2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise

Students can Download Maths Chapter 5 Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise

2nd PUC Maths Chapter 5 Continuity and Differentiability NCERT Text Book Questions and Answers Miscellaneous Exercise

Differentiate w.r.t. x the function in Exercise 1 to 11.

Question 1.
(3x2 – 9x + 5)9
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 1

Question 2.
sin3 x + cos6 x
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 2

KSEEB Solutions

Question 3.
(5x)3cos2x
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 3
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 4

Question 4.
\(\sin ^{-1}(x \sqrt{x}), 0 \leq x \leq 1\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 5

Question 5.
\(\frac{\cos ^{-1}\left(\frac{x}{2}\right)}{\sqrt{2 x+7}},-2<x<2\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 6
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 7

KSEEB Solutions

Question 6.
\(\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], 0<x<\frac{\pi}{2}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 8

Question 7.
(log x)log x ,x > 1
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 9

Question 8.
cos (a cos x + b sin x), for some constant a and b.
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 10

Question 9.
\((\sin x-\cos x)^{(\sin x \cdot \cos x)}, \frac{\pi}{4}<x<\frac{3 \pi}{4}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 11

KSEEB Solutions

Question 10.
xx + xn + ax + an, for some fixed a > x and x > 0
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 12

Question 11.
\(x^{x^{2}-3}+(x-3)^{x^{2}}, \text { for } x>3\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 13
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 14

Question 12.
Find
\(\frac{d y}{d x}, \text { if } y=12(1-\cos t), x=10(t-\sin t),-\frac{\pi}{2}<t<\frac{\pi}{2}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 15

KSEEB Solutions

Question 13.
Find
\(\frac{d y}{d x}, \text { if } y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}},-1 \leq x \leq 1\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 16

Question 14.
If \(x \sqrt{1+y}+y \sqrt{1+x}=0 \text { for },-1<x<1, \text { prove that } \frac{d y}{d x}=-\frac{1}{(1+x)^{2}}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 17
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 18

Question 15.
If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that
\(\frac{\left[1+\left(\frac{\mathrm{dy}}{\mathrm{d} \mathbf{x}}\right)^{2}\right]^{\frac{3}{2}}}{\frac{\mathbf{d}^{2} \mathbf{y}}{\mathbf{d x}^{2}}}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 19

KSEEB Solutions

Question 16.
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that
\(\frac{d y}{d x}=\frac{\cos ^{2}(a+y)}{\sin a}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 20
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 21

Question 17.
If x = a (cos t + t sin t) and y = a (sin t -1 cos t), find \(\frac{d^{2} y}{d x^{2}}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 22

Question 18.
If f (x) = |x|3, show that fn (x) exists for all real x and find it.
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 23

KSEEB Solutions

Question 19.
Using mathematical induction prove that \(\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}\) for all positive integers n.
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 24

Question 20.
Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.
Answer:
sin (A + B) = sin A cos B + cos A sin B
Diff : w:r.to B
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 25
cos(A +b) = cos A cos B – sin A sin B
hence cosine formulae

Question 21.
Does there exist a function which is continuous everywhere but not differentiable at exactly two points ? Justify your answer.
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 26
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 27

KSEEB Solutions

Question 22.
It
\(\left|\begin{array}{ccc}{\mathbf{f}(\mathbf{x})} & {\mathbf{g}(\mathbf{x})} & {\mathbf{h}(\mathbf{x})} \\{\mathbf{1}} & {\mathbf{m}} & {\mathbf{n}} \\{\mathbf{a}} & {\mathbf{b}} & {\mathbf{c}}\end{array}\right|, \text { prove that } \quad \frac{\mathbf{d y}}{\mathbf{d x}}=\left|\begin{array}{ccc}{\mathbf{f}^{\prime}(\mathbf{x})} & {\mathbf{g}^{\prime}(\mathbf{x})} & {\mathbf{h}^{\prime}(\mathbf{x})} \\{1} & {\mathbf{m}} & {\mathbf{n}} \\{\mathbf{a}} & {\mathbf{b}} & {\mathbf{c}}\end{array}\right|\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 28

Question 23.
\(y=e^{a \cos ^{-1} x},-1 \leq x \leq 1\)\(\text { show that }\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 29

2nd PUC Maths Chapter 5 Continuity and Differentiability Miscellaneous Exercise Additional Questions and Answer

Question 1.
\(\text { If } x=a\left(\cos \theta+\log \tan \frac{\theta}{2}\right), y=a \sin \theta, \text { find } \frac{d y}{d x} \text { at } \theta=\frac{\pi}{4}\) (DB 2011)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 30

Question 2.
\(f(x)=\left\{\begin{aligned}\frac{\sin (a+1) x+\sin x}{x}, & x<0 \\c &, x=0 \\\frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}}, & x>0\end{aligned}\right.\)
If f(x) is continuous at x = 0 ,find a,b,c. (CBSE 2011, 2008, 2005)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 31

2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 32

KSEEB Solutions

Question 3.
\(\text { If } y=\cos ^{-1}\left\{\frac{3 x+4 \sqrt{1-x^{2}}}{5}\right\} \text { find } \frac{d y}{d x}\) (CBSE 2010)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 33

Question 4.
\(\text { If } x^{y}=e^{x \cdot y}, \text { show that } \frac{d y}{d x}=\frac{\log x}{\{\log (x e)\}^{2}}\)(CBSE 2009)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 34

Question 5.
Differentiate
\(\sin ^{-1}\left(\frac{t}{\sqrt{1+t^{2}}}\right) \text { w:r:to }\)
\(\cos ^{-1}\left(\frac{1}{\sqrt{1+t^{2}}}\right)\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 35
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 36

KSEEB Solutions

Question 6.
\(\text { If } y=\sin ^{-1}\left[\frac{1-\sqrt{x}}{1+\sqrt{x}}\right]+\sec ^{-1}\left[\cfrac{1+\sqrt{x}}{1-\sqrt{x}}\right], \text { find } \frac{d y}{d x}\)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 37

Question 7.
Find the derivative of |x|
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 38

Question 8.
Diff: log10 x w : r : to logx10
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 39

Question 9.
\(\text { Let } f(x)=\left\{\begin{array}{ll}{\frac{3|x|+4 \tan x}{x},} & {x \neq 0} \\{k} & {x=0}\end{array}\right.\) is continuos at x = o,find K
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 40

Question 10.
\(\text { If } y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots \ldots \infty}}} \text { prove }\) that \((2 y-1) \frac{d y}{d x}=\cos x\)(Kerala CET)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 41

Question 11.
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 42
\(\text { prove that, } \frac{d y}{d x}=\cfrac{(1+y) \cos x+y \sin x}{1+2 y+\cos x-\sin x}\) (Kerala CET)
Answer:
2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 43

KSEEB Solutions

Question 12.
\(y=(\sqrt{x})^{(\sqrt{x})^{(\sqrt{x}}} \text { show that } \frac{d y}{d x}=\frac{y^{2}}{x(2-y \log x)}\)  (Kerala CET)
Answer:

2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability Miscellaneous Exercise 44

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