Consider function f (x) = x^3 - 8x^2 + 20x -13 Number of positive integers x for which f (x) is a pr

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1.Relation and Function

normal

Consider the function $f (x) = x^3 - 8x^2 + 20x -13$
Number of positive integers $x$ for which $f (x)$ is a prime number, is

$1$

$2$

$3$

$4$

Solution

$f (x) = (x – 1)(x^2 – 7x + 13)$
for $f (x)$ to be prime at least one of the factors must be one.
Hence $x – 1 = 1 \,\, \Rightarrow x = 2$ or
$x^2 – 7x + 13 = 1 \,\, \Rightarrow x^2 – 7x + 12 = 0 \,\, \Rightarrow x = 3$ or $4$
$\Rightarrow x = 2, 3, 4$

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Solution

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Consider the function $f (x) = x^3 - 8x^2 + 20x -13$
Number of positive integers $x$ for which $f (x)$ is a prime number, is

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as

$f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 .$ Then, the value of $\sum_{\mathrm{k}=1}^{20} \frac{1}{\sin (\mathrm{k}) \sin (\mathrm{k}+\mathrm{f}(\mathrm{k}))}$ is equal to: