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 $\frac{3}{2}$ and $10^{\text {th }}$ term is $\alpha$ and $11^{\text {th }}$ term is $\beta$ and $S_{11}=88$. Find the value of $q-2 p$.

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YESPYQ · Accepted Answer

Correct answer: c) 474. $\begin{array}{rlrl} d=3 / 2 & a_{10}=\alpha =a+9 d \ a_{11} & =\beta=a+10 d \ 3_{11}=\frac{11}{2}(2 a+10 d) & =88 \ (a+5 d) & =8 \ a+\frac{15}{2} & =8 \Rightarrow a=1 / 2 \end{array}$ $\begin{gathered} \alpha=\frac{1}{2}+9 \times \frac{3}{2}=14 \ \beta=\frac{1}{2}+10 \times \frac{3}{2}=\frac{31}{2} \ 3 x^2-p x+q=0 \ \alpha+\beta=\frac{p}{3} \ \frac{14+81}{2}=\frac{p}{3} \Rightarrow \frac{28+31}{2}=\frac{p}{3} \ \frac{59}{2}=\frac{p}{3} \Rightarrow p=\frac{59 \times 3}{2}=\frac{177}{2} \end{gathered}$ $\begin{aligned} & \alpha \beta=q / 3 \ & 14 \times \frac{31}{2}=\frac{q}{3} \ & 217 \times 3=q \ & q=651 \ & q-2 p \ & 651-2\left(\frac{177}{2}\right)=651-177=474 \end{aligned}$

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}{…

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