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JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Basics of Mathematics

JEE Advanced · Mathematics · Basics of Mathematics

Let equation x^ _2 x-4 =32 has two real solutions x_1 and x_2...

Let equation

x^{\log _2 x-4}=32

has two real solutions

x_1

and

x_2 \left(x_1>x_2\right), then which of the following is correct?

Options

A $x_1 \cdot x_2=32$
B $x_1+x_2=\frac{65}{2}$
CCharacteristic of $\log _3 \left(x_1\right) is 3$
DMantissa of $\log _2 \left(x_2\right) is 0$

Correct answer

D. Mantissa of $\log _2 \left(x_2\right) is 0$

Step-by-step solution

\left(\log _2 x-4\right) \cdot \log _2 x=5

Let

\log _2 x=a

\begin{array}{ll}\Rightarrow & a^2-4 a-5=0 \quad \Rightarrow \quad(a-5)(a+1)=0 \\ \Rightarrow & a=5 \text { or }a=-1 \Rightarrow \quad x=32 \text { or }}x=\frac{1}{2}\end{array}

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