Let α , β ( α  = β ) be the values of m , for which the equations x + y + z = 1 ; x + 2 y + 4 z = m and x + 4 y + 10 z = m 2 have infinitely many solutions. Then the value of ∑ n = 1 10 ( n α + n β

Question

Let α , β ( α  = β ) be the values of m , for which the equations x + y + z = 1 ; x + 2 y + 4 z = m and x + 4 y + 10 z = m 2 have infinitely many solutions. Then the value of ∑ n = 1 10 ​ ( n α + n β ) is equal to :

TestHub · Accepted Answer

Δ ​ = ​ 1 1 1 ​ 1 2 4 ​ 1 4 10 ​ ​ = 1 ( 20 − 16 ) − 1 ( 10 − 4 ) + 1 ( 4 − 2 ) = 4 − 6 + 2 = 0 ​ For infinite solutions ​ Δ x ​ = Δ y ​ = Δ z ​ = 0 m 2 − 3 x + 2 = 0 m = 1 , 2 α = 1 , β = 2 ∴ n = 1 ∑ 10 ​ ( n α + n β ) = n = 1 ∑ 10 ​ n 1 + n = 1 ∑ 10 ​ n 2 = 2 10 ( 11 ) ​ + 6 10 ( 11 ) ( 21 ) ​ = 55 + 385 = 440 ​

Mathematics - Determinant Question with Solution | TestHub

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