Question:
<p>A quadratic polynomial whose product and sum of zeroes are <span class="math-tex">\(\frac{1}{3}\)</span> and <span class="math-tex">\(\sqrt 2 \)</span> respectively is</p>
<p style="display:inline">3x<sup>2</sup> - x + <span class="math-tex">\(3\sqrt{2}x\)</span></p>
<p style="display:inline">3x<sup>2</sup> + x - <span class="math-tex">\(3\sqrt{2}x\)</span></p>
<p style="display:inline">3x<sup>2</sup> + <span class="math-tex">\(3\sqrt{2}x\)</span> + 1</p>
<p style="display:inline">3x<sup>2</sup> - <span class="math-tex">\(3\sqrt{2}x\)</span> + 1</p>
Step-by-Step Solution
Key Concept: A quadratic polynomial is constructed by relating the sum and product of zeroes to the coefficients using the ratios $-b/a$ and $c/a$ and substituting them into the general form $ax^2 + bx + c$.
<p>Given: <span class="math-tex">\(\alpha + \beta\)</span> = <span class="math-tex">\(\frac{{\sqrt 2 }}{1}\)</span> = <span class="math-tex">\(\frac{{ - \left( { - \sqrt 2 } \right)}}{1}\)</span> = <span class="math-tex">\(\frac{{ - \left( { - 3\sqrt 2 } \right)}}{3}\)</span><br />
And <span class="math-tex">\(\alpha \beta = \frac{c}{a} = \frac{1}{3}\)</span> On comparing, we get, a = 3, b = <span class="math-tex">\(-3\sqrt{2}\)</span>, c = 1<br />
Putting these values in the general form of a quadratic polynomial ax<sup>2</sup> + bx + c,<br />
we have 3x<sup>2</sup> - <span class="math-tex">\(3\sqrt{2}\)</span> + 1</p>
Correct Answer: D