Q. 345.0( 1 Vote )

# 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)*

*(CBSE 2013)*

Answer :

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 ^{– 1}B

C_{11 =} (– 1)^{1 + 1} (3 – 4) = – 1

C_{12} = (– 1)^{1 + 2} (5 – 4) = – 1

C_{13} = (– 1)^{1 + 3} (5 – 3) = 2

C_{21} = (– 1)^{2 + 1} (3 – 2) = – 1

C_{22} = (– 1)^{2 + 2} (4 – 2) = 2

C_{23} = (– 1)^{2 + 3} (4 – 3 ) = – 1

C_{31} = (– 1)^{3 + 1} (12 – 6) = 6

C_{32} = (– 1)^{3 + 2} (16 – 10) = – 6

C_{33} = (– 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.

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The determinant is equal to zero.

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