The roots of the equation x^5 - 40x^4 + px^3 | vedclass

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

Question

(D) Let the roots be $\frac{a}{r^2}, \frac{a}{r}, a, ar, ar^2$. From the sum of roots,$\frac{a}{r^2} + \frac{a}{r} + a + ar + ar^2 = 40$. From the sum

Vedclass · Accepted Answer

$32$

Vedclass · Answer

(D) Let the roots be $\frac{a}{r^2}, \frac{a}{r}, a, ar, ar^2$. From the sum of roots,$\frac{a}{r^2} + \frac{a}{r} + a + ar + ar^2 = 40$. From the sum of reciprocals,$\frac{r^2}{a} + \frac{r}{a} + \frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} = 10$. This can be written as $\frac{1}{a} (r^2 + r + 1 + \frac{1}{r} + \frac{1}{r^2}) = 10$. Also,the sum of roots is $a(\frac{1}{r^2} + \frac{1}{r} + 1 + r + r^2) = 40$. Dividing the two equations: $\frac{a(r^2 + r + 1 + \frac{1}{r} + \frac{1}{r^2})}{\frac{1}{a}(r^2 + r + 1 + \frac{1}{r} + \frac{1}{r^2})} = \frac{40}{10}$ $\Rightarrow a^2 = 4$ $\Rightarrow a = 2$ (assuming positive roots). The product of the roots is $s = -(\frac{a}{r^2} \cdot \frac{a}{r} \cdot a \cdot ar \cdot ar^2) = -a^5$. Since $a = 2$,$s = -(2)^5 = -32$. Therefore,$|s| = |-32| = 32$.

The roots of the equation $x^5 - 40x^4 + px^3 + qx^2 + rx + s = 0$ are in $G.P.$ The sum of their reciprocals is $10$. Then the value of $|s|$ is

Similar Questions

If $1+\cos x+\cos ^2 x+\cos ^3 x+\ldots \text{ to } \infty = 4+2 \sqrt{3}$,then $x=$

If in a $G.P.$ of $64$ terms,the sum of all the terms is $7$ times the sum of the odd terms of the $G.P.$,then the common ratio of the $G.P.$ is equal to

The sum of infinite terms of a $G.P.$ is $x$ and on squaring each term of it,the sum becomes $y$. Then the common ratio of this series is:

Let $a_{n}$ be the $n^{\text{th}}$ term of a $G$.$P$. of positive terms. If $\sum_{n=1}^{100} a_{2n+1} = 200$ and $\sum_{n=1}^{100} a_{2n} = 100$,then $\sum_{n=1}^{200} a_{n}$ is equal to:

If $\frac{a+bx}{a-bx} = \frac{b+cx}{b-cx} = \frac{c+dx}{c-dx}$ and $x \neq 0$,then show that $a, b, c,$ and $d$ are in $G.P.$

Vedclass Test Series

Exam Paper Generator

Online Exam Module