Question

Match the statements/expressions in List 1 with the open intervals in List 2.

Answer 1

A) $(x-3{)}^2{y}^{\prime }+y=0\Rightarrow (x-3{)}^2\frac{dy}{dx}+y=0\Rightarrow \int \frac{dx}{{(x-3)}^2}=-\int \frac{dy}{y}\Rightarrow \frac{1}{x-3}=\ln |y|+C$
Hence domain is $R-\left\{3\right\}$
B) ${\int }_1^5(x-1)(x-2)(x-3)(x-4)(x-5)dx$
Substituting $x=t+3$, we get
${\int }_{-2}^2(t+2)(t+1)t(t-1)(t-2)dt={\int }_{-2}^{2}t(t^2-1)(t^2-4)dt=0$
C) $f\left(x\right)={\cos }^{2}x+sinx=\frac{5}{4}-{(\sin x-\frac{1}{2})}^2$
So $f\left(x\right)$ is maximum for $sinx=\frac{1}{2}\Rightarrow x=\frac{\pi }{6}$
D) $f\left(x\right)={\tan }^{-1}(\sin x+\cos x)$ is increasing when $\Rightarrow f^{\prime }\left(x\right)>0\Rightarrow cosx>sinx$