## Chapter 8 – Solution of Simultaneous Linear Equations Exercise Ex. 8.1

Using A^{-1}, solve the system of

linear equations

X – 2y = 10, 2x + y + 3z = 8 and -2y + z

= 7

The management committee of a

residential colony decided to award some of its members (say x) for honesty,

some (say y) for helping and 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 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. 6000. Three times the award money for Hard work added to that given

for honesty amounts to Rs. 11000. 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 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.

The school can include

an award for creativity and extra-curricular activities.

Two institutions decided to award

their employees for the three values of resourcefulness, competence and

determination in the form of prizes at the rate of Rs. *x*, Rs. *y* and Rs. *z* respectively per person. The first

institution decided to award respectively 4, 3 and 2 employees with a total prize money of Rs. 37000 and the second

institution decided to award respectively 5, 3 and 4 employees with a total

prize money of Rs. 47000. If all the three prizes per person together amount

to Rs. 12000, then using matrix method find the value of *x*, *y* and *z*. What values are described in these

equations?

Two factories decided

to award their employees for three values of (a) adaptable to new techniques,

(b) careful and alert in difficult situations and (c) keeping calm in tense

situations, at the rate of Rs. *x*,

Rs. *y* and Rs. *z* per person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of Rs. 29000. The second factory

decided to honour respectively 5, 2 and 3 employees

with the prize money of Rs. 30500. If the three prizes per person together

cost Rs 9500, then

(i)

represent the above situation by matrix equation and

form linear equations using matrix multiplication.

(ii) Solve these

equations using matrices.

(iii) Which values are

reflected in the questions?

Keeping calm in a tense

situation is more rewarding than carefulness, and carefulness is more

rewarding than adaptability.

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 Rs. *x* each Rs. *y* each and

Rs. *z* each for the three respective

values to 3, 2 and 1 students respectively with a total

award money of Rs. 1,600. School B wants to spend Rs 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 Rs 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 Rs. *x* each, Rs. *y* each and Rs. *z* each for the three respectively values to its 3, 2 and 1

students with a total award money of Rs. 1,000. School *Q* wants to spend Rs. 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 Rs.

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 Rs. *x* each, Rs. *y* each and Rs. *z* each

for the three respectively values to its 3, 2 and 1 students with a total

award money of Rs. 2,200. School *Q*

wants to spend Rs. 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 Rs. 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 Rs. 7000 is

deposited in three different saving bank accounts with annual interest rates

5%, 8% and 8.5% respectively. The total annual interest from these three

accounts is Rs. 550. Equal amounts have been deposited in the 5% and 8%

savings accounts. Find the amount deposited in each of the three accounts,

with the help of matrices.

Let the amount

deposited be x, y and z respectively.

As per the data in the

question, we get

## Chapter 8 – Solution of Simultaneous Linear Equations Exercise Ex. 8.2

## Chapter 8 – Solution of Simultaneous Linear Equations Exercise Ex. 8VSAQ

## Chapter 8 – Solution of Simultaneous Linear Equations Exercise MCQ

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

a. a unique solution

b. no solution

c. an infinite number of solutions

d. zero solution as the only solution

a. 3

b. 2

c. 1

d. 0

a.

b.

c.

d.

The number of solutions of the system of equations:

, is

a. 3

b. 2

c. 1

d. 0

The system of linear equations:

Has a unique solution if

a. k ≠ 0

b. -1 < k < 1

c. -2 < k < 2

d. 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.

a. more than two solutions

b. one trivial and one non-trivial solutions

c. no solution

d. 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 solutions

b. unique solution

c. there is no solution

d. finitely many solutions

For the system of equations :

x + 2y + 3x = 1

2x + y + 3z = 2

5x + 5y + 9z = 4

a. there is only one solution

b. there exists infinitely many solution

c. there is no solution

d. 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

a. μ only

b. λ only

c. λ and μ both

d. 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