2 Dr. Roy decided to reward his best students, Joyeeta and Sunalini. He split 8000 between them in the ratio of 3:1. Workout how much each of them got.
Answers
Step-by-step explanation:
Let x,y and z be the prize amount per student for Discipline, Politeness and Punctuality respectively.
3x+2y+z=1000
4x+y+3z=1500
x+y+z=600
These three equations can be written as
⎣
⎢
⎢
⎡
3
4
1
2
1
1
1
3
1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1000
1500
600
⎦
⎥
⎥
⎤
AX=B
∣A∣=3(1–3)–2(4–3)+1(4–1)
=−6−2+3=−5
Hence, the unique solution given by X=A
−1
B
C
11
=(−1)
1+1
∣
∣
∣
∣
∣
∣
1
1
3
1
∣
∣
∣
∣
∣
∣
=1−3=−2
C
12
=(−1)
1+2
∣
∣
∣
∣
∣
∣
4
1
3
1
∣
∣
∣
∣
∣
∣
=−(4−3)=−1
C
13
=(−1)
1+3
∣
∣
∣
∣
∣
∣
4
1
1
1
∣
∣
∣
∣
∣
∣
=4−1=3
C
21
=(−1)
2+1
∣
∣
∣
∣
∣
∣
2
1
1
1
∣
∣
∣
∣
∣
∣
=−(2−1)=−1
C
22
=(−1)
2+2
∣
∣
∣
∣
∣
∣
3
1
1
1
∣
∣
∣
∣
∣
∣
=3−1=2
C
23
=(−1)
2+3
∣
∣
∣
∣
∣
∣
3
1
2
1
∣
∣
∣
∣
∣
∣
=−(3−2)=−1
C
31
=(−1)
3+1
∣
∣
∣
∣
∣
∣
2
1
1
3
∣
∣
∣
∣
∣
∣
=6−1=5
C
32
=(−1)
3+2
∣
∣
∣
∣
∣
∣
3
4
1
3
∣
∣
∣
∣
∣
∣
=−(9−4)=−5
C
33
=(−1)
3+3
∣
∣
∣
∣
∣
∣
3
4
2
1
∣
∣
∣
∣
∣
∣
=3−8=−5
adjA=
⎣
⎢
⎢
⎡
⎣
⎢
⎢
⎡
−2
−1
5
−1
2
−5
3
−1
−5
⎦
⎥
⎥
⎤
⎦
⎥
⎥
⎤
T
=
⎣
⎢
⎢
⎡
−2
−1
3
−1
2
−1
5
−5
−5
⎦
⎥
⎥
⎤
X=A
−1
B=
∣A∣
1
(adj(A))B
X=−
5
1
⎣
⎢
⎢
⎡
−2
−1
3
−1
2
−1
5
−5
−5
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
1000
1500
600
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
5
−1
⎣
⎢
⎢
⎡
−2000−1500+3000
−1000+3000−3000
3000−1500−3000
⎦
⎥
⎥
⎤
=
5
−1
⎣
⎢
⎢
⎡
−500
−1000
−1500
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
100
200
300
⎦
⎥
⎥
⎤
∴x=100,y=200,z=300
VALUE:Excellence in extra curricular activities should be another value considered for an award.