Question:
If one root of the polynomial $f(x)=5 x^{2}+13 x+k$ is reciprocal of the other, then the value of $k$ is
(a) 0
(b) 5
(c) $\frac{1}{6}$
(d) 6
Solution:
If one zero of the polynomial $f(x)=5 x^{2}+13 x+k$ is reciprocal of the other. So $\beta=\frac{1}{\alpha} \Rightarrow \alpha \beta=1$
Now we have
$\alpha \times \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$
$=\frac{k}{5}$
Since $\alpha \beta=1$
Therefore we have
$\alpha \beta=\frac{k}{5}$
$1=\frac{k}{5}$
$\Rightarrow k=5$
Hence, the correct choice is (b)