Quadratic Equations
Roots and coefficients with probability
Grade 11
Question:
<p>A quadratic equation is chosen from the set of all quadratic equations which are unchanged by squaring their roots. The chance that the chosen equation has equal roots, is</p>
<p>(a) \(\frac{1}{2}\)</p>
<p>(b) \(\frac{1}{3}\)</p>
<p>(c) \(\frac{1}{4}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Find all quadratic equations whose roots remain unchanged when squared, then count those with equal roots.
<p><strong>Solution:</strong> Let $\alpha$ and $\beta$ be the roots of the quadratic equation.</p><p>According to the question: $\alpha \cdot \beta = \alpha^2 \cdot \beta^2$ and $\alpha\beta = \alpha^2 + \beta^2 - 2\alpha\beta$</p><p>This gives: $2\alpha\beta = 1$ or $\alpha\beta = 0$</p><p>So: $\alpha = 1, \beta = 1; \alpha = 1, \beta = \frac{1}{5}; \alpha = 1, \beta = 0; \alpha = 0, \beta = 0$</p><p><strong>Total number of quadratic equations unchanged by squaring their roots:</strong> $n(S) = 4$</p><p><strong>Number of equations with equal roots:</strong> $n(E) = 2$</p><p><strong>Required probability:</strong> $\frac{n(E)}{n(S)} = \frac{2}{4} = \frac{1}{2}$</p>
Correct Answer: A