QJEEM21Maa90d: JEE Main 2021 Mathematics - Class 12 JEEM21Maa90d Solution: Consider the function where P(x) is a polynomial such that P''…

$$f(x)=\{\begin{array}{cc}\frac{P(x)}{\sin (x-2)},& x\ne 2\\ 7,& x=2\end{array}$$

P''(x) = const. $\Rightarrow$ P(x) is a 2 degree polynomial

f(x) is cont. at x = 2

f(2⁺) = f(2 ^$-$ )

$\mathop {\lim }\limits_{x \to {2^ + }} {{P(x)} \over {\sin (x - 2)}} = 7$

$\mathop {\lim }\limits_{x \to {2^ + }} {{(x - 2)(ax + b)} \over {\sin (x - 2)}} = 7 \Rightarrow 2a + b = 7$

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

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

a = 2, b = 3

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