Polynomials
Grade Class 10
Question:
<p>If <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> are zeroes of the polynomial <span class="math-tex">\(5 x^2+3 x-7\)</span>, the value of <span class="math-tex">\(\frac{1}{\alpha}+\frac{1}{\beta}\)</span> is</p>
<p style="display:inline"><span class="math-tex">\(-\frac 57\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac 37\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac 35\)</span></p>
<p style="display:inline"><span class="math-tex">\(-\frac 37\)</span></p>
Step-by-Step Solution
Key Concept: Relate symmetric expressions of roots to the sum (-b/a) and product (c/a) of the zeroes of the quadratic polynomial.
<p><span class="math-tex">$ p(x)=5 x^2+3 x-7 $</span><br />
<span class="math-tex">$ \alpha+\beta=\frac{-b}{a}=\frac{-3}{5} $</span><br />
<span class="math-tex">$ \alpha \beta=\frac{c}{a}=\frac{-7}{5}$</span><br />
Now <span class="math-tex">$ \frac{1}{\alpha}+\frac{1}{\beta}$</span><br />
<span class="math-tex">$ =\frac{\beta+\alpha}{\alpha \beta}$</span><br />
<span class="math-tex">$ =\frac{\frac {-3}{ 5}}{\frac {-1}{5}} $</span><br />
<span class="math-tex">$ =\frac{-3}{-7}=\frac{3}{7}$</span></p>
Correct Answer: B