The roots of the quadratic equation $3 x^2-\mathrm{p} x+\mathrm{q}=0$ are $10^{ ext {th }}$ and $11^{ ext {th }}$ 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 ________ -.

Question

The roots of the quadratic equation $3 x^2-\mathrm{p} x+\mathrm{q}=0$ are $10^{	ext {th }}$ and $11^{	ext {th }}$ 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 ________ -.

Aicharya · Accepted Answer

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

The roots of the quadratic equation $3 x^2-\mathrm{p} x+\mathrm{q}=0$ are $10^{\text {th }}$ and $11^{\text…

Solution