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  4. If f(x)= prodk = 1999 (x2 - 47 x + k) , then product of all real roots of f(x)=0 is

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Q. If $f\left(x\right)=\prod_{k = 1}^{999} \left(x^{2} - 47 x + k\right)$ , then product of all real roots of $f\left(x\right)=0$ is

NTA AbhyasNTA Abhyas 2020Complex Numbers and Quadratic Equations

Solution:

Consider $x^{2}-47x+k=0$
For real roots, $47^{2}-4k\geq 0\Rightarrow k\leq 552$
$\therefore \, \, \, k=1, 2, 3\ldots .,552$
Product of real roots $=1\times 2\times 3\times 4\times \ldots ..\times 552=552!$