Question

If $\alpha ,\beta$ and $\gamma$ are such that $\alpha +\beta +\gamma =2,{\alpha }^2+{\beta }^2+{\gamma }^2=6,{\alpha }^3+{\beta }^3+{\gamma }^3=8$ then ${\alpha }^4+{\beta }^4+{\gamma }^4$ is equal to

• A. $7$
• B. $12$
• C. $18$
• D. $36$

Answer 1

The correct option is C

$18$

Explanation for the correct option:

Step 1. Find the value of ${\alpha }^4+{\beta }^4+{\gamma }^4$:

Given, $\alpha +\beta +\gamma =2,{\alpha }^2+{\beta }^2+{\gamma }^2=6,{\alpha }^3+{\beta }^3+{\gamma }^3=8$

Now,

${\left(\alpha +\beta +\gamma \right)}^2=2^2$

$\Rightarrow$${\alpha }^2+{\beta }^2+{\gamma }^2+2(\alpha \beta +\beta \gamma +\gamma \alpha )=4$

$\Rightarrow$ $2(\alpha \beta +\beta \gamma +\gamma \alpha )=4-\left(a^2+{\beta }^2+{\gamma }^2\right)$

Step 2. Put the value of $\left(a^2+{\beta }^2+{\gamma }^2\right)$, we get:

$\begin{array}{rcl}2(\alpha \beta +\beta \gamma +\gamma \alpha )& =& 4-\left(a^2+{\beta }^2+{\gamma }^2\right)\\ & =& 4-6\\ & =& 2\end{array}$

As we know,

${\alpha }^3+{\beta }^3+{\gamma }^3-3\alpha \beta \gamma =(\alpha +\beta +\gamma )({\alpha }^2+{\beta }^2+{\gamma }^2-\alpha \beta -\beta \gamma -\gamma \alpha )$

Step 3. Put the values of :$\alpha +\beta +\gamma ,{\alpha }^2+{\beta }^2+{\gamma }^2,{\alpha }^3+{\beta }^3+{\gamma }^3$ in above equation, we get

${\alpha }^3+{\beta }^3+{\gamma }^3-3\alpha \beta \gamma =(\alpha +\beta +\gamma )({\alpha }^2+{\beta }^2+{\gamma }^2-\alpha \beta -\beta \gamma -\gamma \alpha )$

$\Rightarrow$ $8-3\alpha \beta \gamma =2[6-(-1\left)\right]$

$\Rightarrow$ $8-3\alpha \beta \gamma =14$

$\Rightarrow$ $\alpha \beta \gamma =8-14$

$\Rightarrow$ $\alpha \beta \gamma =-2$

Step 4. The value of :${\alpha }^4+{\beta }^4+{\gamma }^4={({\alpha }^2+{\beta }^2+{\gamma }^2)}^2-2\sum _{}{\alpha }^2\beta$

$$\begin{gathered} ={({\alpha }^2+{\beta }^2+{\gamma }^2)}^2-2\left[{\left(\sum \beta \gamma \right)}^2-2\alpha \beta \gamma \sum \alpha \right] \\ =(6{)}^2-2\left[{(-1)}^2-2{(-2)}^2\right] \\ =36-2\times 9 \\ =36-18 \\ =\mathbf{18} \end{gathered}$$

Hence, Option ‘C’ is Correct.