NCERT Exemplar Solutions
Class 11 - Mathematics - Chapter 13: LIMITS AND DERIVATIVESShort Answer Type
Class 11 - Mathematics - Chapter 13: LIMITS AND DERIVATIVES
Question. 1
Evaluate \(\displaystyle\lim_{x\to3}\dfrac{x^{2}-9}{x-3}\).
Answer:
6
Question. 2
Evaluate \(\displaystyle\lim_{x\to\tfrac{1}{2}}\dfrac{4x^{2}-1}{2x-1}\).
Answer:
2
Question. 3
Evaluate \(\displaystyle\lim_{h\to0}\dfrac{\sqrt{x+h}-\sqrt{x}}{h}\).
Answer:
\(\dfrac{1}{2\sqrt{x}}\)
Question. 4
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{(x+2)^{1/3}-2^{1/3}}{x}\).
Answer:
\(\dfrac{1}{3}2^{-2/3}\)
Question. 5
Evaluate \(\displaystyle\lim_{x\to1}\dfrac{(1+x)^{6}-1}{(1+x)^{2}-1}\).
Answer:
3
Question. 6
Evaluate \(\displaystyle\lim_{x\to-1}\dfrac{x^{3}+27}{x+1}\).
Answer:
3
Question. 7
Evaluate \(\displaystyle\lim_{x\to1}\dfrac{x^{4}-\sqrt{x}}{\sqrt{x}-1}\).
Answer:
7
Question. 8
Evaluate \(\displaystyle\lim_{x\to2}\dfrac{x^{2}-4}{\sqrt{3x-2}-\sqrt{x+2}}\).
Answer:
8
Question. 9
Evaluate \(\displaystyle\lim_{x\to\sqrt{2}}\dfrac{x^{4}-4}{x^{2}+3\sqrt{2}x-8}\).
Answer:
\(\dfrac{8}{5}\)
Question. 10
Evaluate \(\displaystyle\lim_{x\to1}\dfrac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}\).
Answer:
1
Question. 11
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sqrt{1+x^{3}}-\sqrt{1-x^{3}}}{x^{2}}\).
Answer:
0
Question. 12
Evaluate \(\displaystyle\lim_{x\to-3}\dfrac{x^{3}+27}{x^{5}+243}\).
Answer:
\(\dfrac{1}{15}\)
Question. 13
Evaluate \(\displaystyle\lim_{x\to\tfrac{1}{2}}\left(\dfrac{8x-3}{2x-1}-\dfrac{4x^{2}+1}{4x^{2}-1}\right)\).
Answer:
\(\dfrac{7}{2}\)
Question. 14
Find \(n\in\mathbb{N}\) if \(\displaystyle\lim_{x\to2}\dfrac{x^{n}-2^{n}}{x-2}=80\).
Answer:
\(n=5\)
Question. 15
Evaluate \(\displaystyle\lim_{x\to a}\dfrac{\sin3x}{\sin7x}\).
Answer:
\(\dfrac{3}{7}\)
Question. 16
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin^{2}2x}{\sin^{2}4x}\).
Answer:
\(\dfrac{1}{4}\)
Question. 17
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{1-\cos2x}{x^{2}}\).
Answer:
2
Question. 18
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{2\sin x-\sin2x}{x^{3}}\).
Answer:
1
Question. 19
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{1-\cos mx}{1-\cos nx}\).
Answer:
\(\dfrac{m^{2}}{n^{2}}\)
Question. 20
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{3}}\sqrt{2}\left(\dfrac{\pi}{3}-x\right)\sqrt{1-\cos6x}\).
Answer:
3
Question. 21
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{4}}\dfrac{\sin x-\cos x}{x-\tfrac{\pi}{4}}\).
Answer:
\(\sqrt{2}\)
Question. 22
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{6}}\dfrac{\sqrt{3}\sin x-\cos x}{x-\tfrac{\pi}{6}}\).
Answer:
2
Question. 23
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin2x+3x}{2x+\tan3x}\).
Answer:
1
Question. 24
Evaluate \(\displaystyle\lim_{x\to a}\dfrac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}\).
Answer:
\(2\sqrt{a}\cos a\)
Question. 25
Evaluate \(\displaystyle\lim_{x\to\tfrac{\pi}{6}}\dfrac{\cot^{2}x-3}{\cosec x-2}\).
Answer:
4
Question. 26
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sqrt{2}-\sqrt{1+\cos x}}{\sin^{2}x}\).
Answer:
\(\dfrac{1}{4\sqrt{2}}\)
Question. 27
Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin x-2\sin3x+\sin5x}{x}\).
Answer:
0
Question. 28
If \(\displaystyle\lim_{x\to1}\dfrac{x^{4}-1}{x-1}=\lim_{x\to k}\dfrac{x^{3}-k^{3}}{x^{2}-k^{2}}\), then find \(k\).
Answer:
\(k=\dfrac{3}{8}\)
Question. 29
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{x^{4}+x^{3}+x^{2}+1}{x}\).
Answer:
\(3x^{2}+2x+1-\dfrac{1}{x^{2}}\)
Question. 30
Differentiate with respect to \(x\): \(\displaystyle f(x)=\left(x+\dfrac{1}{x}\right)^{3}\).
Answer:
\(3\left(x+\dfrac{1}{x}\right)^{2}\left(1-\dfrac{1}{x^{2}}\right)\)
Question. 31
Differentiate with respect to \(x\): \(\displaystyle f(x)=(3x+5)(1+\tan x)\).
Answer:
\(3(1+\tan x)+(3x+5)\sec^{2}x\)
Question. 32
Differentiate with respect to \(x\): \(\displaystyle f(x)=(\sec x-1)(\sec x+1)=\sec^{2}x-1\).
Answer:
\(2\sec^{2}x\tan x\)
Question. 33
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{3x+4}{5x^{2}-7x+9}\).
Answer:
\(\dfrac{55-40x-15x^{2}}{(5x^{2}-7x+9)^{2}}\)
Question. 34
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{x^{5}-\cos x}{\sin x}\).
Answer:
\(\dfrac{(5x^{4}+\sin x)\sin x-(x^{5}-\cos x)\cos x}{\sin^{2}x}\)
Question. 35
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{x^{2}\cos\tfrac{\pi}{4}}{\sin x}\) (note \(\cos\tfrac{\pi}{4}=\tfrac{1}{\sqrt{2}}\)).
Answer:
\(\dfrac{\sqrt{2}\,x\sin x- x^{2}\cos x}{2\sin^{2}x}\) (equivalently, compute using quotient rule with constant \(\cos\tfrac{\pi}{4}\))
Question. 36
Differentiate with respect to \(x\): \(\displaystyle f(x)=(ax^{2}+\cot x)(p+q\cos x)\).
Answer:
\( (2ax-\csc^{2}x)(p+q\cos x)+(ax^{2}+\cot x)(-q\sin x)\)
Question. 37
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{a+b\sin x}{c+d\cos x}\).
Answer:
\(\dfrac{(b\cos x)(c+d\cos x)-(a+b\sin x)(-d\sin x)}{(c+d\cos x)^{2}}\)
Question. 38
Differentiate with respect to \(x\): \(\displaystyle f(x)=(\sin x+\cos x)^{2}\).
Answer:
\(2(\sin x+\cos x)(\cos x-\sin x)\)
Question. 39
Differentiate with respect to \(x\): \(\displaystyle f(x)=(2x-7)^{2}(3x+5)^{3}\).
Answer:
Use product rule: \(2(2x-7)(2)(3x+5)^{3}+(2x-7)^{2}\cdot3(3x+5)^{2}\cdot3\).
Question. 40
Differentiate with respect to \(x\): \(\displaystyle f(x)=x^{2}\sin x+2x\sin x-2\sin2x\).
Answer:
\(2x\sin x+x^{2}\cos x+2\sin x+2x\cos x-4\cos2x\)
Question. 41
Differentiate with respect to \(x\): \(\displaystyle f(x)=\sin^{3}x\cos^{3}x\).
Answer:
\(3\sin^{2}x\cos^{3}x\cos x+3\sin^{3}x\cos^{2}x(-\sin x)\) (apply product rule or write \( (\sin x\cos x)^{3}\)).
Question. 42
Differentiate with respect to \(x\): \(\displaystyle f(x)=\dfrac{1}{ax^{2}+bx+c}\).
Answer:
\(-\dfrac{2ax+b}{(ax^{2}+bx+c)^{2}}\)