Question

Probability of getting positive integral roots of the equation \(x^2-n=0\) for the integer \(n,\ 1\le n \le 40\) is

• A. \(\frac{1}{5}\)
• B. \(\frac{1}{10}\)
• C. \(\frac{3}{20}\)
• D. \(\frac{1}{20}\)

Hint:

For integer roots of \(x^2=n\), \(n\) must be a perfect square.

Answer 1

The Correct Option is C

Step 1: Condition for positive integral roots.
Equation:
\[ x^2-n=0 \Rightarrow x=\pm\sqrt{n} \]
Roots are integers only if \(n\) is a perfect square.
Step 2: Count perfect squares between 1 and 40.
Perfect squares \(\le 40\):
\[ 1,4,9,16,25,36 \]
Total = 6 numbers.
Step 3: Total possible values of \(n\).
\[ n=1,2,3,\ldots,40 \Rightarrow 40 \text{ outcomes} \]
Step 4: Probability.
\[ P=\frac{6}{40}=\frac{3}{20} \]
Final Answer:
\[ \boxed{\frac{3}{20}} \]