Show that the following systems of linear equations is consistent and also find their solutions:
2x + 3y = 5
6x + 9y = 15

Ex. 8.1 | Q 3.3 | Page 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

Ex. 8.1 | Q 3.4 | Page 15

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

Ex. 8.1 | Q 3.5 | Page 15

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

Ex. 8.1 | Q 3.6 | Page 15

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

Ex. 8.1 | Q 4.1 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13

Ex. 8.1 | Q 4.2 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10

Ex. 8.1 | Q 4.3 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5

Ex. 8.1 | Q 4.4 | Page 15

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

Ex. 8.1 | Q 4.5 | Page 15

Show that each one of the following systems of linear equation is inconsistent:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3

Ex. 8.1 | Q 4.6 | Page 15

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

Ex. 8.10 | Q 5 | Page 15
If $A = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}\text{ and }B = \begin{bmatrix}2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5\end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear equations: x − y = 3, 2x + 3y + 4z = 17, y + 2z = 7
Ex. 8.1 | Q 6 | Page 15

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

Ex. 8.1 | Q 7 | Page 16

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

Ex. 8.1 | Q 8.1 | Page 16
If $A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}$ , find A−1. Using A−1, solve the system of linear equations  x − 2y = 10, 2x + y + 3z = 8, −2y + z = 7.
Ex. 8.1 | Q 8.2 | Page 16
If $A = \begin{bmatrix}3 & - 4 & 2 \\ 2 & 3 & 5 \\ 1 & 0 & 1\end{bmatrix}$ , find A−1 and hence solve the following system of equations:
Ex. 8.1 | Q 8.3 | Page 16
$A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}7 & 2 & - 6 \\ - 2 & 1 & - 3 \\ - 4 & 2 & 5\end{bmatrix}$, find AB. Hence, solve the system of equations: x − 2y = 10, 2x + y + 3z = 8 and −2y + z = 7
Ex. 8.1 | Q 8.4 | Page 16

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

Ex. 8.1 | Q 8.5 | Page 16

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

Ex. 8.1 | Q 8.6 | Page 16

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.

Ex. 8.1 | Q 8.7 | Page 16

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.

Ex. 8.1 | Q 9 | Page 16

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.

Ex. 8.1 | Q 10 | Page 16

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.

Ex. 8.1 | Q 11 | Page 16

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.

Ex. 8.1 | Q 12 | Page 16

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.

Ex. 8.1 | Q 13 | Page 17

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.

Ex. 8.1 | Q 14 | Page 17

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.

Ex. 8.1 | Q 15 | Page 17

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. xy 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 xy and z. What values are described in this equations?

Ex. 8.1 | Q 16 | Page 17

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?

Ex. 8.1 | Q 17 | Page 17

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.

Ex. 8.1 | Q 18 | Page 17

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.

Ex. 8.1 | Q 19 | Page 17

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.

Ex. 8.1 | Q 20 | Page 18

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.

Ex. 8.1 | Q 21 | Page 18

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]

Ex. 8.2 | Q 1 | Page 20

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

Ex. 8.2 | Q 2 | Page 20

2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0

Ex. 8.2 | Q 3 | Page 20

3x − y + 2z = 0
4x + 3y + 3z = 0
5x + 7y + 4z = 0

Ex. 8.2 | Q 4 | Page 20

x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0

Ex. 8.2 | Q 5 | Page 20

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

Ex. 8.2 | Q 6 | Page 20

x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0

Ex. 8.2 | Q 7 | Page 21

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

Ex. 8.2 | Q 8 | Page 21

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]

Ex. 8.2 | Q 1 | Page 21
If $\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ - 1 \\ 0\end{bmatrix}$, find x, y and z.
Ex. 8.2 | Q 2 | 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.

Ex. 8.2 | Q 3 | Page 21

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.

Ex. 8.2 | Q 4 | Page 21

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

Ex. 8.2 | Q 5 | Page 21
If $\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}2 \\ - 1 \\ 3\end{bmatrix}$, find x, y, z.
Ex. 8.2 | Q 6 | Page 21
If $A = \begin{bmatrix}2 & 4 \\ 4 & 3\end{bmatrix}, X = \binom{n}{1}, B = \binom{ 8}{11}$  and AX = B, then find n.

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

Ex. 8.2 | Q 1 | Page 21

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

Ex. 8.2 | Q 2 | Page 21

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

Q 3 | Page 22

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

• $\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}$

• $\begin{bmatrix}- 1 \\ - 2 \\ - 3\end{bmatrix}$

• $\begin{bmatrix}- 1 \\ - 2 \\ - 3\end{bmatrix}$

• $\begin{bmatrix}- 1 \\ 2 \\ 3\end{bmatrix}$

• $\begin{bmatrix}0 \\ 2 \\ 1\end{bmatrix}$

Q 4 | Page 22

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

Q 5 | Page 22

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

Q 6 | Page 22

Consider the system of equations:
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 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)

Q 7 | Page 22

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

$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1, \frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, - \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \text { has }$
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions
• no solution

• unique solution

• infinitely many solutions

•  finitely many solutions

Q 8 | Page 22

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

Q 9 | Page 22

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

Q 10 | Page 23

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

Ex. 8.1Ex. 8.10Ex. 8.2Others

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) RD Sharma solutions for Class 12 Mathematics chapter 8 - Solution of Simultaneous Linear Equations

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

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