If alpha and beta are the roots of x^2-2x+4=0 | vedclass

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

Question

(C) Given,$\alpha$ and $\beta$ are the roots of $x^2-2x+4=0$. From the properties of roots,we have: $\alpha+\beta = 2$ ...$(i)$ $\alpha\beta = 4$ ...$

Vedclass · Accepted Answer

$128$

Vedclass · Answer

(C) Given,$\alpha$ and $\beta$ are the roots of $x^2-2x+4=0$. From the properties of roots,we have: $\alpha+\beta = 2$ ...$(i)$ $\alpha\beta = 4$ ...$(ii)$ Now,we can write $\alpha^6+\beta^6 = (\alpha^2)^3 + (\beta^2)^3$. Alternatively,note that the roots of $x^2-2x+4=0$ are $x = \frac{2 \pm \sqrt{4-16}}{2} = 1 \pm \sqrt{3}i$. In polar form,$1 \pm \sqrt{3}i = 2(\frac{1}{2} \pm i\frac{\sqrt{3}}{2}) = 2e^{\pm i\pi/3}$. Thus,$\alpha = 2e^{i\pi/3}$ and $\beta = 2e^{-i\pi/3}$. Then,$\alpha^6 = (2e^{i\pi/3})^6 = 2^6 e^{i2\pi} = 64(1) = 64$. Similarly,$\beta^6 = (2e^{-i\pi/3})^6 = 2^6 e^{-i2\pi} = 64(1) = 64$. Therefore,$\alpha^6+\beta^6 = 64+64 = 128$.

If $\alpha$ and $\beta$ are the roots of $x^2-2x+4=0$,then the value of $\alpha^6+\beta^6$ is

Similar Questions

If the polynomial $P(x) = x^2 + ax + b$ has factors $(x - a)(x - b)$,where $a, b \in R$,then the value of $P(2)$ is:

If $\alpha$ and $\beta$ are the roots of the equation $a x^2+b x+c=0$ and if $p x^2+q x+r=0$ has roots $\frac{1-\alpha}{\alpha}$ and $\frac{1-\beta}{\beta}$,then $r$ is equal to

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$,then the equation whose roots are $2 + \alpha$ and $2 + \beta$ is:

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4x+1=0$,then $(\alpha+\beta)^{-1}+(\beta+\gamma)^{-1}+(\gamma+\alpha)^{-1}$ is equal to

Suppose the quadratic polynomial $p(x)=ax^2+bx+c$ has positive coefficients $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$,then what could be the possible value of $\alpha+\beta+\alpha\beta$ if $0 \leq \alpha+\beta+\alpha\beta \leq 8$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module