NCERT Exemplar Solutions
Class 11 - Mathematics - Chapter 13: LIMITS AND DERIVATIVES

Long Answer Questions

Differentiate with respect to x: \(\cos\big(x^{2}+1\big)\).

Differentiate with respect to x: \(\dfrac{ax+b}{cx+d}\).

Differentiate with respect to x: \(x^{2/3}\).

Differentiate with respect to x: \(x\cos x\).

Evaluate: \(\displaystyle \lim_{y\to 0} \dfrac{(x+y)\sec(x+y)-x\sec x}{y}\).

Evaluate: \(\displaystyle \lim_{x\to 0} \dfrac{\sin(\alpha+\beta)x+\sin(\alpha-\beta)x+\sin 2\alpha x}{\cos 2\beta x-\cos 2\alpha x}\cdot x\).

Evaluate: \(\displaystyle \lim_{x\to \tfrac{\pi}{4}} \dfrac{\tan^{3}x-\tan x}{\cos\big(x+\tfrac{\pi}{4}\big)}\).

Evaluate: \(\displaystyle \lim_{x\to\pi} \dfrac{1-\sin\tfrac{x}{2}}{\cos\tfrac{x}{2}\big(\cos\tfrac{x}{4}-\sin\tfrac{x}{4}\big)}\).

Evaluate (show whether limit exists): \(\displaystyle \lim_{x\to 4} \dfrac{|x-4|}{x-4}\).

Evaluate (find constant): Let \(f(x)=\begin{cases}\dfrac{k\cos x}{\pi-2x}&\text{when }x\neq\dfrac{\pi}{2},\\[6pt]3&\text{when }x=\dfrac{\pi}{2}.\end{cases}\) If \(\displaystyle\lim_{x\to\tfrac{\pi}{2}} f(x)=f\big(\tfrac{\pi}{2}\big)\), find \(k\).

Evaluate (find constant): Let \(f(x)=\begin{cases}x+2,&x\le 1,\\[6pt]cx^{2},&x>-1.\end{cases}\) Find \(c\) if \(\displaystyle\lim_{x\to -1} f(x)\) exists.