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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Functions

NDA 2024 · Mathematics · Functions

If f(2 x)=4 x^2+1 , then for how many real values of x will f(2 x)...

If f(2x)=4x2+1f(2 x)=4 x^2+1, then for how many real values of xx will f(2x)f(2 x) be the GM of f(x)f(x) and f(4x)f(4 x) ?

Options

  1. AFour
  2. BTwo
  3. COne
  4. DNone

Correct answer

C. One

Step-by-step solution

Given f(2x)=4x2+1⇒f(x)=x2+1f(4x)=16x2+1(f(2x))2=f(x)×f(4x)(4x2+1)2=(x2+1)×(16x2+1)(f(2x) is GM of f(x)f(4x))16x4+1+8x2=16x2+16x2+1+x2⇒9x2=0x=0\begin{aligned} & f(2 x)=4 x^2+1 \Rightarrow f(x)=x^2+1 \\ & f(4 x)=16 x^2+1 \\ & (f(2 x))^2=f(x) \times f(4 x) \\ & \left(4 x^2+1\right)^2=\left(x^2+1\right) \times\left(16 x^2+1\right) \\ & (f(2 x) \text{ is } G M \text{ of } f(x) & f(4 x)) \\ & 16 x^4+1+8 x^2=16 x^2+16 x^2+1+x^2 \\ & \Rightarrow 9 x^2=0 \\ & x=0 \end{aligned}\therefore \quad$ One real value of x is possible.
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