The roots of the quadratic equation are and terms of an arithmetic progression with common difference . If the sum of the first 11 terms of this arithmetic progression is 88, then q – 2p is equal to __________. [2025]

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Solution

Q.
The roots of the quadratic equation $3 x^{2} - p x + q = 0$ are $10^{th}$ and $11^{th}$ 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 – 2p is equal to __________. [2025]

Ans.
(474)

We have, $S_{11} = \frac{11}{2} (2 a + 10 d) = 88$

$\Rightarrow a + 5 d = 8$

$\Rightarrow a = 8 - 5 \times \frac{3}{2} = \frac{1}{2}$ $[∵ d = \frac{3}{2}]$

Let $α, β$ are the roots of the given quadratic equation.

$∴ α = T_{10} = a + 9 d = \frac{1}{2} + 9 \times \frac{3}{2} = 14$

and $β = T_{11} = a + 10 d = \frac{1}{2} + 10 \times \frac{3}{2} = \frac{31}{2}$

Sum of roots = $\frac{p}{3} = T_{10} + T_{11} = 14 + \frac{31}{2} = \frac{59}{2} \Rightarrow p = \frac{177}{2}$

and product of roots = $\frac{q}{3} = T_{10} \times T_{11} = 7 \times 31 = 217$

$\Rightarrow q = 651$ .

Now, q – 2p = 651 – 177 = 474.

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