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प्रश्न
Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method
उत्तर
Let the time taken by one man alone be x days and one woman alone be y days.
4x + 4y = `1/3`
2x + 5y = `1/4`
Matrix form `[(4, 4),(2, 5)] [(x),(y)] = [(1/3),(1/4)]`
AX = B
X = `"A"^-1 "B"`
A = `[(4, 4),(2, 5)]`
|A|+ 20 – 8
= 12 ≠ 0.A–1 exists.
adj A = `[(5, -4),(-2, 4)]`
A–1 = `1/|"A"|`
adj A = `1/12[(5, -4),(-2, 4)]`
X = `1/12[(5, -4),(-2, 4)][(1/3),(1/4)]`
= `1/12[(5/3 - 1),((-2)/3 + 1)]`
= `1/12 [(2/3),(1/3)]`
= `[(1/18),(1/36)]`
`[(x),(y)] = [(1/18),(1/36)]`
∴ One man can do 18 days
One woman can do 36 days
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