Polynomials
Grade Class 10
Question:
<p><strong>Assertion (A): </strong>A quadratic polynomial, sum of whose zeroes is 8 and their product is 12 is x<sup>2 </sup>− 20x + 96<br />
<strong>Reason (R):</strong> If <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> be the zeroes of the polynomial f(x), then polynomial is given by f(x) = x<sup>2</sup> - <span class="math-tex">\((\alpha+\beta) x\)</span> + <span class="math-tex">\(\alpha \beta\)</span></p>
<p style="display:inline">Both A and R are true and R is the correct explanation of A.</p>
<p style="display:inline">Both A and R are true but R is not the correct explanation of A.</p>
<p style="display:inline">A is true but R is false.</p>
<p style="display:inline">A is false but R is true.</p>
Step-by-Step Solution
Key Concept: A quadratic polynomial is constructed by substituting the given sum and product of zeroes into the standard form $x^2 - (sum\ of\ zeroes)x + (product\ of\ zeroes)$.
<p>Reason is correct. If <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> be the zeroes of the required polynomial f(x),<br />
then <span class="math-tex">\((\alpha+\beta)\)</span> = 8 and <span class="math-tex">\(\alpha \beta\)</span> = 12<br />
<span class="math-tex">\(\therefore\)</span> f(x) = x<sup>2</sup> - <span class="math-tex">\((\alpha+\beta) \mathrm{x}+\alpha \beta\)</span><br />
<span class="math-tex">\(\Rightarrow\)</span> f(x) = x<sup>2</sup> - 8x + 12<br />
So, Assertion is not correct</p>
Correct Answer: D