"The roots of the quadratic equation $3 x^2-p x+q=0$ are $10^{\text {th }}$ and $11^{\text {th }}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88 , then $q-2 p$ is equal to ________ ."

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$$\begin{aligned} &\begin{aligned} & S_{11}=\frac{11}{2}(2 a+10 d)=88 \\ & a+5 d=8 \\ & a=8-5 \times \frac{3}{2}=\frac{1}{2} \end{aligned}\\ &\text { Roots are }\\ &\begin{aligned} & \mathrm{T}_{10}=\mathrm{a}+9 \mathrm{~d}=\frac{1}{2}+9 \times \frac{3}{2}=14 \\ & \mathrm{~T}_{11}=\mathrm{a}+10 \mathrm{~d}=\frac{1}{2}+10 \times \frac{3}{2}=\frac{31}{2} \\ & \frac{\mathrm{p}}{3}=\mathrm{T}_{10}+\mathrm{T}_{11}=14+\frac{31}{2}=\frac{59}{2} \\ & \mathrm{p}=\frac{177}{2} \\ & \frac{\mathrm{q}}{3}=\mathrm{T}_{10} \times \mathrm{T}_{11}=7 \times 31=217 \\ & \mathrm{q}=651 \\ & \mathrm{q}-2 \mathrm{p} \\ & =651-177 \\ & =474 \end{aligned} \end{aligned}$$

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The roots of the quadratic equation $3 x^2-p x+q=0$ are $10^{\text {th }}$ and $11^{\text {th }}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88 , then $q-2 p$ is equal to ________ .

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The roots of the quadratic equation $3 x^2-p x+q=0$ are $10^{\text {th }}$ and $11^{\text {th }}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88 , then $q-2 p$ is equal to ________ .

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