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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2024 · Mathematics · Quadratic Equation

If f(x)=9 x-8 x such that g(x)=f(x)-1 , then which one of the...

If f(x)=9x−8xf(x)=9 x-8 \sqrt{x} such that g(x)=f(x)−1g(x)=f(x)-1, then which one of the following is correct?

Options

  1. Ag(x)=0g(x)=0 has no real roots
  2. Bg(x)=0g(x)=0 has only one real root which is an integer
  3. Cg(x)=0g(x)=0 has two real roots which are integers
  4. Dg(x)=0g(x)=0 has only one real root which is not an integer

Correct answer

B. g(x)=0g(x)=0 has only one real root which is an integer

Step-by-step solution

Given, f(x)=9x−8x,g(x)=9x−8x−1f(x)=9 x-8 \sqrt{x}, g(x)=9 x-8 \sqrt{x}-1 Lets take x=t2,\mathrm{x}=\mathrm{t}^2,g(t2)=9t2−8t−1t=8±1018⇒t=1818 or  −19t=1 or −19⇒x=1,181\begin{aligned} & g\left(t^2\right)=9 t^2-8 t-1 \\ & t=\frac{8 \pm 10}{18} \Rightarrow t=\frac{18}{18} \text{\text{ or } } \frac{-1}{9} \\ & t=1 \text{ or } \frac{-1}{9} \\ & \Rightarrow x=1, \frac{1}{81} \end{aligned}∴g(x)=0\therefore g(x)=0 has only one real root which is an integer.
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