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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Probability

NDA 2022 · Mathematics · Probability

A number x is chosen at random from first n natural numbers. What...

A number xx is chosen at random from first nn natural numbers. What is the probability that the number chosen satisfies x+1x>2x+\frac{1}{x} > 2 ?

Options

  1. A1n\frac{1}{n}
  2. B1(2n)\frac{1}{(2 n)}
  3. C(n−1)n\frac{(n-1)}{n}
  4. D11

Correct answer

C. (n−1)n\frac{(n-1)}{n}

Step-by-step solution

x+1x>2⇒x2+1x−2>0⇒x2−2x+1x>0⇒(x−1)2x>0−−∞0x∈(0,∞)−{1} and x∈N∴n={2,3,4,….n)\begin{aligned} & x+\frac{1}{x} > 2 \Rightarrow \frac{x^2+1}{x}-2 > 0 \\ & \Rightarrow \frac{x^2-2 x+1}{x} > 0 \Rightarrow \frac{(x-1)^2}{x} > 0 \\ & \frac{-}{-\infty} 0 \\ & x \in(0, \infty)-\{1\} \text{ and } x \in \mathbb{N} \\ & \therefore \quad n=\{2,3,4, \ldots. n)\end{aligned}∴n(E)=n−1P(E)=n−1n\begin{aligned} & \therefore \quad n(E)=n-1 \\ & P(E)=\frac{n-1}{n}\end{aligned}$
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