The root of the quadratic equation 3x^2 – kx + 14 = 0 are in the ratio 7 : 6 then k = ………………

Correct option is (D) 13

Let \(7\alpha\;and\;6\alpha\) are roots of quadratic equation \(3x^2-kx + 14= 0.\) 

\(\therefore\) Product of roots \(=\frac{14}3\)

\(\Rightarrow\) \(7\alpha\times6\alpha\) \(=\frac{14}3\)

\(\Rightarrow\) \(\alpha^2=\frac{14}{3\times42}\)

\(=\frac1{3\times3}=\frac19\)

\(\Rightarrow\) \(\alpha=\pm\frac13\)        _______________(1)

Now, sum of roots \(=\frac{-(-k)}3=\frac{k}3\)

\(\Rightarrow\) \(7\alpha+6\alpha\) \(=\frac k3\)

\(\Rightarrow\) \(13\alpha\) \(=\frac k3\)

\(\Rightarrow\) \(\frac k3=\pm\frac{13}3\)            (From (1))

\(\Rightarrow k=\pm13\)

\(\Rightarrow\) k = 13 or k = -13

Correct option is D) 13