If α ,β ,gamma are roots of equation {x^3} + x + 1 = 0 , then value of {α ^3}{β ^3}{gamma ^3}

English

Gujarati

Hindi

4-2.Quadratic Equations and Inequations

easy

If $\alpha ,\beta ,\gamma $are the roots of the equation ${x^3} + x + 1 = 0$, then the value of ${\alpha ^3}{\beta ^3}{\gamma ^3}$

A

$0$

B

$-3$

C

$3$

D

$-1$

Solution

(d) We know that the roots of the equation

$a{x^3} + b{x^2} + cx + d = 0$ follows $\alpha \beta \gamma = – d/a$

Comparing above equation with given equationwe get $d = 1,\,a = 1$

So, $\alpha \beta \gamma = – 1$ or ${\alpha ^3}{\beta ^3}{\gamma ^3} = – 1$.

Standard 11

Mathematics

Share

Similar Questions

The maximum value $M$ of $3^x+5^x-9^x+15^x-25^x$, as $x$ varies over reals, satisfies

normal

(KVPY-2012)

If the inequality $kx^2 -2x + k \geq 0$ holds good for atleast one real $'x'$ , then the complete set of values of $'k'$ is

normal

Let $\mathrm{S}=\left\{x \in R:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $\mathrm{S}$ is :

hard

(JEE MAIN-2024)

Let $\alpha $ and $\beta $ be the roots of the quadratic equation ${x^2}\,\sin \,\theta – x\,\left( {\sin \,\theta \cos \,\,\theta + 1} \right) + \cos \,\theta = 0\,\left( {0 < \theta < {{45}^o}} \right)$ , and $\alpha < \beta $. Then $\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + \frac{{{{\left( { – 1} \right)}^n}}}{{{\beta ^n}}}} \right)} $ is equal to

hard

(JEE MAIN-2019)

Let $\alpha$ and $\beta$ be the two disinct roots of the equation $x^3 + 3x^2 -1 = 0.$ The equation which has $(\alpha \beta )$ as its root is equal to

normal