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 the $n^{th}$ term of the geometric progression $5, - \frac{5}{2}, \frac{5}{4}, - \frac{5}{8}, \dots$ is $\frac{5}{1024}$,then the value of $n$ is:

The arithmetic mean of two numbers $b$ and $c$ is $a$,and $g_1$ and $g_2$ are two geometric means between them. If $g_1^3 + g_2^3 = kabc$,then $k = \dots$

If $x, y, z$ are in geometric progression and $a^x = b^y = c^z$,then . . . . . .

What will $Rs. 500$ amount to in $10$ years after its deposit in a bank which pays an annual interest rate of $10\%$ compounded annually?

If the $10^{th}$ term of a geometric progression is $9$ and the $4^{th}$ term is $4$,then its $7^{th}$ term is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module