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

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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

Question 2.
sin3 x + cos6 x

Question 3.
(5x)3cos2x

Question 4.
$$\sin ^{-1}(x \sqrt{x}), 0 \leq x \leq 1$$

Question 5.
$$\frac{\cos ^{-1}\left(\frac{x}{2}\right)}{\sqrt{2 x+7}},-2<x<2$$

Question 6.
$$\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], 0<x<\frac{\pi}{2}$$

Question 7.
(log x)log x ,x > 1

Question 8.
cos (a cos x + b sin x), for some constant a and b.

Question 9.
$$(\sin x-\cos x)^{(\sin x \cdot \cos x)}, \frac{\pi}{4}<x<\frac{3 \pi}{4}$$

Question 10.
xx + xn + ax + an, for some fixed a > x and x > 0

Question 11.
$$x^{x^{2}-3}+(x-3)^{x^{2}}, \text { for } x>3$$

Question 12.
Find
$$\frac{d y}{d x}, \text { if } y=12(1-\cos t), x=10(t-\sin t),-\frac{\pi}{2}<t<\frac{\pi}{2}$$

Question 13.
Find
$$\frac{d y}{d x}, \text { if } y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}},-1 \leq x \leq 1$$

Question 14.
If $$x \sqrt{1+y}+y \sqrt{1+x}=0 \text { for },-1<x<1, \text { prove that } \frac{d y}{d x}=-\frac{1}{(1+x)^{2}}$$

Question 15.
If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that
$$\frac{\left[1+\left(\frac{\mathrm{dy}}{\mathrm{d} \mathbf{x}}\right)^{2}\right]^{\frac{3}{2}}}{\frac{\mathbf{d}^{2} \mathbf{y}}{\mathbf{d x}^{2}}}$$

Question 16.
If cos y = x cos (a + y), with cos a ≠ ± 1, prove that
$$\frac{d y}{d x}=\frac{\cos ^{2}(a+y)}{\sin a}$$

Question 17.
If x = a (cos t + t sin t) and y = a (sin t -1 cos t), find $$\frac{d^{2} y}{d x^{2}}$$

Question 18.
If f (x) = |x|3, show that fn (x) exists for all real x and find it.

Question 19.
Using mathematical induction prove that $$\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$$ for all positive integers n.

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.
sin (A + B) = sin A cos B + cos A sin B
Diff : w:r.to B

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.

Question 22.
It
$$\left|\begin{array}{ccc}{\mathbf{f}(\mathbf{x})} & {\mathbf{g}(\mathbf{x})} & {\mathbf{h}(\mathbf{x})} \\{\mathbf{1}} & {\mathbf{m}} & {\mathbf{n}} \\{\mathbf{a}} & {\mathbf{b}} & {\mathbf{c}}\end{array}\right|, \text { prove that } \quad \frac{\mathbf{d y}}{\mathbf{d x}}=\left|\begin{array}{ccc}{\mathbf{f}^{\prime}(\mathbf{x})} & {\mathbf{g}^{\prime}(\mathbf{x})} & {\mathbf{h}^{\prime}(\mathbf{x})} \\{1} & {\mathbf{m}} & {\mathbf{n}} \\{\mathbf{a}} & {\mathbf{b}} & {\mathbf{c}}\end{array}\right|$$

Question 23.
$$y=e^{a \cos ^{-1} x},-1 \leq x \leq 1$$$$\text { show that }\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$$

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)

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)

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)

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)

Question 5.
Differentiate
$$\sin ^{-1}\left(\frac{t}{\sqrt{1+t^{2}}}\right) \text { w:r:to }$$
$$\cos ^{-1}\left(\frac{1}{\sqrt{1+t^{2}}}\right)$$

Question 6.
$$\text { If } y=\sin ^{-1}\left[\frac{1-\sqrt{x}}{1+\sqrt{x}}\right]+\sec ^{-1}\left[\cfrac{1+\sqrt{x}}{1-\sqrt{x}}\right], \text { find } \frac{d y}{d x}$$

Question 7.
Find the derivative of |x|

Question 8.
Diff: log10 x w : r : to logx10

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

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)
$$\text { prove that, } \frac{d y}{d x}=\cfrac{(1+y) \cos x+y \sin x}{1+2 y+\cos x-\sin x}$$ (Kerala CET)
$$y=(\sqrt{x})^{(\sqrt{x})^{(\sqrt{x}}} \text { show that } \frac{d y}{d x}=\frac{y^{2}}{x(2-y \log x)}$$  (Kerala CET)