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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2017 · Mathematics · Quadratic Equation

It is given that the roots of the equation x^2-4 x- _3 P=0 are...

It is given that the roots of the equation x2−4x−log⁡3P=0\mathrm{x}^2-4 \mathrm{x}-\log _3 \mathrm{P}=0 are real. For this, the minimum value of P is

Options

  1. A127\frac{1}{27}
  2. B164\frac{1}{64}
  3. C181\frac{1}{81}
  4. D11

Correct answer

C. 181\frac{1}{81}

Step-by-step solution

x2−4x−log⁡3P=0\mathrm{x}^2-4 \mathrm{x}-\log _3 \mathrm{P}=0 We know, roots are real if discriminant is greater than or equal to 0 . i.e., b2−4ac≥0⇒b2≥4ac\mathrm{b}^2-4 \mathrm{ac} \geq 0 \Rightarrow \mathrm{ b}^2 \geq 4 \mathrm{ac} In the given equation, a=1,b=−4,c=−log⁡3Pa=1, b=-4, c=-\log _3 \mathrm{P}. ∴b2≥4ac⇒(−4)2≥4(1)(−log⁡3P)\therefore \mathrm{b}^2 \geq 4 \mathrm{ac} \Rightarrow(-4)^2 \geq 4(1)\left(-\log _3 \mathrm{P}\right)∴b2≥4ac⇒(−4)2≥4(1)(−log⁡3P)⇒16≥−4log⁡3P⇒4≥−log⁡3P\begin{aligned} & \therefore \mathrm{b}^2 \geq 4 \mathrm{ac} \Rightarrow(-4)^2 \geq 4(1)\left(-\log _3 \mathrm{P}\right) \\ & \Rightarrow 16 \geq-4 \log _3 \mathrm{P} \\ & \Rightarrow 4 \geq-\log _3 \mathrm{P}\end{aligned}⇒4≥log⁡3(1P)(∵sin⁡θ−log⁡a=log⁡1a)⇒34≥1P⇒81≥1P⇒P≥181\Rightarrow 4 \geq \log _3\left(\frac{1}{\mathrm{P}}\right) \quad\left(\because \sin \theta-\log \mathrm{a}=\log \frac{1}{\mathrm{a}}\right)\Rightarrow 3^4 \geq \frac{1}{P} \quad \Rightarrow 81 \geq \frac{1}{P} \Rightarrow P \geq \frac{1}{81}. ∴\therefore The minimum value of P is 181\frac{1}{81}.
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