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$Let f (x) = |\begin{array}{ccc} x & 1 & 1 \\ \sin 2 π x & 2 x^{2} & 1 \\ x^{3} & 3 x^{4} & 1 \end{array}| . If f (x) be an odd function and its odd values is equal g (x), then find$ $the value of λ$
$If λ f (1) g (1) = - 2$

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$5$

$2$

$1$

$\frac{1}{2}$

answer is D.

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Detailed Solution

$f (- x) = - f (x) = g (x)$

$∴ f (x) \cdot g (x) = - (f (x))^{2} or f (1) g (1) = - (f (1))^{2} = - {|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 2 & 1 \\ 1 & 3 & 1 \end{array}|}^{2} = - 4$

$\Rightarrow λ f (1) g (1) = - 2 \Rightarrow λ (- 4) = - 2 \Rightarrow λ = \frac{1}{2}$

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