Question

Let $\alpha, \beta, \gamma$ and $\delta$ be 4 distinct roots of the equation $x^4-4 x+3=x\left(x^3-f^{\prime}(1) x^2+f^{\prime \prime}(1) x-4\right)+f(1)$ and $f ( x )$ is a monic polynomial of degree 3 .
If $\int \frac{ dx }{\frac{ f ( x )-3}{ x -1}+4}=\frac{1}{2} g ( x )+ C$, where $C$ is constant of integration and $g (3)=\frac{\pi}{4}$, then the value of $g(5)+g(7)$ is

• A. $\frac{\pi}{2}$
• B. $\frac{3 \pi}{4}$
• C. $\pi$
• D. $\frac{5 \pi}{4}$

Answer 1

Answer: (B)

$ x^4-4 x+3=x^4-f^{\prime}(1) x^3+f^{\prime \prime}(1) x^2-4 x+f(1)$
which is a cubic equation and it has 4 distinct roots
$\therefore$ It is an identity
$f ^{\prime}(1)=0, f ^{\prime \prime}(1)=0, f (1)=3 $
$\therefore f ( x )=( x -1)^3+3$
(ii)$\int \frac{ dx }{( x -1)^2+4}=\frac{1}{2} \tan ^{-1}\left(\frac{ x -1}{2}\right)+ c $
$\therefore g ( x )=\tan ^{-1}\left(\frac{ x -1}{2}\right) $
$\Rightarrow g (5)+ g (7)=\left(\tan ^{-1} 2+\tan ^{-1} 3\right)=\frac{3 \pi}{4} $.