If one real root of the quadratic equation 81 | vedclass

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

Question

(D) Let the roots of the quadratic equation $81x^2 + kx + 256 = 0$ be $\alpha$ and $\alpha^3$.From the relation between roots and coefficients:Sum of

Vedclass · Accepted Answer

$-300$

Vedclass · Answer

(D) Let the roots of the quadratic equation $81x^2 + kx + 256 = 0$ be $\alpha$ and $\alpha^3$.From the relation between roots and coefficients:Sum of roots: $\alpha + \alpha^3 = -\frac{k}{81}$  $(1)$Product of roots: $\alpha \cdot \alpha^3 = \alpha^4 = \frac{256}{81}$  $(2)$From $(2)$,$\alpha^4 = (\frac{4}{3})^4$,so $\alpha = \pm \frac{4}{3}$.Case $1$: If $\alpha = \frac{4}{3}$,then $\alpha + \alpha^3 = \frac{4}{3} + (\frac{4}{3})^3 = \frac{4}{3} + \frac{64}{27} = \frac{36+64}{27} = \frac{100}{27}$.Substituting into $(1)$: $\frac{100}{27} = -\frac{k}{81} \implies k = -\frac{100 	imes 81}{27} = -300$.Case $2$: If $\alpha = -\frac{4}{3}$,then $\alpha + \alpha^3 = -\frac{4}{3} - \frac{64}{27} = -\frac{100}{27}$.Substituting into $(1)$: $-\frac{100}{27} = -\frac{k}{81} \implies k = 300$.Since $300$ is not in the options,the value is $-300$.

If one real root of the quadratic equation $81x^2 + kx + 256 = 0$ is the cube of the other root,then a value of $k$ is

Similar Questions

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $a/c, b/a, c/b$ are in

If the sum of the squares of the roots of the equation $x^2 - (\sin \alpha - 2)x - (1 + \sin \alpha) = 0$ is least,then $\alpha$ is

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what is the value of $\frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2}$?

If one root of $5x^2 + 13x + k = 0$ is the reciprocal of the other,then $k = $?

If $f(x) = 2x^3 + mx^2 - 13x + n$ and $2, 3$ are roots of the equation $f(x) = 0$,then the values of $m$ and $n$ are $-$

Vedclass Test Series

Exam Paper Generator

Online Exam Module