Q15 of 75 Page 8

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

Let the numbers are x, y, z be the cash awards for Resourcefulness, Competence, and Determination respectively


4x + 3y + 2z = 37000 …… (i)


Also,


5x + 3y + 4z = 47000 …… (ii)


Again,


x + y + z = 12000 …… (iii)



A X = B


|A| = 4(3 – 4) – 3(5 – 4) + 2(5 – 3)


= 4(– 1) – 3(1) + 2(2)


= – 4 – 3 + 4


= – 3


Hence, the unique solution given by x = A – 1B


C11 = (– 1)1 + 1 (3 – 4) = – 1


C12 = (– 1)1 + 2 (5 – 4) = – 1


C13 = (– 1)1 + 3 (5 – 3) = 2


C21 = (– 1)2 + 1 (3 – 2) = – 1


C22 = (– 1)2 + 2 (4 – 2) = 2


C23 = (– 1)2 + 3 (4 – 3 ) = – 1


C31 = (– 1)3 + 1 (12 – 6) = 6


C32 = (– 1)3 + 2 (16 – 10) = – 6


C33 = (– 1)3 + 3 (12 – 15) = – 3


Adj A =


X = A – 1 B =


X =


X =


=


Hence, x = 4000, y = 5000 and z = 3000


Thus, The value X, Y, Z describes the amount of prizes per person for Resourcefulness, Competence and Determination.


More from this chapter

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13

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping 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. A part from these values, namely, honesty cooperation and supervision, suggest one more value which the management must include for awards.

 14

A school wants to award its student for the values of Honesty, Regularity and Hard work with a total cash award of ₹6000. Three times the award money for Hard work added to that given for honesty amounts to ₹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 the matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, and suggest one more value which the school must include for awards.

 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 ₹x, ₹y and ₹z per persons respectively. The first factory decided to honour respectively 2, 4 and 3 employees with 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 a matrix equation and form linear equations using matrix multiplication.


ii. Solve these equations using matrices.


iii. Which values are reflected in the questions?

 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 total award money of ₹16,00. 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 the award.