The sum of distinct values of $\lambda$ for which the system of equations $(\lambda-1)x+(3\lambda+1)y+2\lambda z=0$ $(\lambda-1)x+(4\lambda-2)y+(\lambda+3)z=0$ $2x+(3\lambda+1)y+3(\lambda-1)z=0$, has non-zero solutions, is ______.

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Determinants

The sum of distinct values of

λ

for which the system of equations

(λ - 1) x + (3 λ + 1) y + 2 λ z = 0

(λ - 1) x + (4 λ - 2) y + (λ + 3) z = 0

2 x + (3 λ + 1) y + 3 (λ - 1) z = 0

,
has non-zero solutions, is ______.

Answer: 3

Solution

A homogeneous system has non-zero solutions only when the coefficient determinant vanishes. Reducing the determinant of the three coefficient rows (for instance by

R_{2} \to R_{2} - R_{1}

R_{3} \to R_{3} - R_{1}

and then combining columns) factorises it to

6 λ (λ - 3)^{2} = 0

, whose roots are

λ = 0

and

λ = 3

. The sum of the distinct values is

0 + 3 = 3

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Integer answer

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