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The sum of distinct values of for which the system of equations (λ-1)x+(3λ+1)y+2λz=0(λ-1)x+(4λ-2)y+(λ+3)z=02x+(3λ+1)y+3(λ-1)z=0has non zero solutions, is................

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answer is 3.

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Detailed Solution

λ-13λ+12λλ-14λ-2λ+323λ+13(λ-1) =0 ⇒2λ10λ6λλ-14λ-2λ+323λ+13λ-3=0R1+R2+R3⇒2λ153λ-14λ-2λ+323λ+13λ-3=0 2λ100λ-1-λ+3-2λ+623λ-93λ-9=0C2-5C1C3-3C1⇒λ(λ-3)2100λ-1-1-2233=0⇒λ(λ-3)2=0⇒λ=0 or 3 Sum =3

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The sum of distinct values of for which the system of equations (λ-1)x+(3λ+1)y+2λz=0(λ-1)x+(4λ-2)y+(λ+3)z=02x+(3λ+1)y+3(λ-1)z=0has non zero solutions, is................