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Determinant zero for a non-zero solution of homogeneous linear equations

This type of question, asked 2 times: 26 Apr 2025 S1, 15 May 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    26 Apr 2025 S1

    130
    If a‾=λxi^+yj^+4zk^\overline{a}=\lambda x\hat{i}+y\hat{j}+4z\hat{k}, b‾=yi^+xj^+3yk^\overline{b}=y\hat{i}+x\hat{j}+3y\hat{k} and c‾=−zi^−2zj^−(λ+1)xk^\overline{c}=-z\hat{i}-2z\hat{j}-(\lambda+1)x\hat{k} are the sides of the triangle ABC, where x,y,zx,y,z are not all zero, such that a‾+b‾−c‾=0‾\overline{a}+\overline{b}-\overline{c}=\overline{0}, then the value of λ\lambda is

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  2. Question 2

    15 May 2026 S2

    73
    If the planes ax=y+zax = y + z, by=z+xby = z + x, cz=x+ycz = x + y pass through a common line, then the value of (a+b+c−abc)=⋯(a + b + c - abc) = \cdots

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