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JEE Main202310 Apr 2023Evening ShiftMathematicsDeterminantsActual

Let S be the set of values of λ , for which the system of equations 6 λ x - 3 y + 3 z = 4 λ 2 , 2 x + 6 λ y + 4 z = 1 and 3 x + 2 y + 3 λ z = λ has no solution. Then, 12 ∑ λ ∈ S λ is equal to _______.

Correct answer

0

Step-by-step solution

Given, System of equations, 6 λ x - 3 y + 3 z = 4 λ 2 2 x + 6 λ y + 4 z = 1 3 x + 2 y + 3 λ z = λ For no solution △ = 0 , So, 6 λ - 3 3 2 6 λ 4 3 2 3 λ = 0 ⇒ 9 λ 3 - 7 λ - 2 = 0 ⇒ ( λ - 1 ) ( 3 λ + 1 ) ( 3 λ + 2 ) = 0 Hence, λ ∈ 1 , - 1 3 , - 2 3 Now putting the value of λ in 12 ∑ λ ∈ S λ we get, 12 ∑ λ ∈ S λ = 12 × 1 + 1 3 + 2 3 = 24

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