Chapter 8 Solution of Simultaneous Linear Equations Ex. 8.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 2(vii)

Solution 2(vii)

Question 2(viii)

Solution 2(viii)

Question 2(ix)

Solution 2(ix)

Question 2(x)

Solution 2(x)

Question 2(xi)

Solution 2(xi)

Question 2(xii)

Solution 2(xii)

Question 2(xiii)

Solution 2(xiii)

Question 2(xiv)

Solution 2(xiv)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 4(vi)

Solution 4(vi)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 8(iv)

Using A^-1, solve the system of linear equations

X – 2y = 10, 2x – y – z = 8 and -2y + z = 7Solution 8(iv)

Question 8(v)

Solution 8(v)

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

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.Solution 13

Question 14

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.Solution 14

The school can include an award for creativity and extra-curricular activities.Question 15

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?Solution 15

Question 16

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?Solution 16

Keeping calm in a tense situation is more rewarding than carefulness, and carefulness is more rewarding than adaptability.Question 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 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.Solution 17

Question 18

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.Solution 18

Question 19

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.Solution 19

Question 20

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.Solution 20

Let the amount deposited be x, y and z respectively.

As per the data in the question, we get

Question 8(vi)

If  find A^-1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.Solution 8(vi)

Therefore, A is invertible.

Let C_ij be the co-factors of the elements a_ij.

Now, the given system of equations is expressible as

Here we have |A^T| = |A| = -16 ≠ 0

Therefore, the given system of equations is consistent with a unique solution given by

Question 8(vii)

Use the product  to solve the system of equations x + 3z = -9, -x + 2y – 2z = 4, 2x – 3y + 4z = -3.Solution 8(vii)

Let 

Now, 

Now, the given system of equations is expressible as

Here we have |B^T| = |B| = -1 ≠ 0

Therefore, the given system of equations is consistent with a unique solution given by

Hence, x = 36, y = 5 and z = -15.Question 21

A shopkeeper has 3 varieties of pens ‘A’, ‘B’ and ‘C’. Meenu purchased 1 pen of each variety for a total of Rs. 21. Jeen purchased 4 pens of ‘A’ variety, 3 pens of ‘B’ variety and 2 pens of ‘C’ variety for Rs. 60. Using matrix method find the cost of each pen.Solution 21

From the given information, we can form a matrix as follows

Applying R₂→ R₂ – 4R₁, R₃→ R₃ – 6R₁

Applying R₃→ R₃ + (-4R₁)

From the above matrix form, we get

A + B + C = 21 … (i)

-B – 2C = -24 … (ii)

5C = 40

⇒ C = 8 … (iii)

Putting the value of C in (ii), we get

B = 8

Substituting B and C in (i), we get

C = 5

Hence, cost of variety ‘A’ pen is Rs. 8, cost of variety B pen is Rs. 8 and cost of variety ‘C’ pen is Rs. 5.

Chapter 8 Solution of Simultaneous Linear Equations Ex. 8.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8