Consider the function f(x) = (P(x))/( (x - 2)), x ≠ 2 , and f(x) = 7, x = 2 where P(x) is a polynomial such that P ^ " (x) is always a constant and P(3) = 9 . If f(x) is continuous at x = 2, then P(5) is equal to

Correct answer: 0 We have, f(x) = \ array ll (P(x))/( (x - 2)), 7, array array ll x ≠ 2 x = 2 array P ^ " (x) = Constant ⇒ P(x) is a 2 degree polynomial Let P(x) = (x - 2)(ax + b) f(x) is continuous at x = 2 f( 2 ^ + ) = f( 2 ^ - ) Now, _ x → 2 ^ + (P(x))/( (x - 2)) = 7 ⇒ _ x → 2 ^ + ((x - 2)(ax + b))/( (x - 2)) = 7 ⇒ 2a + b = 7 ⇒ P(3) = (3 - 2)(3a + b) = 9 ⇒ 3a + b = 9 ⇒ a = 2, b = 3 Hence, P(5) = (5 - 2)(2 . 5 + 3) = 3 . 13 = 39 .

JEE Main 202125 JulyEvening Shift

Exam: JEE Main
Year: 2021
Date: 25 July
Shift: Evening Shift
Subject: Mathematics
Chapter: Continuity and Differentiability

JEE Main 2021 · 25 July · Evening Shift · Mathematics · Continuity and Differentiability

Consider the function f(x) = (P(x))/( (x - 2)), x ≠ 2 , and f(x) =...

Consider the function

f\left(x\right) = \dfrac{P\left(x\right)}{\sin \left(x - 2\right)}, x \ne 2

, and

f\left(x\right) = 7, x = 2

where

P\left(x\right)

is a polynomial such that

{P}^{"}\left(x\right)

is always a constant and

P\left(3\right) = 9 .

f\left(x\right)

is continuous at

x = 2,

then

P\left(5\right)

is equal to __________.

Correct answer

Step-by-step solution

We have, $f\left(x\right) = \left\{$ $\begin{array}{ll}\dfrac{P\left(x\right)}{\sin \left(x - 2\right)}, \\ 7,\end{array}$ $\right$ $\begin{array}{ll}x \ne 2 \\ x = 2\end{array}$ $ $$ {P}^{"}\left(x\right) = $Constant$ ⇒ P\left(x\right) $is a$ 2 $degree polynomialLet$ P\left(x\right) = \left(x - 2\right)\left(ax + b\right) $$ f\left(x\right) $is continuous at$ x = 2 $$ f\left({2}^{+}\right) = f\left({2}^{-}\right) $Now,$ \lim_{x \to {2}^{+}}\dfrac{P\left(x\right)}{\sin \left(x - 2\right)} = 7 $$ ⇒ \lim_{x \to {2}^{+}}\dfrac{\left(x - 2\right)\left(ax + b\right)}{\sin \left(x - 2\right)} = 7 $$ ⇒ 2a + b = 7 $$ ⇒ P\left(3\right) = \left(3 - 2\right)\left(3a + b\right) = 9 $$ ⇒ 3a + b = 9 $$ ⇒ a = 2, b = 3 $Hence,$ P\left(5\right) = \left(5 - 2\right)\left(2 . 5 + 3\right) $$ = 3 . 13 $$ = 39 .

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