Let x_1, x_2, ldots, x_6 be the roots of the | vedclass

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

Question

(C) The given equation is $x^6+2 x^5+4 x^4+8 x^3+16 x^2+32 x+64=0$.This is a geometric progression with $7$ terms.Multiplying by $(x-2)$,we get $(x-2)

Vedclass · Accepted Answer

$|x_i|=2$ for all values of $i$

Vedclass · Answer

(C) The given equation is $x^6+2 x^5+4 x^4+8 x^3+16 x^2+32 x+64=0$.This is a geometric progression with $7$ terms.Multiplying by $(x-2)$,we get $(x-2)(x^6+2 x^5+4 x^4+8 x^3+16 x^2+32 x+64) = 0$.This simplifies to $x^7 - 2^7 = 0$,which means $x^7 = 128$.The roots of this equation are $x_k = 2 e^{i \frac{2k\pi}{7}}$ for $k = 0, 1, 2, 3, 4, 5, 6$.However,the original equation is the sum of the geometric series excluding the term for $x=2$ (since $x=2$ makes the sum $64+64+64+64+64+64+64 = 448 
eq 0$).Thus,the roots are $x_k = 2 e^{i \frac{2k\pi}{7}}$ for $k = 1, 2, 3, 4, 5, 6$.For all these roots,$|x_k| = |2 e^{i \frac{2k\pi}{7}}| = 2 |e^{i \frac{2k\pi}{7}}| = 2(1) = 2$.Therefore,$|x_i|=2$ for all values of $i$.

Let $x_1, x_2, \ldots, x_6$ be the roots of the polynomial equation $x^6+2 x^5+4 x^4+8 x^3+16 x^2+32 x+64=0$. Then,

Similar Questions

The sum of the first three terms of a $G.P.$ is $16$ and the sum of the next three terms is $128$. Determine the first term,the common ratio,and the sum to $n$ terms of the $G.P.$

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2$,then the value of $\frac{a^2+a-14}{a+1}$ is

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

The money invested in a company is compounded continuously. If ₹ $200$ invested today becomes ₹ $400$ in $6$ years, then at the end of $33$ years it will become ₹ (in $\sqrt{2}$)

The sum of the first two terms of a geometric progression is $12$. The sum of the third and fourth terms is $48$. The terms of the geometric progression alternate between positive and negative. What is the first term?

Vedclass Test Series

Exam Paper Generator

Online Exam Module