If $z_{1}, z_{2}, z_{3},$ and $z_{4}$ are the complex roots of the equation $z^{4} + 3 z^{2} + 1 = 0$ , then the value of $\prod_{i = 1}^{4} 4 + (z_{i}^{2})$ is

9

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25

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27

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Solution

The correct option is C $25$
$(z^{4} + 3 z^{2} + 1) = 0$
$(z^{2} + 4)^{2} + μ (2^{2} + 4) = 0$
$2^{4} + 2^{2} (8 + μ) + (16 + 4 μ + 1) = 0$
$8 + μ = 3$ $16 + 4 μ + 1 = 1$
$μ = - 5$ $c = 5$
$\frac{(2^{2} + 4)^{2}}{x^{2}} - \frac{5 (2^{2} + 4)}{x} + 5 = 0$ Product of roots
$x_{1} x_{2} = 5$
$∴ z_{1} z_{2} z_{3} z_{4} = 5 \times 5 = 25$ (c).

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If z1,z2,z3, and z4 are the complex roots of the equation z4+3z2+1=0, then the value of ∏4i=14+(z2i) is

The correct option is C 25(z4+3z2+1)=0(z2+4)2+μ(22+4)=024+22(8+μ)+(16+4μ+1)=08+μ=3 16+4μ+1=1μ=−5 c=5(22+4)2x2−5(22+4)x+5=0 Product of rootsx1x2=5∴z1z2z3z4=5×5=25 (c).

If $z_{1}, z_{2}, z_{3},$ and $z_{4}$ are the complex roots of the equation $z^{4} + 3 z^{2} + 1 = 0$ , then the value of $\prod_{i = 1}^{4} 4 + (z_{i}^{2})$ is