The roots of the quadratic equation 3x^2 - px | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given the sum of the first $11$ terms $S_{11} = 88$ and common difference $d = \frac{3}{2}$.Using the formula $S_n = \frac{n}{2}(2a + (n-1)d)$:$88

Vedclass · Accepted Answer

$474$

Vedclass · Answer

(A) Given the sum of the first $11$ terms $S_{11} = 88$ and common difference $d = \frac{3}{2}$.Using the formula $S_n = \frac{n}{2}(2a + (n-1)d)$:$88 = \frac{11}{2}(2a + 10 	imes \frac{3}{2})$$8 = \frac{1}{2}(2a + 15) \implies 16 = 2a + 15 \implies 2a = 1 \implies a = \frac{1}{2}$.The $10^{	ext{th}}$ and $11^{	ext{th}}$ terms are:$T_{10} = a + 9d = \frac{1}{2} + 9(\frac{3}{2}) = \frac{1}{2} + \frac{27}{2} = 14$.$T_{11} = a + 10d = \frac{1}{2} + 10(\frac{3}{2}) = \frac{1}{2} + 15 = \frac{31}{2}$.For the quadratic equation $3x^2 - px + q = 0$,the sum of roots is $\frac{p}{3} = T_{10} + T_{11} = 14 + \frac{31}{2} = \frac{59}{2} \implies p = \frac{177}{2}$.The product of roots is $\frac{q}{3} = T_{10} 	imes T_{11} = 14 	imes \frac{31}{2} = 7 	imes 31 = 217 \implies q = 651$.Calculating $q - 2p = 651 - 2(\frac{177}{2}) = 651 - 177 = 474$.

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

Similar Questions

The cubic equation whose roots are the squares of the roots of $x^3-2x^2+10x-8=0$ is

Let $\alpha$ and $\beta$ be the roots of the equation $5x^{2} + 6x - 2 = 0$. If $S_{n} = \alpha^{n} + \beta^{n}$ for $n = 1, 2, 3, \dots$,then:

If $\alpha \ne \beta$ but $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$,then the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

Let $f(x)$ be a quadratic polynomial such that $f(-2) + f(3) = 0$. If one of the roots of $f(x) = 0$ is $-1$,then the sum of the roots of $f(x) = 0$ is equal to

Let $A = \left| \begin{matrix} 2 & e^{i \pi} \\ -1 & i^{2012} \end{matrix} \right|$,$C = \left. \frac{d}{dx} \left( \frac{1}{x} \right) \right|_{x=1}$,and $D = \int_{e^2}^{1} \frac{dx}{x}$. If the sum of two roots of the equation $Ax^3 + Bx^2 + Cx - D = 0$ is equal to zero,then $B$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module