CBSE 2026 · Region 2 · Set 1 · Q33 · 5 marks
Find the sub intervals in which $\displaystyle \mathrm{f}(x)=\cot ^{-1}(\sin x+\cos x), x \in(0, \pi)$ is increasing and decreasing.A rectangle of perimeter $\displaystyle 36$ cm is revolved around one of its sides to sweep out a cylinder of maximum volume.
Find the dimensions of the rectangle.
Find the sub intervals in which $\displaystyle \mathrm{f}(x)=\cot ^{-1}(\sin x+\cos x), x \in(0, \pi)$ is increasing and decreasing.
A rectangle of perimeter $\displaystyle 36$ cm is revolved around one of its sides to sweep out a cylinder of maximum volume.
Find the dimensions of the rectangle.
Marking-scheme solution
$\displaystyle f(x)=\cot^{-1}(\sin x+\cos x), x \in(0, \pi)$
$\displaystyle f^{\prime}(x)=\dfrac{-(\cos x-\sin x)}{1+(\sin x+\cos x)^{2}}=\dfrac{\sin x-\cos x}{1+(\sin x+\cos x)^{2}}$
For critical points of $\displaystyle f(x)$, put $\displaystyle f^{\prime}(x)=0$
$\displaystyle \Rightarrow \sin x=\cos x \Rightarrow x=\dfrac{\pi}{4}$
$\displaystyle f^{\prime}(x)>0$ when $\displaystyle x \in\left(\dfrac{\pi}{4}, \pi\right) \Rightarrow f(x)$ is increasing in $\displaystyle \left(\dfrac{\pi}{4}, \pi\right)$.
$\displaystyle f^{\prime}(x)<0$ when $\displaystyle x \in\left(0, \dfrac{\pi}{4}\right) \Rightarrow f(x)$ is decreasing in $\displaystyle \left(0, \dfrac{\pi}{4}\right)$.
Let the lengths of the sides of the rectangle be $\displaystyle x$ and $\displaystyle (18-x)$.
Let it be revolved around the side of length $\displaystyle (18-x)$, so that $\displaystyle x$ becomes the radius of the cylinder.
Volume of cylinder, $\displaystyle V=\pi x^{2}(18-x)=\pi\left(18 x^{2}-x^{3}\right)$
$\displaystyle \dfrac{dV}{dx}=\pi\left(36 x-3 x^{2}\right)$
For critical points, put $\displaystyle \dfrac{dV}{dx}=0$
$\displaystyle \Rightarrow x=12$ cm $\displaystyle (\because x \neq 0)$
Now $\displaystyle \dfrac{d^{2} V}{dx^{2}}=\pi(36-6 x)$ ; $\displaystyle \left.\dfrac{d^{2} V}{dx^{2}}\right|_{x=12\ \text{cm}}=-36\pi<0$
So, volume is maximum when $\displaystyle x=12$ cm.
Hence the dimensions of rectangle are $\displaystyle 12$ cm and $\displaystyle 6$ cm.
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CBSE Class 12 Mathematics past-paper question from the 2026board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.