Polynomials
Grade Class 10
Question:
<p>If <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> are the zeroes of the polynomial ax<sup>2 </sup>- 5x + c and <span class="math-tex">\(\alpha+\beta=\alpha \beta\)</span> = 10, then:</p>
<p style="display:inline">a = 5, c = <span class="math-tex">\(\frac{1}{2}\)</span></p>
<p style="display:inline">a = <span class="math-tex">\(\frac{1}{2}\)</span>, c = 5</p>
<p style="display:inline">a = <span class="math-tex">\(\frac{5}{2}\)</span>, c = 1</p>
<p style="display:inline">a = 1, c = <span class="math-tex">\(\frac{5}{2}\)</span></p>
Step-by-Step Solution
Key Concept: Use the relationship between roots and coefficients, specifically alpha + beta = -b/a and alpha * beta = c/a, to determine the unknown constants by comparing them to the given polynomial.
<p>P(x) = x<sup>2</sup> - (sum of roots)x + (product of roots)<br />
x<sup>2 </sup>- (10)x + 10 = 0<br />
<span class="math-tex">$\frac{1}{2}$</span>x<sup>2</sup> - 5x + 5 = 0 ...(1)<br />
Given, ax<sup>2</sup> - 5x + c = 0 ...(2)<br />
comparing (1) and (2), we get,<br />
a = <span class="math-tex">$\frac{1}{2}$</span>, c = 5</p>
Correct Answer: B