Question

$\sqrt{4},\sqrt[4]{4},\sqrt[8]{4},\sqrt[16]{4},\dots \dots \dots$up to ∞ are roots of the equation

• (A) x² - 4 = 0
• (B) x² - 4x + 6 = 0
• (C) x² - 5x + 4 = 0
• (D) x² - 3x + 2 = 0

Answer 1

The Correct Option is C

Explanation:
$4^{1/2},4^{1/4},4^{1/8},4^{1/16},\dots$ are given roots, thenSum of roots$$\begin{array}{l}=4^{1/2}+4^{1/4}+4^{1/8}+\dots \\ =5\end{array}$$Product of roots$$\begin{array}{l}=4^{1/2}\cdot 4^{1/4}\cdot 4^{1/8}\dots \\ =4^{1/2+1/4+1/8+\dots }\\ =4^{\frac{1/2}{1-1/2}}=4\end{array}$$$\therefore$ Required equation is $x^2-(\text{ sum of roots })x+(\text{ product of roots })=0$$$\Rightarrow x^2-5x+4=0$$