First slide

Theory of equations

Question

If $\cos^{4} θ + α and \sin^{4} θ + α$ are the roots of the equation $x^{2} + 2 bx + b = 0 and \cos^{2} θ + β, \sin^{2} θ + β$ are the roots of the equation $x^{2} + 4 x + 2 = 0$ , then values of b are

Moderate

A

2
B

-1
C

-2
D

1

Solution

We have, $\cos^{2} θ - \sin^{2} θ = \cos 2 θ$
$\begin{array}{l} \Rightarrow \cos^{4} θ - \sin^{4} θ = \cos 2 θ \\ \Rightarrow (- 2 b)^{2} - 4 b = (- 4)^{2} - 4 \times 2 \end{array}$
(since L.H.S. is difference of roots of first equation and R.H.S. is difference of roots of second equation)
$\begin{array}{l} or 4 b^{2} - 4 b = 16 - 8 = 8 \\ or 4 b^{2} - 4 b - 8 = 0 \\ or b^{2} - b - 2 = 0 \\ or (b + 1) (b - 2) = 0 \\ or b = 2, - 1 \end{array}$

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