If \(\alpha\) and \(\beta\) are the roots of the equation \(3 x^2-p x+q=0\) Let an A.P whose common diff. is \(\frac{3}{…

\(\begin{array}{rlrl}d=3 / 2 & a_{10}=\alpha =a+9 d \\a_{11} & =\beta=a+10 d \\3_{11}=\frac{11}{2}(2 a+10 d) & =88 \\(a+5 d) & =8 \\a+\frac{15}{2} & =8 \Rightarrow a=1 / 2\end{array}\)

\(\begin{gathered}\alpha=\frac{1}{2}+9 \times \frac{3}{2}=14 \beta=\frac{1}{2}+10 \times \frac{3}{2}=\frac{31}{2} 3 x^2-p x+q=0 \alpha+\beta=\frac{p}{3} \frac{14+81}{2}=\frac{p}{3} \Rightarrow \frac{28+31}{2}=\frac{p}{3} \frac{59}{2}=\frac{p}{3} \Rightarrow p=\frac{59 \times 3}{2}=\frac{177}{2}\end{gathered}\)

\(\begin{aligned}& \alpha \beta=q / 3 \\& 14 \times \frac{31}{2}=\frac{q}{3} \\& 217 \times 3=q \\& q=651 \\& q-2 p \\& 651-2\left(\frac{177}{2}\right)=651-177=474\end{aligned}\)