Question

If the function $f\left(x\right)=x^3-9x^2+24x+c$ has three real and distinct roots $\alpha$, $\beta$ and $\gamma$, then the value of $\left[\alpha \right]+\left[\beta \right]+\left[\gamma \right]$ is
Note: $\left[x\right]$ denotes greatest integer less than or equal to $x$.

• A. 5, 6, 7
• B. 6, 7, 8
• C. 7, 8, 9
• D. none of the above

Answer 1

The correct option is C 7, 8, 9

Here, $\alpha +\beta +\gamma =9$

$\Rightarrow \left[\alpha \right]+\left[\beta \right]+\left[\gamma \right]+\left\{\alpha \right\}+\left\{\beta \right\}+\left\{\gamma \right\}=9$

$\left[\alpha \right]+\left[\beta \right]+\left[\gamma \right]$ is an integer , $\left\{\alpha \right\}+\left\{\beta \right\}+\left\{\gamma \right\}$ has to be an integer.

Also, $0\le \left\{\alpha \right\},\left\{\beta \right\},\left\{\gamma \right\}<1\Rightarrow 0\le \left\{\alpha \right\}+\left\{\beta \right\}+\left\{\gamma \right\}<3=>\left\{\alpha \right\}+\left\{\beta \right\}+\left\{\gamma \right\}=0,1,2$

$\Rightarrow \left[\alpha \right]+\left[\beta \right]+\left[\gamma \right]=7,8,9$