Q33 of 44 Page 4

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.(CBSE 2013)

Let the numbers are x, y, z be the cash awards for Honesty, Regularity and Hard Work


x + y + z = 6000 …… (i)


Also,


x + 3z = 11000 …… (ii)


Again,


x – 2y + z = 0 …… (iii)



A X = B


|A| = 1(0 + 6) – 1(1 – 3) + 1(– 2 – 0)


= 1(6) – 1(– 2) – 2


= 6 + 2 – 2


= 6


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


C11 = (– 1)1 + 1 (0 + 6) = 6


C12 = (– 1)1 + 2 (1 – 3) = 2


C13 = (– 1)1 + 3 (– 2 – 0) = – 2


C21 = (– 1)2 + 1 (1 + 2) = – 3


C22 = (– 1)2 + 2 (1 – 1) = 0


C23 = (– 1)2 + 3 (– 2 – 1 ) = 3


C31 = (– 1)3 + 1 (3 – 0) = 3


C32 = (– 1)3 + 2 (3 – 1) = – 2


C33 = (– 1)3 + 3 (0 – 1) = – 1


Adj A =


X = A – 1 B =


X =


X =


=


Hence, x = 500, y = 2000 and z = 3500

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31

Find A – 1, if . Hence, solve the following system of linear equations:

x + 2y + 5z = 10, x – y – z = – 2,


2x + 3y – z = – 11 (CBSE 2012)

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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. (CBSE 2013)

 34

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?(CBSE 2013)

 35

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? (CBSE 2013)