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Column – I Column – II
A) Number of distinct real roots of $x^{4} + 4 x^{3} + 12 x^{2} + 4 x + 1 = 0$ is P) 4
B) Number of distinct real roots of $x^{8} - x^{7} + x^{2} - x + 15 = 0$ is Q) 3
C) Number of distinct real roots of the equation $x^{7} + 20 x^{6} - x^{5} - 576 = 0$ is R) 0
D) If $α$ is greatest positive root of $\sqrt{6} . x^{4} - (\sqrt{3} + \frac{3}{2} \sqrt{2}) x^{2} + \frac{3}{2} = 0$ then value of $[4 α^{4}]$ is ([.] denotes GIF) S) 2

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A – R, B – R, C – Q, D – Q

A – R, B – S, C – P, D – Q

A – P, B – R, C – Q, D – S

A – S, B – R, C – Q, D – Q

answer is B.

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Detailed Solution

$0 < x < 1 \Rightarrow f (x) = (15 - x) + (x^{2} - x^{7}) + x^{8} > 0 x > 1 \Rightarrow f (x) = (x^{8} - x^{7}) + (x^{2} - x) + 15 > 0$
$X = 1$ is not root. Also there can not be negative root
C) No. of real roots = 0
D) $x^{2} = \frac{\sqrt{3}}{2} (o r) \frac{1}{\sqrt{2}} \Rightarrow x = \sqrt[4]{\frac{3}{4}} \Rightarrow α = \sqrt[4]{\frac{3}{4}} \Rightarrow 4 α^{4} = 3$

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