Consider the function f(x)= P(x) (x-2) , x 2 7, x=2 where P(x) is… - Vidyadip

MCQ

Consider the function

$f(x)=\frac{\mathrm{P}(\mathrm{x})}{\sin (\mathrm{x}-2)}, \quad \mathrm{x} \neq 2$

$\quad \quad \quad \quad 7, \quad\quad\quad \mathrm{x}=2$

where $P(x)$ is a polynomial such that $P^{\prime \prime}(x)$ is always a constant and $P(3)=9$. If $f(x)$ is continuous at $x=2$, then $P(5)$ is equal to $.....$

A
$41$
B
$40$
✓
$39$
D
$71$

✓

Answer

Correct option: C.

$39$

c
$f(x)=\frac{\mathrm{P}(\mathrm{x})}{\sin (\mathrm{x}-2)}, \quad \mathrm{x} \neq 2$

$\quad \quad \quad \quad 7, \quad\quad\quad \mathrm{x}=2$

$\mathrm{P}^{\prime \prime}(\mathrm{x})=$ const. $\Rightarrow \mathrm{P}(\mathrm{x})$ is a 2 degree polynomial

$f(x)$ is cont. at $x=2$

$f\left(2^{+}\right)=f\left(2^{-}\right)$

$\lim _{x \rightarrow 2^{+}} \frac{\mathrm{P}(\mathrm{x})}{\sin (\mathrm{x}-2)}=7$

$\lim _{x \rightarrow 2^{+}} \frac{(x-2)(a x+b)}{\sin (x-2)}=7 \Rightarrow 2 a+b=7$

$P(x)=(x-2)(a x+b)$

$P(3)=(3-2)(3 a+b)=9 \Rightarrow 3 a+b=9$

$a=2, b=3$

$P(5)=(5-2)(2.5+3)=3.13=39$

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