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# R.D. Sharma solutions for Class 12 Mathematics chapter 15 - Mean Value Theorems

## Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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## Chapter 15 - Mean Value Theorems

#### Page 17

Q 1 | Page 17

Differentiate the following functions from first principles e−x.

Q 2 | Page 17

Differentiate the following functions from first principles e3x.

Q 3 | Page 17

Differentiate the following functions from first principles eax+b.

Q 4 | Page 17

Differentiate the following functions from first principles ecos x.

Q 5 | Page 17

Differentiate the following functions from first principles  $e^\sqrt{2x}$.

Q 6 | Page 17

Differentiate the following functions from first principles log cos x ?

Q 7 | Page 17

​Differentiate the following function from first principles $e^\sqrt{\cot x}$ .

Q 8 | Page 17

Differentiate the following functions from first principles x2ex ?

Q 9 | Page 17

Differentiate the following functions from first principles log cosec x ?

Q 10 | Page 17

Differentiate the following functions from first principles sin−1 (2x + 3) ?

#### Pages 37 - 38

Q 1 | Page 37

Differentiate sin (3x + 5) ?

Q 2 | Page 37

Differentiate tan2 x ?

Q 3 | Page 37

Differentiate tan (x° + 45°) ?

Q 4 | Page 37

Differentiate sin (log x) ?

Q 5 | Page 37

Differentiate $e^{\sin} \sqrt{x}$ ?

Q 6 | Page 37

Differentiate etan x ?

Q 7 | Page 37

Differentiate sin2 (2x + 1) ?

Q 8 | Page 37

Differentiate log7 (2x − 3) ?

Q 9 | Page 37

Differentiate tan 5x° ?

Q 10 | Page 37

Differentiate ${2^x}^3$ ?

Q 11 | Page 37

Differentiate $3^{e^x}$ ?

Q 12 | Page 37

Differentiate logx 3 ?

Q 13 | Page 37

Differentiate $3^{x^2 + 2x}$ ?

Q 14 | Page 37

Differentiate $\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}$ ?

Q 15 | Page 37

Differentiate $3^{x \log x}$ ?

Q 16 | Page 37

Differentiate $\sqrt{\frac{1 + \sin x}{1 - \sin x}}$ ?

Q 17 | Page 37

Differentiate $\sqrt{\frac{1 - x^2}{1 + x^2}}$ ?

Q 18 | Page 37

Differentiate (log sin x)?

Q 19 | Page 37

Differentiate $\sqrt{\frac{1 + x}{1 - x}}$ ?

Q 20 | Page 37

Differentiate $\sin \left( \frac{1 + x^2}{1 - x^2} \right)$ ?

Q 22 | Page 37

Differentiate $e^{3 x} \cos 2x$ ?

Q 23 | Page 37

Differentiate $e^{\tan 3 x}$ ?

Q 24 | Page 37

Differentiate $e^\sqrt{\cot x}$ ?

Q 25 | Page 37

Differentiate $\log \left( \frac{\sin x}{1 + \cos x} \right)$ ?

Q 26 | Page 37

Differentiate $\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}$ ?

Q 27 | Page 37

Differentiate $\tan \left( e^{\sin x }\right)$ ?

Q 28 | Page 37

Differentiate $\log \left( x + \sqrt{x^2 + 1} \right)$ ?

Q 29 | Page 37

Differentiate $\frac{e^x \log x}{x^2}$ ?

Q 30 | Page 37

Differentiate $\log \left( cosec x - \cot x \right)$ ?

Q 31 | Page 37

Differentiate $\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}$ ?

Q 32 | Page 37

Differentiate $\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)$ ?

Q 33 | Page 37

Differentiate $\tan^{- 1} \left( e^x \right)$ ?

Q 34 | Page 37

Differentiate $e^{\sin^{- 1} 2x}$ ?

Q 35 | Page 37

Differentiate $\sin \left( 2 \sin^{- 1} x \right)$ ?

Q 36 | Page 37

Differentiate $e^{\tan^{- 1}} \sqrt{x}$ ?

Q 37 | Page 37

Differentiate $\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}$ ?

Q 38 | Page 37

Differentiate $\log \left( \tan^{- 1} x \right)$?

Q 39 | Page 37

Differentiate $\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}$ ?

Q 40 | Page 37

Differentiate $x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3$ ?

Q 41 | Page 37

Differentiate $\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)$ ?

Q 42 | Page 37

Differentiate $\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}$ ?

Q 43 | Page 37

Differentiate $\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}$ ?

Q 44 | Page 37

Differentiate  $e^x \log \sin 2x$ ?

Q 46 | Page 37

Differentiate $\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}$ ?

Q 47 | Page 37

Differentiate $\left( \sin^{- 1} x^4 \right)^4$ ?

Q 48 | Page 37

Differentiate $\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)$ ?

Q 49 | Page 37

Differentiate $\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}$ ?

Q 50 | Page 37

Differentiate $3 e^{- 3x} \log \left( 1 + x \right)$ ?

Q 51 | Page 37

Differentiate $\frac{x^2 + 2}{\sqrt{\cos x}}$ ?

Q 52 | Page 38

Differentiate $\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}$ ?

Q 53 | Page 38

$\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}$ ?

Q 54 | Page 38

Differentiate $e^{ax} \sec x \tan 2x$ ?

Q 55 | Page 38

Differentiate $\log \left( \cos x^2 \right)$ ?

Q 56 | Page 38

Differentiate $\cos \left( \log x \right)^2$ ?

Q 57 | Page 38

Differentiate $\log \sqrt{\frac{x - 1}{x + 1}}$ ?

Q 58 | Page 38

If $y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}$ ,show that $\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}$ ?

Q 59 | Page 38

If $y = \sqrt{x + 1} + \sqrt{x - 1}$ , prove that $\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y$ ?

Q 60 | Page 38

If $y = \frac{x}{x + 2}$  , prove tha $x\frac{dy}{dx} = \left( 1 - y \right) y$ ?

Q 61 | Page 38

If $y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)$prove that $\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}$ ?

Q 62 | Page 38

If  $y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}$  prove that $\frac{dy}{dx} = \sec 2x$ ?

Q 63 | Page 38

If $y = \sqrt{x} + \frac{1}{\sqrt{x}}$, prove that  $2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}$ ?

Q 64 | Page 38

If $y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}$ ,  prove that $\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}$ ?

Q 65 | Page 38

If $y = \frac{e^x - e^{- x}}{e^x + e^{- x}}$ .prove that $\frac{dy}{dx} = 1 - y^2$ ?

Q 66 | Page 38

If  $y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)$ , prove that $\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)$ ?

Q 67 | Page 38

If $y = e^x \cos x$ ,prove that $\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)$ ?

Q 68 | Page 38

If $y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)$ , prove that $\frac{ dy }{ dx } = 2 cosec 2x$ ?

Q 69 | Page 38

If $y = x \sin^{- 1} x + \sqrt{1 - x^2}$ ,prove that $\frac{dy}{dx} = \sin^{- 1} x$ ?

Q 70 | Page 38

If $y = \sqrt{x^2 + a^2}$ prove that  $y\frac{dy}{dx} - x = 0$ ?

Q 71 | Page 38

If $y = e^x + e^{- x}$ prove that  $\frac{dy}{dx} = \sqrt{y^2 - 4}$ ?

Q 72 | Page 38

If $y = \sqrt{a^2 - x^2}$ prove that  $y\frac{dy}{dx} + x = 0$ ?

Q 73 | Page 38

If xy = 4, prove that $x\left( \frac{dy}{dx} + y^2 \right) = 3 y$ ?

Q 74 | Page 38

Prove that $\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}$ ?

#### Pages 62 - 64

Q 1 | Page 62

Differentiate $\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1$ ?

Q 2 | Page 62

Differentiate $\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1$ ?

Q 3 | Page 63

Differentiate  $\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1$  ?

Q 4 | Page 63

Differentiate $\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1$ ?

Q 5 | Page 63

Differentiate $\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a$ ?

Q 6 | Page 63

Differentiate $\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}$ ?

Q 7 | Page 63

Differentiate $\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1$  ?

Q 8 | Page 63

Differentiate $\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1$ ?

Q 9 | Page 63

Differentiate $\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}$ ?

Q 10 | Page 63

Differentiate $\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}$ ?

Q 11 | Page 63

Differentiate $\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}$ ?

Q 12 | Page 63

Differentiate $\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1$ ?

Q 13 | Page 63

Differentiate $\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a$ ?

Q 14 | Page 63

Differentiate $\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1$ ?

Q 15 | Page 63

Differentiate $\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1$ ?

Q 16 | Page 63

Differentiate $\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}$ ?

Q 17 | Page 63

Differentiate $\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0$ ?

Q 18 | Page 63

Differentiate $\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0$ ?

Q 19 | Page 63

Differentiate $\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1$ ?

Q 20 | Page 63

Differentiate $\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0$ ?

Q 21 | Page 63

Differentiate $\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi$ ?

Q 22 | Page 63

Differentiate $\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)$ ?

Q 23 | Page 63

Differentiate $\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty$ ?

Q 24 | Page 63

Differentiate $\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R$ ?

Q 25 | Page 63

Differentiate $\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)$ ?

Q 26 | Page 63

Differentiate  $\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)$ ?

Q 27 | Page 63

Differentiate $\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)$ ?

Q 28 | Page 63

Differentiate $\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)$ ?

Q 29 | Page 63

Differentiate $\tan^{- 1} \left( \frac{x - a}{x + a} \right)$ ?

Q 30 | Page 63

Differentiate $\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)$ ?

Q 31 | Page 64

Differentiate $\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}$ ?

Q 32 | Page 63

Differentiate

$\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}$ ?

Q 33 | Page 64

Differentiate $\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}$ ?

Q 34 | Page 64

Differentiate $\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)$ ?

Q 35 | Page 64

If  $y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,$ prove that  $\frac{dy}{dx} = \frac{4}{1 + x^2}$ ?

Q 36 | Page 64

If $y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty$ prove that  $\frac{dy}{dx} = \frac{2}{1 + x^2}$ ?

Q 37 | Page 64

Differentiate the following with respect to x

$\left( i \right) \cos^{- 1} \left( \sin x \right)$

(ii)  $\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)$

Q 38 | Page 64

If  $y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}$,  show that $\frac{dy}{dx}$ is independent of x. ?

Q 39 | Page 64

If $y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0$ ,prove that $\frac{dy}{dx} = \frac{4}{1 + x^2}$ ?

Q 40 | Page 64

If  $y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}$ ?

Q 41 | Page 64

If $y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}$ ?

Q 42 | Page 64

If  $y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, 0 < x < \frac{1}{2}, \text{ find } \frac{dy}{dx} .$ ?

Q 43 | Page 64

If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?

Q 44 | Page 64

If $y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx}$ ?

Q 45 | Page 64

If $y = \tan^{- 1} \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \right), \text{find } \frac{dy}{dx}$ ?

Q 46 | Page 64

If $y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}$ ?

Q 47 | Page 64

Differentiate $\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left( 36 \right)^x} \right\}$  with respect to x ?

Q 48 | Page 64

If $y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}$ $\frac{dy}{dx}$ ?

#### Pages 74 - 75

Q 1 | Page 74

Find $\frac{dy}{dx}$ in each of the following cases $xy = c^2$  ?

Q 2 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases $y^3 - 3x y^2 = x^3 + 3 x^2 y$ ?

Q 3 | Page 74

Find   $\frac{dy}{dx}$ in each of the following cases  $x^{2/3} + y^{2/3} = a^{2/3}$ ?

Q 4 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases  $4x + 3y = \log \left( 4x - 3y \right)$ ?

Q 5 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases  $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ?

Q 6 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases $x^5 + y^5 = 5 xy$ ?

Q 7 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases $\left( x + y \right)^2 = 2axy$ ?

Q 8 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases $\left( x^2 + y^2 \right)^2 = xy$ ?

Q 9 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases $\tan^{- 1} \left( x^2 + y^2 \right) = a$ ?

Q 10 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases $e^{x - y} = \log \left( \frac{x}{y} \right)$ ?

Q 11 | Page 74

Find  $\frac{dy}{dx}$ in each of the following cases $\sin xy + \cos \left( x + y \right) = 1$ ?

Q 12 | Page 74

If $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)$ , prove that $\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}$ ?

Q 13 | Page 75

If $y \sqrt{1 - x^2} + x \sqrt{1 - y^2} = 1$ ,prove that $\frac{dy}{dx} = - \sqrt{\frac{1 - y^2}{1 - x^2}}$ ?

Q 14 | Page 75

If $xy = 1$ prove that $\frac{dy}{dx} + y^2 = 0$ ?

Q 15 | Page 75

If $x y^2 = 1,$ prove that $2\frac{dy}{dx} + y^3 = 0$ ?

Q 16 | Page 75

If $x \sqrt{1 + y} + y \sqrt{1 + x} = 0$ , prove that $\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0$  ?

Q 17 | Page 75

If $\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)$ Prove that $\frac{dy}{dx} = \frac{x + y}{x - y}$ ?

Q 18 | Page 75

If $\sec \left( \frac{x + y}{x - y} \right) = a$ Prove that  $\frac{dy}{dx} = \frac{y}{x}$ ?

Q 19 | Page 75

If $\tan^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = a$ Prove that  $\frac{dy}{dx} = \frac{x}{y}\frac{\left( 1 - \tan a \right)}{\left( 1 + \tan a \right)}$ ?

Q 20 | Page 75

If $xy \log \left( x + y \right) = 1$ ,Prove that $\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}$ ?

Q 21 | Page 75

If $y = x \sin \left( a + y \right)$ ,Prove that $\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}$ ?

Q 22 | Page 75

If $x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0$ Prove that $\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}$ ?

Q 23 | Page 75

If $y = x \sin y$ , Prove that $\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}$ ?

Q 24 | Page 75

If $y \sqrt{x^2 + 1} = \log \left( \sqrt{x^2 + 1} - x \right)$ ,Show that $\left( x^2 + 1 \right) \frac{dy}{dx} + xy + 1 = 0$ ?

Q 25 | Page 75

If $\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}$ ?

Q 26 | Page 75

If  $\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}$ ?

Q 27 | Page 75

$If e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0$ ?

Q 28 | Page 75

If $\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}$ ?

Q 29 | Page 75

If $\sin^2 y + \cos xy = k,$ find  $\frac{dy}{dx}$ at $x =$ $y = \frac{\pi}{4} .$ ?

Q 30 | Page 75

If $y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}$ ?

Q 31 | Page 75

If $\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}$ ?

#### Pages 88 - 90

Q 1 | Page 88

Differentiate $x^{1/x}$ ?

Q 2 | Page 88

Differentiate $x^{\sin x}$  ?

Q 3 | Page 88

Differentiate $\left( 1 + \cos x \right)^x$ ?

Q 4 | Page 88

Differentiate $x^{\cos^{- 1} x}$ ?

Q 5 | Page 88

Differentiate $\left( \log x \right)^x$ ?

Q 6 | Page 88

Differentiate $\left( \log x \right)^{\cos x}$ ?

Q 7 | Page 88

Differentiate $\left( \sin x \right)^{\cos x}$ ?

Q 8 | Page 88

Differentiate $e^{x \log x}$ ?

Q 9 | Page 88

Differentiate  $\left( \sin x \right)^{\log x}$ ?

Q 10 | Page 88

Differentiate ${10}^{ \log \sin x }$ ?

Q 11 | Page 88

Differentiate $\left( \log x \right)^{ \log x }$ ?

Q 12 | Page 88

Differentiate ${10}^\left( {10}^x \right)$ ?

Q 13 | Page 88

Differentiate  $\sin \left( x^x \right)$ ?

Q 14 | Page 88

Differentiate $\left( \sin^{- 1} x \right)^x$ ?

Q 15 | Page 88

Differentiate $x^{\sin^{- 1} x}$  ?

Q 16 | Page 88

Differentiate $\left( \tan x \right)^{1/x}$ ?

Q 17 | Page 88

Differentiate $x^{\tan^{- 1} x }$  ?

Q 18.1 | Page 88

Differentiate  $\left( x^x \right) \sqrt{x}$ ?

Q 18.2 | Page 88

Differentiate $x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}$ ?

Q 18.3 | Page 88

Differentiate  $x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}$  ?

Q 18.4 | Page 88

Differentiate $\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}$ ?

Q 18.5 | Page 88

Differentiate$\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)$ ?

Q 18.6 | Page 88

Differentiate $e^{\sin x }+ \left( \tan x \right)^x$ ?

Q 18.7 | Page 88

Differentiate $\left( \cos x \right)^x + \left( \sin x \right)^{1/x}$ ?

Q 18.8 | Page 88

Differentiate  $x^{x^2 - 3} + \left( x - 3 \right)^{x^2}$ ?

Q 19 | Page 89

Find  $\frac{dy}{dx}$ $y = e^x + {10}^x + x^x$ ?

Q 20 | Page 89
Find $\frac{dy}{dx}$  $y = x^n + n^x + x^x + n^n$ ?
Q 21 | Page 89

find  $\frac{dy}{dx}$  $y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}$ ?

Q 22 | Page 89

Find  $\frac{dy}{dx}$  $y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}$ ?

Q 23 | Page 89

Find  $\frac{dy}{dx}$ $y = e^{3x} \sin 4x \cdot 2^x$ ?

Q 24 | Page 89

Find  $\frac{dy}{dx}$ $y = \sin x \sin 2x \sin 3x \sin 4x$ ?

Q 25 | Page 89

Find  $\frac{dy}{dx}$ $y = x^{\sin x} + \left( \sin x \right)^x$ ?

Q 26 | Page 89

Find  $\frac{dy}{dx}$  $y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}$ ?

Q 27 | Page 89

Find $\frac{dy}{dx}$ $y = \left( \tan x \right)^{\cot } x + \left( \cot x \right)^{\tan x}$ ?

Q 28 | Page 89

Fine $\frac{dy}{dx}$ $y = \left( \sin x \right)^x + \sin^{- 1} \sqrt{x}$ ?

Q 29.1 | Page 89

Find $\frac{dy}{dx}$ $y = x^{\cos x} + \left( \sin x \right)^{\tan x}$ ?

Q 29.2 | Page 89

Find $\frac{dy}{dx}$  $y = x^x + \left( \sin x \right)^x$ ?

Q 30 | Page 89

Find $\frac{dy}{dx}$ $y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)$ ?

Q 31 | Page 89

Find $\frac{dy}{dx}$ $y = x^x + x^{1/x}$ ?

Q 32 | Page 89

Find $\frac{dy}{dx}$ $y = x^{\log x }+ \left( \log x \right)^x$ ?

Q 33 | Page 89

If $x^{13} y^7 = \left( x + y \right)^{20}$ prove that $\frac{dy}{dx} = \frac{y}{x}$ ?

Q 34 | Page 89

If $x^{16} y^9 = \left( x^2 + y \right)^{17}$ ,prove that $x\frac{dy}{dx} = 2 y$ ?

Q 35 | Page 89

If $y = \sin \left( x^x \right)$ prove that  $\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)$ ?

Q 36 | Page 89

If $x^x + y^x = 1$, prove that $\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}$ ?

Q 37 | Page 89

If $x^y \cdot y^x = 1$ , prove that $\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}$ ?

Q 38 | Page 89

If $x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}$ ?

Q 39 | Page 89

If $x^m y^n = 1$ , prove that $\frac{dy}{dx} = - \frac{my}{nx}$ ?

Q 40 | Page 89

If $y^x = e^{y - x}$ ,prove that $\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}$ ?

Q 41 | Page 89

If $\left( \sin x \right)^y = \left( \cos y \right)^x ,$, prove that $\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}$ ?

Q 42 | Page 89

If $\left( \cos x \right)^y = \left( \tan y \right)^x$ , prove that $\frac{dy}{dx} = \frac{\log \tan y + y \tan x}{ \log \cos x - x \sec y \ cosec\ y }$ ?

Q 43 | Page 89

If $e^x + e^y = e^{x + y}$ , prove that

$\frac{dy}{dx} + e^{y - x} = 0$ ?

Q 44 | Page 90

If $e^y = y^x ,$ prove that$\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}$ ?

Q 45 | Page 90

If $e^{x + y} - x = 0$ ,prove that $\frac{dy}{dx} = \frac{1 - x}{x}$ ?

Q 46 | Page 90

If $y = x \sin \left( a + y \right)$ , prove that $\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}$ ?

Q 47 | Page 90

If  $x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0$ , prove that $\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}$ ?

Q 48 | Page 90

If  $\left( \sin x \right)^y = x + y$ , prove that $\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}$ ?

Q 49 | Page 90

If $xy \log \left( x + y \right) = 1$ , prove that  $\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}$ ?

Q 50 | Page 90

If $y = x \sin y$ , prove that  $\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}$ ?

Q 51 | Page 90

Find the derivative of the function f (x) given by  $f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)$ and hence find f' (1) ?

Q 52 | Page 90

If $y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .$ ?

Q 53 | Page 90

If $y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}$ ?

Q 54 | Page 90

If  $xy = e^{x - y} , \text{ find } \frac{dy}{dx}$ ?

Q 55 | Page 90

If $y^x + x^y + x^x = a^b$ ,find $\frac{dy}{dx}$ ?

Q 56 | Page 90

If  $\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}$ ?

Q 57 | Page 90
$\text{ If }\cos y = x\cos\left( a + y \right),\text{ where } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}$ ?
Q 58 | Page 90
$\text{ If } \left( x - y \right) e^\frac{x}{x - y} = a,\text{ prove that y }\frac{dy}{dx} + x = 2y$ ?
Q 59 | Page 90
$\text{ If } x = e^{x/y} , \text{ prove that } \frac{dy}{dx} = \frac{x - y}{x\log x}$ ?
Q 60 | Page 90
$\text{ If }y = x^{\tan x} + \sqrt{\frac{x^2 + 1}{2}}, \text{ find} \frac{dy}{dx}$ ?

Q 61 | Page 90
$\text{ If y } = 1 + \frac{\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{{\beta}/{x}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)} + \frac{{\gamma}/{x^2}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)\left( \frac{1}{x} - \gamma \right)}, \text{ find } \frac{dy}{dx}$ ?

#### Pages 98 - 99

Q 1 | Page 98

If $y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}$ prove that $\frac{dy}{dx} = \frac{1}{2 y - 1}$ ?

Q 2 | Page 98

If $y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}$ , prove that $\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}$ ?

Q 3 | Page 98

If  $y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + . . to \infty}}}$  $\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}$ ?

Q 4 | Page 98

If  $y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}$ , prove that $\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}$ ?

Q 5 | Page 98

$y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}}$,prove that $\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}$ ?

Q 6 | Page 98

If $y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}$, prove that $\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}$ ?

Q 7 | Page 99

If $y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}$, prove that  $\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}$$+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}$ ?

Q 8 | Page 99

If $y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}$,prove that $\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}$?

#### Pages 103 - 104

Q 1 | Page 103

Find $\frac{dy}{dx}$ $x = a t^2 \text{ and } y = 2\ at$ ?

Q 2 | Page 103

Find $\frac{dy}{dx}$, When $x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)$ ?

Q 3 | Page 103

If $\frac{dy}{dx}$ when $x = a \cos \theta \text{ and } y = b \sin \theta$ ?

Q 4 | Page 103

Find $\frac{dy}{dx}$,when $x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)$ ?

Q 5 | Page 103

Find $\frac{dy}{dx}$ , when $x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta$ ?

Q 6 | Page 103

Find $\frac{dy}{dx}$ ,When $x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta = \frac{\pi}{2}$ ?

Q 7 | Page 103

Find $\frac{dy}{dx}$ ,when $x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}$ ?

Q 8 | Page 103

Find $\frac{dy}{dx}$ , when $x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}$ ?

Q 9 | Page 103

Find $\frac{dy}{dx}$, when $x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)$ ?

Q 10 | Page 103

Find $\frac{dy}{dx}$ ,When $x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)$ ?

Q 11 | Page 103

Find $\frac{dy}{dx}$ when $x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}$ ?

Q 12 | Page 103

Find $\frac{dy}{dx}$ , when  $x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R$ ?

Q 13 | Page 103

Find  $\frac{dy}{dx}$ , when  $x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}$ ?

Q 14 | Page 103

If  $x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta$  $\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)$ ?

Q 15 | Page 103

If $x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,$ prove that $\frac{dy}{dx} = - \frac{y \log x}{x \log y}$ ?

Q 16 | Page 103

If $x = \cos t \text{ and y } = \sin t,$ prove that  $\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}$ ?

Q 17 | Page 103

If  $x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)$ ,prove that  $\frac{dy}{dx} = \frac{x}{y}$?

Q 18 | Page 103
If $x = \sin^{- 1} \left( \frac{2 t}{1 + t^2} \right) \text{ and y } = \tan^{- 1} \left( \frac{2 t}{1 - t^2} \right), - 1 < t < 1$ porve that $\frac{dy}{dx} = 1$ ?

Q 19 | Page 103

If  $x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}$ , find$\frac{dy}{dx}$ ?

Q 20 | Page 103

If $x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}$ ?

Q 21 | Page 103

If $x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}$ ?

Q 22 | Page 104

If $x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .$ ?

Q 23 | Page 104

If $x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} at \theta = \frac{\pi}{3}$ ?

Q 24 | Page 104

If  $x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)$ , show that at  $t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}$ ?

Q 25 | Page 104

$\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx} at t = \frac{\pi}{4}$ ?

Q 26 | Page 104

If  $x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}$ ?

Q 27 | Page 104
$\sin x = \frac{2t}{1 + t^2}, \tan y = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}$ ?
Q 28 | Page 104

Write the derivative of sinx with respect to cosx ?

#### Pages 112 - 113

Q 1 | Page 112

Differentiate x2 with respect to x3

Q 2 | Page 112

Differentiate log (1 + x2) with respect to tan−1 x ?

Q 3 | Page 112

Differentiate (log x)x with respect to log x ?

Q 4.1 | Page 112

Differentiate  $\sin^{- 1} \sqrt{1 - x^2}$ with respect to $\cos^{- 1} x, \text { if}$$x \in \left( 0, 1 \right)$  ?

Q 4.2 | Page 112

Differentiate  $\sin^{- 1} \sqrt{1 - x^2}$ with respect to $\cos^{- 1} x, \text { if}$ $x \in \left( - 1, 0 \right)$ ?

Q 5.1 | Page 112
Differentiate $\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)$ with respect to $\sqrt{1 - 4 x^2}$ , if $x \in \left( - \frac{1}{2 \sqrt{2}}, \frac{1}{\sqrt{2 \sqrt{2}}} \right)$ ?
Q 5.2 | Page 112

Differentiate $\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)$ with respect to $\sqrt{1 - 4 x^2}$ , if $x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)$ ?

Q 5.3 | Page 112

Differentiate $\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)$ with respect to $\sqrt{1 - 4 x^2}$ , if $x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)$ ?

Q 6 | Page 112

Differentiate$\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)$ with respect to $\sin^{-1} \left( \frac{2x}{1 + x^2} \right)$, If $- 1 < x < 1, x \neq 0 .$ ?

Q 7.1 | Page 112

Differentiate $\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)$ with respect to  $\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)$ $x \in \left( 0, \frac{1}{\sqrt{2}} \right)$ ?

Q 7.2 | Page 112

Differentiate $\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)$ with respect to  $\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)$ $x \in \left( \frac{1}{\sqrt{2}}, 1 \right)$ ?

Q 8 | Page 112

Differentiate $\left( \cos x \right)^{\sin x }$ with respect to $\left( \sin x \right)^{\cos x }$?

Q 9 | Page 112

Differentiate $\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)$ with respect to $\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { if } 0 < x < 1$ ?

Q 10 | Page 113

Differentiate $\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)$ with respect to $\sqrt{1 + a^2 x^2}$ ?

Q 11 | Page 113

Differentiate $\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)$ with respect to $\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$ ?

Q 12 | Page 113

Differentiate $\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)$ with respect to $\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1$ ?

Q 13 | Page 113

Differentiate $\tan^{- 1} \left( \frac{x - 1}{x + 1} \right)$ with respect to $\sin^{- 1} \left( 3x - 4 x^3 \right), \text { if }- \frac{1}{2} < x < \frac{1}{2}$ ?

Q 14 | Page 113

Differentiate $\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)$ with  respect to $\sec^{- 1} x$ ?

Q 15 | Page 113

Differentiate $\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)$ with respect to $\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1$ ?

Q 16 | Page 113

Differentiate $\cos^{- 1} \left( 4 x^3 - 3x \right)$ with respect to $\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), if \frac{1}{2} < x < 1$ ?

Q 17 | Page 113

Differentiate $\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)$ with respect to $\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$ ?

Q 18 | Page 113

$\sin^{- 1} \sqrt{1 - x^2}$ with respect to $\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1$ ?

Q 19 | Page 113

Differentiate $\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)$ with respect to $\sqrt{1 a^2 x^2}, if - \frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}$ ?

Q 20 | Page 113

Differentiate $\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)$ with respect to $\sqrt{1 - x^2},\text {if} - 1 < x < 1$ ?

#### Pages 116 - 118

Q 1 | Page 117

If f (x) = loge (loge x), then write the value of f' (e) ?

Q 2 | Page 117

If $f\left( x \right) = x + 1$ , then write the value of $\frac{d}{dx} \left( fof \right) \left( x \right)$ ?

Q 3 | Page 117

If $f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e$ ?

Q 4 | Page 117

If $f\left( 1 \right) = 4, f'\left( 1 \right) = 2$ find the value of the derivative of  $\log \left( f\left( e^x \right) \right)$ w.r. to x at the point x = 0 ?

Q 5 | Page 117

If $f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)$ then find $\frac{dy}{dx} \text { at } x = 1$ ?

Q 6 | Page 117

Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and f' (3) = 9, write the value of g' (9).

Q 7 | Page 117

If $y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ ,Then, write the value of $\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right)$ ?

Q 8 | Page 117

If $\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx}$ ?

Q 9 | Page 117

If $\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}$ ?

Q 10 | Page 118

If $y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)$ write the value of $\frac{dy}{dx}\text { for } x > 1$ ?

Q 11 | Page 118

If $f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}$ write the value of $\frac{dy}{dx} \text{ at x } = 0$ ?

Q 12 | Page 118

If $y = x \left| x \right|$ , find $\frac{dy}{dx} \text{ for } x < 0$ ?

Q 13 | Page 118

If $y = \sin^{- 1} x + \cos^{- 1} x$ ,find $\frac{dy}{dx}$ ?

Q 14 | Page 118

If $x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}$ ?

Q 15 | Page 118

If $- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}$ ?

Q 16 | Page 116

If $y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e$ ?

Q 17 | Page 118

If $y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}$  ?

Q 18 | Page 118

If $y = \log_a x, \text{ find } \frac{dy}{dx}$ ?

Q 19 | Page 118

If $y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx}$ ?

Q 20 | Page 118

If $y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}$ ?

Q 21 | Page 118

If $y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)$ then write the value of $\frac{dy}{dx}$ ?

Q 22 | Page 118

If $\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . .$  to ∞, then find the value of  $\frac{dy}{dx}$ ?

Q 23 | Page 118

If $u = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ and v} = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)$ where $- 1 < x < 1$, then write the value of $\frac{du}{dv}$ ?

Q 24 | Page 118

If $f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2$ , then find the value of f' (1) ?

Q 25 | Page 118

If $y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx}$ ?

Q 26 | Page 118

If f (x) is an even function, then write whether f' (x) is even or odd ?

Q 27 | Page 118

If f (x) is an odd function, then write whether f' (x) is even or odd ?

Q 28 | Page 118

If $x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}$ ?

#### Pages 119 - 122

Q 1 | Page 119

If f (x) = logx2 (log x), the f' (x) at x = e is
(a) 0
(b) 1
(c) 1/e
(d) 1/2 e

Q 2 | Page 119

The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is

(a) $\frac{x}{\log x}$

(b)  $\frac{\log x}{x}$

(c) $\left( x \log x \right)^{- 1}$

(d) none of these

Q 3 | Page 119

The derivative of the function

$\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }$
(a) (2/3)1/2
(b) (1/3)1/2
(c) 31/2
(d) 61/2
Q 4 | Page 119

Differential coefficient of sec

$\sec \left( \tan^{- 1} x \right)$ is
(a)  $\frac{x}{1 + x^2}$
(b)  $x \sqrt{1 + x^2}$
(c) $\frac{1}{\sqrt{1 + x^2}}$
(d) $\frac{x}{\sqrt{1 + x^2}}$

Q 5 | Page 119

If $f\left( x \right) = \tan^{- 1} \sqrt{\frac{1 + \sin x}{1 - \sin x}}, 0 \leq x \leq \pi/2, \text{ then } f' \left( \pi/6 \right) \text{ is }$

(a) − 1/4
(b) − 1/2
(c) 1/4
(d) 1/2

Q 6 | Page 119

If $y = \left( 1 + \frac{1}{x} \right)^x , \text{ then} \frac{dy}{dx} =$

(a) $\left( 1 + \frac{1}{x} \right)^x \left( 1 + \frac{1}{x} \right) - \frac{1}{x + 1}$

(b) $\left( 1 + \frac{1}{x} \right)^x \log \left( 1 + \frac{1}{x} \right)$

(c) $\left( x + \frac{1}{x} \right)^x \left\{ \log \left( x + 1 \right) - \frac{x}{x + 1} \right\}$

(d) $\left( x + \frac{1}{x} \right)^x \left\{ \log \left( 1 + \frac{1}{x} \right) + \frac{1}{x + 1} \right\}$

Q 7 | Page 119

If $x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}$ is

(a) $\frac{1 + x}{1 + \log x}$

(b) $\frac{1 - \log x}{1 + \log x}$

(c) not defined

(d) $\frac{\log x}{\left( 1 + \log x \right)^2}$

Q 8 | Page 119

Given  $f\left( x \right) = 4 x^8 , \text { then }$

(a) $f'\left( \frac{1}{2} \right) = f'\left( - \frac{1}{2} \right)$

(b) $f\left( \frac{1}{2} \right) = - f'\left( - \frac{1}{2} \right)$

(c) $f\left( - \frac{1}{2} \right) = f\left( - \frac{1}{2} \right)$

(d) $f\left( \frac{1}{2} \right) = f'\left( - \frac{1}{2} \right)$

Q 9 | Page 119

If $x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =$

(a) $\tan^2 \theta$

(b) $\sec^2 \theta$

(c) $\sec \theta$

(d) $\left| \sec \theta \right|$

Q 10 | Page 120

If $y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =$

(a) $- \frac{2}{1 + x^2}$

(b) $\frac{2}{1 + x^2}$

(c) $\frac{1}{2 - x^2}$

(d) $\frac{2}{2 - x^2}$

Q 11 | Page 120

The derivative of $\sec^{- 1} \left( \frac{1}{2 x^2 + 1} \right) \text { w . r . t }. \sqrt{1 + 3 x} \text { at } x = - 1/3$

(a) does not exist
(b) 0
(c) 1/2
(d) 1/3

Q 12 | Page 120

For the curve $\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }$

(a) 1/2
(b) 1
(c) −1
(d) 2
Q 13 | Page 120

If $\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =$

(a) 2
(b) − 2
(c) 1
(d) − 1]

Q 14 | Page 120

Let  $\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =$

(a) 1/2
(b) x
(c) $\frac{1 - x^2}{x^2 - 4}$

(d) 1

Q 15 | Page 120

$\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }$

(a) 1/2
(b) − 1/2
(c) 1
(d) − 1

Q 16 | Page 120
$\frac{d}{dx} \left[ \log \left\{ e^x \left( \frac{x - 2}{x + 2} \right)^{3/4} \right\} \right]$ equals
(a) $\frac{x^2 - 1}{x^2 - 4}$
(b) 1
(c)$\frac{x^2 + 1}{x^2 - 4}$
(d)  $e^x \frac{x^2 - 1}{x^2 - 4}$
Q 17 | Page 120

If $y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =$

(a) $\frac{\sin x}{2 y - 1}$

(b) $\frac{\sin x}{1 - 2 y}$

(c) $\frac{\cos x}{1 - 2 y}$

(d) $\frac{\cos x}{2 y - 1}$

Q 18 | Page 120

If $3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =$

(a) $- \frac{y}{x}$

(b) $\frac{3 \sin \left( xy \right) + 4 \cos \left( xy \right)}{3 \cos \left( xy \right) - 4 \sin \left( xy \right)}$

(c) $\frac{3 \cos \left( xy \right) + 4 \sin \left( xy \right)}{4 \cos \left( xy \right) - 3 \sin \left( xy \right)}$

(d) none of these

Q 19 | Page 120

If $\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}$

(a) $\frac{\sin a}{\sin a \sin^2 \left( a + y \right)}$

(b) $\frac{\sin^2 \left( a + y \right)}{\sin a}$

(c) $\sin a \sin^2 \left( a + y \right)$

(d) $\frac{\sin^2 \left( a - y \right)}{\sin a}$

Q 20 | Page 121

The derivative of $\cos^{- 1} \left( 2 x^2 - 1 \right)$ with respect to  $\cos^{- 1} x$  is

(a) 2

(b) $\frac{1}{2 \sqrt{1 - x^2}}$

(c) $2/x$

(d) $1 - x^2$

Q 21 | Page 121

If $f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)$ is equal to

(a) $1 \text { for x } < - 3$

(b) $- 1\text { for x} < - 3$

(c) $1\text { for all } x \in R$

(d) none of these

Q 22 | Page 121

If $f\left( x \right) = \left| x^2 - 9x + 20 \right|$  then f' (x) is equal to

(a) $- 2x + 9\text { for all } x \in R$

(b) $2x - 9 \text { if }4 < x < 5$

(c) $- 2x + 9, \text { if }4 < x < 5$

(d) none of these

Q 23 | Page 121

If $f\left( x \right) = \sqrt{x^2 - 10x + 25}$  then the derivative of f (x) in the interval [0, 7] is

(a) 1
(b) −1
(c) 0
(d) none of these

Q 24 | Page 121

If $f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)$  is equal to

(a) 1
(b) −1
(c) 0
(d) none of these

Q 25 | Page 121

If $f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}$ the f' (x) is equal to

(a) 1
(b) 0
(c) $x^{l + m + n}$

(d) none of these

Q 26 | Page 121

If $y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}$ then $\frac{dy}{dx}$  is equal to

(a) 1
(b) $\left( a + b + c \right)^{x^{a + b + c - 1}}$
(c) 0
(d) none of these
Q 27 | Page 121

If  $\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)$ then $\frac{dy}{dx}$ is equal to

(a) $\frac{x^2}{y^2} \sqrt{\frac{1 - y^6}{1 - x^6}}$

(b) $\frac{y^2}{x^2}\sqrt{\frac{1 - y^6}{1 + x^6}}$

(c) $\frac{x^2}{y^2}\sqrt{\frac{1 - x^6}{1 - y^6}}$

(d) none of these

Q 28 | Page 121

If $y = \log \sqrt{\tan x}$ then the value of

$\frac{dy}{dx}\text { at }x = \frac{\pi}{4}$ is given by

(a) ∞
(b) 1
(c) 0
(d) $\frac{1}{2}$

Q 29 | Page 121

If $\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}$ is equal to

(a) $\frac{x^2 - y^2}{x^2 + y^2}$

(b)  $\frac{y}{x}$

(c) $\frac{x}{y}$

(d) none of these

Q 30 | Page 121

If $\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}$ is equal to

(a) $\frac{\cos^2 \left( a + y \right)}{\cos a}$

(b)$\frac{\cos a}{\cos^2 \left( a + y \right)}$

(c) $\frac{\sin^2 y}{\cos a}$

(d) none of these

Q 31 | Page 122

If $y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =$

(a) $\frac{4 x^3}{1 - x^4}$

(b) $- \frac{4x}{1 - x^4}$

(c) $\frac{1}{4 - x^4}$

(d) $- \frac{4 x^3}{1 - x^4}$

Q 32 | Page 122

If $y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }$

(a) $\frac{\cos x}{2y - 1}$

(b) $\frac{\cos x}{1 - 2y}$

(c)  $\frac{\sin x}{1 - 2y}$

(d) $\frac{\sin x}{2y - 1}$

Q 33 | Page 122

If $y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}$ is equal to

(a) $\frac{1}{2}$

(b) 0
(c) 1
(d) none of these

#### Pages 16 - 18

Q 1.1 | Page 16

Find the second order derivatives of the following function  x3 + tan x ?

Q 1.2 | Page 16

Find the second order derivatives of the following function sin (log x) ?

Q 1.3 | Page 16

Find the second order derivatives of the following function  log (sin x) ?

Q 1.4 | Page 16

Find the second order derivatives of the following function ex sin 5x  ?

Q 1.5 | Page 16

Find the second order derivatives of the following function e6x cos 3x  ?

Q 1.6 | Page 16

Find the second order derivatives of the following function x3 log ?

Q 1.7 | Page 16

Find the second order derivatives of the following function tan−1 x ?

Q 1.8 | Page 16

Find the second order derivatives of the following function x cos x ?

Q 1.9 | Page 16

Find the second order derivatives of the following function  log (log x)  ?

Q 2 | Page 16

If y = ex cos x, show that $\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x$ ?

Q 3 | Page 16

If y = x + tan x, show that  $\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0$ ?

Q 4 | Page 16

If y = x3 log x, prove that $\frac{d^4 y}{d x^4} = \frac{6}{x}$ ?

Q 5 | Page 16

If y = log (sin x), prove that $\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x$ ?

Q 6 | Page 16

If y = 2 sin x + 3 cos x, show that $\frac{d^2 y}{d x^2} + y = 0$ ?

Q 7 | Page 16

If $y = \frac{\log x}{x}$ show that $\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}$ ?

Q 8 | Page 16

If x = a sec θ, y = b tan θ, prove that $\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}$ ?

Q 10 | Page 16

If y = ex cos x, prove that $\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)$ ?

Q 11 | Page 16

If x = a cos θ, y = b sin θ, show that $\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}$ ?

Q 12 | Page 16

If x = a (1 − cos3 θ), y = a sin3 θ, prove that $\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}$ ?

Q 13 | Page 16

If x = a (θ + sin θ), y = a (1 + cos θ), prove that $\frac{d^2 y}{d x^2} = - \frac{a}{y^2}$ ?

Q 14 | Page 16

If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find $\frac{d^2 y}{d x^2}$ ?

Q 15 | Page 16

If x = a(1 − cos θ), y = a(θ + sin θ), prove that $\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}$ ?

Q 17 | Page 17

If x = cos θ, y = sin3 θ, prove that $y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)$ ?

Q 18 | Page 17

If y = sin (sin x), prove that $\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0$ ?

Q 19 | Page 17

If x = sin ty = sin pt, prove that $\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0$ ?

Q 20 | Page 17

If y = (sin−1 x)2, prove that (1 − x2)

$\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0$ ?

Q 21 | Page 17

If $y = e^{\tan^{- 1} x}$ prove that (1 + x2)y2 + (2x − 1)y1 = 0 ?

Q 22 | Page 17

If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0 ?

Q 23 | Page 17

If $y = e^{2x} \left( ax + b \right)$  show that  $y_2 - 4 y_1 + 4y = 0$ ?

Q 24 | Page 17

If $x = \sin \left( \frac{1}{a}\log y \right)$ show that (1 − x2)y2 − xy1 − a2y = 0 ?

Q 25 | Page 17

If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?

Q 26 | Page 17

If y = tan−1 x, show that $\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0$ ?

Q 27 | Page 17

If $y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2$ show that $\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2$ ?

Q 28 | Page 17

If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?

Q 29 | Page 17

If y = cot x show that $\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0$ ?

Q 30 | Page 17

Find $\frac{d^2 y}{d x^2}$ where $y = \log \left( \frac{x^2}{e^2} \right)$ ?

Q 31 | Page 17

If y = ae2x + be−x, show that, $\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0$ ?

Q 32 | Page 17

If y = ex (sin + cos x) prove that $\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0$ ?

Q 33 | Page 17

If y = cos−1 x, find $\frac{d^2 y}{d x^2}$ in terms of y alone ?

Q 34 | Page 17

If  $y = e^{a \cos^{- 1}} x$ ,prove that $\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0$ ?

Q 35 | Page 17

If y = 500 e7x + 600 e−7x, show that $\frac{d^2 y}{d x^2} = 49y$ ?

Q 36 | Page 17

If x = 2 cos t − cos 2ty = 2 sin t − sin 2t, find $\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}$ ?

Q 37 | Page 17

If x = 4z2 + 5, y = 6z2 + 7z + 3, find $\frac{d^2 y}{d x^2}$ ?

Q 38 | Page 17

If y log (1 + cos x), prove that $\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0$ ?

Q 39 | Page 17

If y = sin (log x), prove that $x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0$ ?

Q 40 | Page 18

If y = 3 e2x + 2 e3x, prove that  $\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0$ ?

Q 41 | Page 18

If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?

Q 42 | Page 18

If y = cosec−1 xx >1, then show that $x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0$ ?

Q 43 | Page 18

$\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4}$ ?

Q 44 | Page 18

$\text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2}$ ?

Q 45 | Page 18

$\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4}$ ?

Q 46 | Page 18

$\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3}$ ?

Q 47 | Page 18

$\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text { and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2}$ ?

Q 48 | Page 18

If $x = 3 \ cot - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,$ find $\frac{d^2 y}{d x^2}$ ?

Q 49 | Page 18

$\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3}$ ?

Q 50 | Page 18

$\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }$

$\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x$ ?

Q 51 | Page 18

$\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2$ ?

Q 52 | Page 18

$\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0$ Disclaimer: There is a misprint in the question. It must be

$x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0$ instead of 1

$x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0$ ?

Q 53 | Page 18

$\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0$

Disclaimer: There is a misprint in the question,

$\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0$ must be written instead of

$\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0$ ?

#### Page 22

Q 1 | Page 22

If y = a xn + 1 + bxn and $x^2 \frac{d^2 y}{d x^2} = \lambda y$  then write the value of λ ?

Q 2 | Page 22

If x = a cos nt − b sin nt and $\frac{d^2 x}{dt} = \lambda x$  then find the value of λ ?

Q 3 | Page 22

If x = t2 and y = t3, find $\frac{d^2 y}{d x^2}$ ?

Q 4 | Page 22

If x = 2aty = at2, where a is a constant, then find $\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}$ ?

Q 5 | Page 22

If x = f(t) and y = g(t), then write the value of $\frac{d^2 y}{d x^2}$ ?

Q 6 | Page 22

If $y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}$ .....to ∞, then write  $\frac{d^2 y}{d x^2}$ in terms of y ?

Q 7 | Page 22

If y = x + ex, find $\frac{d^2 x}{d y^2}$ ?

Q 8 | Page 22

If y = |x − x2|, then find $\frac{d^2 y}{d x^2}$ ?

Q 9 | Page 22

If $y = \left| \log_e x \right|$ find$\frac{d^2 y}{d x^2}$ ?

#### Pages 22 - 24

Q 1 | Page 22

If x = a cos nt − b sin nt, then $\frac{d^2 x}{d t^2}$ is

(a) n2 x
(b) −n2 x
(c) −nx
(d) nx

Q 2 | Page 22

If x = at2, y = 2 at, then $\frac{d^2 y}{d x^2} =$

(a) $- \frac{1}{t^2}$

(b) $\frac{1}{2 \ at^3}$

(c) $- \frac{1}{t^3}$

(d) $- \frac{1}{ 2 \ at^3}$

Q 3 | Page 23

If y = axn+1 + bx−n, then $x^2 \frac{d^2 y}{d x^2} =$

(a) n (n − 1)y
(b) n (n + 1)y
(c) ny
(d) n2y

Q 4 | Page 23

$\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =$

(a) 220 (cos 2 x − 220 cos 4 x)
(b) 220 (cos 2 x + 220 cos 4 x)
(c) 220 (sin 2 x + 220 sin 4 x)
(d) 220 (sin 2 x − 220 sin 4 x)

Q 5 | Page 23

If x = t2, y = t3, then $\frac{d^2 y}{d x^2} =$

(a) 3/2
(b) 3/4t
(c) 3/2t
(d) 3t/2

Q 6 | Page 23

If y = a + bx2, a, b arbitrary constants, then

(a) $\frac{d^2 y}{d x^2} = 2xy$

(b) $x\frac{d^2 y}{d x^2} = y_1$

(c) $x\frac{d^2 y}{d x^2} - \frac{dy}{dx} + y = 0$

(d) $x\frac{d^2 y}{d x^2} = 2 xy$

Q 7 | Page 23

If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to

(a) $\frac{n\left( n + 1 \right)}{2}$

(b) $\left\{ \frac{n\left( n + 1 \right)}{2} \right\}^2$

(c) $- \left\{ \frac{n\left( n + 1 \right)}{2} \right\}^2$

(d) none of these

Q 8 | Page 23

If y = a sin mx + b cos mx, then $\frac{d^2 y}{d x^2}$   is equal to

(a) −m2y
(b) m2y
(c) −my
(d) my

Q 9 | Page 23

If $f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}$ then (1 − x)2 '' (x) − xf(x) =

(a) 1
(b) −1
(c) 0
(d) none of these

Q 10 | Page 23

If $y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)$, then $\frac{d^2 y}{d x^2} =$

(a) 2
(b) 1
(c) 0
(d) −1

Q 11 | Page 23

Let f(x) be a polynomial. Then, the second order derivative of f(ex) is

(a) f'' (ex) e2x + f'(ex) ex
(b) f'' (ex) ex + f' (ex)
(c) f'' (ex) e2x + f'' (ex) ex
(d) f'' (ex)

Q 12 | Page 23

If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
(a) 0
(b) y
(c) −y
(d) none of these

Q 13 | Page 23

If x = 2 at, y = at2, where a is a constant, then $\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}$ is

(a) 1/2a
(b) 1
(c) 2a
(d) none of these

Q 14 | Page 23

If x = f(t) and y = g(t), then $\frac{d^2 y}{d x^2}$ is equal to

(a) $\frac{f' g'' - g'f''}{\left( f' \right)^3}$

(b) $\frac{f' g'' - g'f''}{\left( f' \right)^2}$

(c)  $\frac{g''}{f''}$

(d) $\frac{f'' g' - g'' f'}{\left( g' \right)^3}$

Q 15 | Page 24

If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
(a) m2y
(b) my
(c) −m2y
(d) none of these

Q 16 | Page 24

If y = (sin−1 x)2, then (1 − x2)y2 is equal to

(a) xy1 + 2
(b) xy1 − 2
(c) −xy1+2
(d) none of these

Q 17 | Page 24

If y = etan x, then (cos2 x)y2 =
(a) (1 − sin 2xy1
(b) −(1 + sin 2x)y1
(c) (1 + sin 2x)y1
(d) none of these

Q 19 | Page 24

If $y = \frac{ax + b}{x^2 + c}$ then (2xy1 + y)y3 =

(a) 3(xy2 + y1)y2
(b) 3(xy1 + y2)y2
(c) 3(xy2 + y1)y1
(d) none of these

Q 20 | Page 24

If $y = \log_e \left( \frac{x}{a + bx} \right)^x$ then x3 y2 =

(a) (xy1 − y)2
(b) (1 + y)2
(c) $\left( \frac{y - x y_1}{y_1} \right)^2$

(d) none of these

Q 21 | Page 24

If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then$\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =$

(a) f(t) − f''(t)
(b) {f(t) − f'' (t)}2
(c) {f(t) + f''(t)}2
(d) none of these

Q 22 | Page 24

If $y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =$ ?

Q 23 | Page 24

If $\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x$ then the value of ar, 0 < r ≤ n, is equal to

(a) $\frac{n!}{r!}$

(b) $\frac{\left( n - r \right)!}{r!}$

(c) $\frac{n!}{\left( n - r \right)!}$

(d) none of these

Q 24 | Page 24

If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to

(a) −(n − 1)2 y
(b) (n − 1)2y
(c) −n2y
(d) n2y

Q 25 | Page 24

If xy − loge y = 1 satisfies the equation $x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0$

(a) −3
(b) 1
(c) 3
(d) none of these

Q 26 | Page 24

If y2 = ax2 + bx + c, then $y^3 \frac{d^2 y}{d x^2}$ is

(a) a constant
(b) a function of x only
(c) a function of y  only
(d) a function of x and y

#### Page 4

Q 1 | Page 4

Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies ?

Q 2 | Page 4

Find the rate of change of the volume of a sphere with respect to its diameter ?

Q 3 | Page 4

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?

Q 4 | Page 4

Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm ?

Q 5 | Page 4

Find the rate of change of the volume of a cone with respect to the radius of its base ?

Q 6 | Page 4

Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm ?

Q 7 | Page 4

Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 cm?

Q 8 | Page 4

The total cost C (x) associated with the production of x units of an item is given by C (x) = 0.007x3 − 0.003x2 + 15x + 4000. Find the marginal cost when 17 units are produced ?

Q 9 | Page 4

The total revenue received from the sale of x units of a product is given by R (x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7 ?

Q 10 | Page 4

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate ?

#### Pages 19 - 21

Q 1 | Page 19

The side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long?

Q 2 | Page 19

An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long?

Q 3 | Page 19

The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square.

Q 4 | Page 19

The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?

Q 5 | Page 19

The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm.

Q 6 | Page 19

A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

Q 7 | Page 19

The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Q 8 | Page 19

A man 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.

Q 9 | Page 19

A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?

Q 10 | Page 19

A man 160 cm tall, walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/sec. How fast is the length of his shadow increasing when he is 1 m away from the pole?

Q 11 | Page 19

A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light?

Q 12 | Page 20

A ladder 13 m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/sec. How fast is the angle θ between the ladder and the ground is changing when the foot of the ladder is 12 m away from the wall.

Q 13 | Page 20

A particle moves along the curve y = x2 + 2x. At what point(s) on the curve are the x and ycoordinates of the particle changing at the same rate?

Q 14 | Page 20

If y = 7x − x3 and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x = 2?

Q 15 | Page 20

A particle moves along the curve y = x3. Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-coordinate.

Q 16.1 | Page 20

Find an angle θ which increases twice as fast as its cosine ?

Q 16.2 | Page 20

Find an angle θ whose rate of increase twice is twice the rate of decrease of its cosine ?

Q 17 | Page 20

The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?
How far is the foot from the wall when it and the top are moving at the same rate?

Q 18 | Page 20

A balloon in the form of a right circular cone surmounted by a hemisphere, having a diametre equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height h, when h = 9 cm.

Q 19 | Page 20

Water is running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 m below the base.

Q 20 | Page 20

A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases ?

Q 21 | Page 20

The surface area of a spherical bubble is increasing at the rate of 2 cm2/s. When the radius of the bubble is 6 cm, at what rate is the volume of the bubble increasing?

Q 22 | Page 20

The radius of a cylinder is increasing at the rate 2 cm/sec. and its altitude is decreasing at the rate of 3 cm/sec. Find the rate of change of volume when radius is 3 cm and altitude 5 cm.

Q 23 | Page 20

The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.

Q 24 | Page 20

Sand is being poured onto a conical pile at the constant rate of 50 cm3/ minute such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep ?

Q 25 | Page 20

A kite is 120 m high and 130 m of string is out. If the kite is moving away horizontally at the rate of 52 m/sec, find the rate at which the string is being paid out.

Q 26 | Page 20

A particle moves along the curve y = (2/3)x3 + 1. Find the points on the curve at which the y-coordinate is changing twice as fast as the x-coordinate ?

Q 27 | Page 20

Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate ?

Q 28 | Page 20

The volume of a cube is increasing at the rate of 9 cm3/sec. How fast is the surface area increasing when the length of an edge is 10 cm?

Q 29 | Page 20

The volume of a spherical balloon is increasing at the rate of 25 cm3/sec. Find the rate of change of its surface area at the instant when radius is 5 cm ?

Q 30.1 | Page 20

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of the perimeter ?

Q 30.2 | Page 20

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of the area of the rectangle ?

Q 31 | Page 21

A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2 cm.

#### Page 24

Q 1 | Page 24

If a particle moves in a straight line such that the distance travelled in time t is given by s = t3 − 6t2+ 9t + 8. Find the initial velocity of the particle ?

Q 2 | Page 24

The volume of a sphere is increasing at 3 cubic centimeter per second. Find the rate of increase of the radius, when the radius is 2 cms ?

Q 3 | Page 24

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. How far is the area increasing when the side is 10 cms?

Q 4 | Page 24

The side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?

Q 5 | Page 24

The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?

Q 6 | Page 24

The side of an equilateral triangle is increasing at the rate of $\frac{1}{3}$ cm/sec. Find the rate of increase of its perimeter ?

Q 7 | Page 24

Find the surface area of a sphere when its volume is changing at the same rate as its radius ?

Q 8 | Page 24

If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere ?

Q 9 | Page 24

The amount of pollution content added in air in a city due to x diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions ?

Q 10 | Page 24

A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10 cm/sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall ?

#### Pages 24 - 26

Q 1 | Page 24

If $V = \frac{4}{3}\pi r^3$ ,  at what rate in cubic units is V increasing when r = 10 and $\frac{dr}{dt} = 0 . 01$ ?

(a) π
(b) 4π
(c) 40π
(d) 4π/3

Q 2 | Page 24

Side of an equilateral triangle expands at the rate of 2 cm/sec. The rate of increase of its area when each side is 10 cm is

(a) $10\sqrt{2} \ {cm}^2 /\sec$

(b)  $10\sqrt{3} {cm}^2 /\sec$

(c) 10 cm2/sec
(d) 5 cm2/sec

Q 3 | Page 24

The radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm is
(a) 8π cm2/sec
(b) 12π cm2/sec
(c) 160π cm2/sec
(d) 200 cm2/sec

Q 4 | Page 24

A cone whose height is always equal to its diameter is increasing in volume at the rate of 40 cm3/sec. At what rate is the radius increasing when its circular base area is 1 m2?
(a) 1 mm/sec
(b) 0.001 cm/sec
(c) 2 mm/sec
(d) 0.002 cm/sec

Q 5 | Page 24

A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π m3/minute. The rate at which the surface of the oil is rising, is
(a) 1 m/minute
(b) 2 m/minute
(c) 5 m/minute
(d) 1.25 m/minute

Q 6 | Page 25

The distance moved by the particle in time t is given by x = t3 − 12t2 + 6t + 8. At the instant when its acceleration is zero, the velocity is
(a) 42
(b) −42
(c) 48
(d) −48

Q 7 | Page 25

The altitude of a cone is 20 cm and its semi-vertical angle is 30°. If the semi-vertical angle is increasing at the rate of 2° per second, then the radius of the base is increasing at the rate of
(a) 30 cm/sec

(b) $\frac{160}{3} cm/\sec$

(c) 10 cm/sec
(d) 160 cm/sec

Q 8 | Page 25

For what values of x is the rate of increase of x3 − 5x2 + 5x + 8 is twice the rate of increase of x ?

(a) $- 3, - \frac{1}{3}$

(b) $- 3, \frac{1}{3}$

(c) $3, - \frac{1}{3}$

(d) $3, \frac{1}{3}$

Q 9 | Page 25

The coordinates of the point on the ellipse 16x2 + 9y2 = 400 where the ordinate decreases at the same rate at which the abscissa increases, are
(a) (3, 16/3)
(b) (−3, 16/3)
(c) (3, −16/3)
(d) (3, −3)

Q 10 | Page 25

The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. The rate of change of lateral surface when the radius = 7 cm and altitude 24 cm is
(a) 54π cm2/min
(b) 7π cm2/min
(c) 27 cm2/min
(d) none of these

Q 11 | Page 25

The radius of a sphere is increasing at the rate of 0.2 cm/sec. The rate at which the volume of the sphere increase when radius is 15 cm, is
(a) 12π cm3/sec
(b) 180π cm3/sec
(c) 225π cm3/sec
(d) 3π cm3/sec

Q 12 | Page 25

The volume of a sphere is increasing at 3 cm3/sec. The rate at which the radius increases when radius is 2 cm, is

(a) $\frac{3}{32\pi}cm/\sec$

(b) $\frac{3}{16\pi}cm/\sec$

(c) $\frac{3}{48\pi}cm/\sec$

(d) $\frac{1}{24\pi}cm/\sec$

Q 13 | Page 25

The distance moved by a particle travelling in straight line in t seconds is given by s = 45t + 11t2 − t3. The time taken by the particle to come to rest is
(a) 9 sec
(b) 5/3 sec
(c) 3/5 sec
(d) 2 sec

Q 14 | Page 25

The volume of a sphere is increasing at the rate of 4π cm3/sec. The rate of increase of the radius when the volume is 288 π cm3, is
(a) 1/4
(b) 1/12
(c) 1/36
(d) 1/9

Q 15 | Page 25

If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to

(a) 1 unit

(b) $\sqrt{2\pi} \text { units }$

(c) $\frac{1}{\sqrt{2\pi}} \text { unit }$

(d)  $\frac{1}{2\sqrt{\pi}} \text { unit}$

Q 16 | Page 25

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to

(a) $\frac{2}{\pi} \text { unit }$

(b) $\frac{1}{\pi} \text { unit }$

(c) $\frac{\pi}{2} \text { units }$

(d) π units

Q 17 | Page 25

Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is

(a) $8\sqrt{3} \ {cm}^2 /hr$

(b) $4\sqrt{3} \ {cm}^2 /hr$

(c) $\frac{\sqrt{3}}{8} \ {cm}^2 /hr$

(d) none of these

Q 18 | Page 25

If s = t3 − 4t2 + 5 describes the motion of a particle, then its velocity when the acceleration vanishes, is

(a) $\frac{16}{9} \text { unit }/\sec$

(b) $- \frac{32}{3} \text { unit }/\sec$

(c) $\frac{4}{3} \text { unit }/\sec$

(d) $- \frac{16}{3} \text { unit }/\sec$

Q 19 | Page 25

The equation of motion of a particle is s = 2t2 + sin 2t, where s is in metres and is in seconds. The velocity of the particle when its acceleration is 2 m/sec2, is

(a) $\pi + \sqrt{3} m\ /\sec$

(b) $\frac{\pi}{3} + \sqrt{3} m/\sec$

(c) $\frac{2\pi}{3} + \sqrt{3} m/\sec$

(d) $\frac{\pi}{3} + \frac{1}{\sqrt{3}} m/\sec$

Q 20 | Page 26

The radius of a circular plate is increasing at the rate of 0.01 cm/sec. The rate of increase of its area when the radius is 12 cm, is
(a) 144 π cm2/sec
(b) 2.4 π cm2/sec
(c) 0.24 π cm2/sec
(d) 0.024 π cm2/sec

Q 21 | Page 26

The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is π, the rate of increase of its area is
(a) π cm2/sec
(b) 2π cm2/sec
(c) π2 cm2/sec
(d) 2π2 cm2/sec2

Q 22 | Page 26

A man 2 metres tall walks away from a lamp post 5 metres height at the rate of 4.8 km/hr. The rate of increase of the length of his shadow is
(a) 1.6 km/hr
(b) 6.3 km/hr
(c) 5 km/hr
(d) 3.2 km/hr

Q 23 | Page 26

A man of height 6 ft walks at a uniform speed of 9 ft/sec from a lamp fixed at 15 ft height. The length of his shadow is increasing at the rate of
(a) 15 ft/sec
(b) 9 ft/sec
(c) 6 ft/sec
(d) none of these

Q 24 | Page 26

In a sphere the rate of change of volume is
(a) π times the rate of change of radius
(b) surface area times the rate of change of diameter
(c) surface area times the rate of change of radius
(d) none of these

Q 25 | Page 26

In a sphere the rate of change of surface area is
(a) 8π times the rate of change of diameter
(b) 2π times the rate of change of diameter
(c) 2π times the rate of change of radius
(d) 8π times the rate of change of radius

Q 26 | Page 26

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
(a) 1 m/hr
(b) 0.1 m/hr
(c) 1.1 m/hr
(d) 0.5 m/hr

#### Pages 9 - 10

Q 1 | Page 9

If y = sin x and x changes from π/2 to 22/14, what is the approximate change in y ?

Q 2 | Page 9

The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume ?

Q 3 | Page 9

A circular metal plate expends under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

Q 4 | Page 9

Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube ?

Q 5 | Page 9

If there is an error of 0.1% in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere ?

Q 6 | Page 9

The pressure p and the volume v of a gas are connected by the relation pv1.4 = const. Find the percentage error in p corresponding to a decrease of 1/2% in v ?

Q 7 | Page 9

The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase (i) in total surface area, and (ii) in the volume, assuming that k is small ?

Q 8 | Page 9

Show that the relative error in computing the volume of a sphere, due to an error in measuring the radius, is approximately equal to three times the relative error in the radius ?

Q 9.01 | Page 9

1 Using differential, find the approximate value of the following:

$\sqrt{25 . 02}$

Q 9.02 | Page 9

Using differential, find the approximate value of the following:  $\left( 0 . 009 \right)^\frac{1}{3}$

Q 9.03 | Page 9

Using differential, find the approximate value of the following: $\left( 0 . 007 \right)^\frac{1}{3}$

Q 9.04 | Page 9

Using differential, find the approximate value of the $\sqrt{401}$ ?

Q 9.05 | Page 9

Using differential, find the approximate value of the $\left( 15 \right)^\frac{1}{4}$ ?

Q 9.06 | Page 9

Using differential, find the approximate value of the $\left( 255 \right)^\frac{1}{4}$ ?

Q 9.07 | Page 9

Using differential, find the approximate value of the $\frac{1}{(2 . 002 )^2}$ ?

Q 9.08 | Page 9

Using differential, find the approximate value of the loge 4.04, it being given that log104 = 0.6021 and log10e = 0.4343 ?

Q 9.09 | Page 9

Using differential, find the approximate value of the loge 10.02, it being given that loge10 = 2.3026 ?

Q 9.1 | Page 9

Using differential, find the approximate value of the  log10 10.1, it being given that log10e = 0.4343 ?

Q 9.11 | Page 9

Using differentials, find the approximate values of the cos 61°, it being given that sin60° = 0.86603 and 1° = 0.01745 radian ?

Q 9.12 | Page 9

Using differential, find the approximate value of the $\frac{1}{\sqrt{25 . 1}}$ ?

Q 9.13 | Page 9

Using differential, find the approximate value of the $\sin\left( \frac{22}{14} \right)$ ?

Q 9.14 | Page 9

Using differential, find the approximate value of the $\cos\left( \frac{11\pi}{36} \right)$ ?

Q 9.15 | Page 9

Using differential, find the approximate value of the $\left( 80 \right)^\frac{1}{4}$ ?

Q 9.16 | Page 9

Using differential, find the approximate value of the $\left( 29 \right)^\frac{1}{3}$ ?

Q 9.17 | Page 9

Using differential, find the approximate value of the $\left( 66 \right)^\frac{1}{3}$ ?

Q 9.18 | Page 9

Using differential, find the approximate value of the $\sqrt{26}$ ?

Q 9.19 | Page 9

Using differential, find the approximate value of the  $\sqrt{37}$ ?

Q 9.2 | Page 9

Using differential, find the approximate value of the  $\sqrt{0 . 48}$ ?

Q 9.21 | Page 9

Using differential, find the approximate value of the $\left( 82 \right)^\frac{1}{4}$ ?

Q 9.22 | Page 9

Using differential, find the approximate value of the $\left( \frac{17}{81} \right)^\frac{1}{4}$ ?

Q 9.23 | Page 9

Using differential, find the approximate value of the $\left( 33 \right)^\frac{1}{5}$ ?

Q 9.24 | Page 9

Using differential, find the approximate value of the $\sqrt{36 . 6}$ ?

Q 9.25 | Page 9

Using differential, find the approximate value of the ${25}^\frac{1}{3}$ ?

Q 9.26 | Page 9

Using differential, find the approximate value of the $\sqrt{49 . 5}$ ?

Q 9.27 | Page 9

Using differential, find the approximate value of the $\left( 3 . 968 \right)^\frac{3}{2}$ ?

Q 9.28 | Page 9

Using differential, find the approximate value of the $\left( 1 . 999 \right)^5$ ?

Q 9.29 | Page 9

Using differential, find the approximate value of the  $\sqrt{0 . 082}$ ?

Q 10 | Page 10

Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2 ?

Q 11 | Page 10

Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15 ?

Q 12 | Page 10

Find the approximate value of log10 1005, given that log10 e = 0.4343 ?

Q 13 | Page 10

If the radius of a sphere is measured as 9 cm with an error of 0.03 m, find the approximate error in calculating its surface area ?

Q 14 | Page 10

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1% ?

Q 15 | Page 10

If the radius of a sphere is measured as 7 m with an error of 0.02 m, find the approximate error in calculating its volume ?

Q 16 | Page 10

Find the approximate change in the value V of a cube of side x metres caused by increasing the side by 1% ?

#### Page 12

Q 1 | Page 12

For the function y = x2, if x = 10 and ∆x = 0.1. Find ∆ y ?

Q 2 | Page 12

If y = loge x, then find ∆y when x = 3 and ∆x = 0.03 ?

Q 3 | Page 12

If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area ?

Q 4 | Page 12

If the percentage error in the radius of a sphere is α, find the percentage error in its volume ?

Q 5 | Page 12

A piece of ice is in the form of a cube melts so that the percentage error in the edge of cube is a, then find the percentage error in its volume ?

#### Page 13

Q 1 | Page 13

If there is an error of 2% in measuring the length of a simple pendulum, then percentage error in its period is
(a)1%
(b) 2%
(c) 3%
(d) 4%

Q 2 | Page 13

If there is an error of a% in measuring the edge of a cube, then percentage error in its surface is
(a) 2a%

(b)$\frac{a}{2} \%$

(c) 3a%

(d) none of these

Q 3 | Page 13

If an error of k% is made in measuring the radius of a sphere, then percentage error in its volume is
(a) k%
(b) 3k%
(c) 2k%
(d) k/3%

Q 4 | Page 13

The height of a cylinder is equal to the radius. If an error of α % is made in the height, then percentage error in its volume is
(a) α %
(b) 2α %
(c) 3α %
(d) none of these

Q 5 | Page 13

While measuring the side of an equilateral triangle an error of k % is made, the percentage error in its area is
(a) k %
(b) 2k %
(c) $\frac{k}{2}\%$

(d) 3k %

Q 6 | Page 13

If loge 4 = 1.3868, then loge 4.01 =
(a) 1.3968
(b) 1.3898
(c) 1.3893
(d) none of these

Q 7 | Page 13

A sphere of radius 100 mm shrinks to radius 98 mm, then the approximate decrease in its volume is
(a) 12000 π mm3
(b) 800 π mm3
(c) 80000 π mm3
(d) 120 π mm3

Q 8 | Page 13

If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius is λ %, then the error in its volume is
(a) λ %
(b) 2 λ %
(c) 3 λ %
(d) none of these

Q 9 | Page 13

The pressure P and volume V of a gas are connected by the relation PV1/4 = constant. The percentage increase in the pressure corresponding to a deminition of 1/2 % in the volume is

(a) $\frac{1}{2} \%$

(b) $\frac{1}{4} \%$

(c)  $\frac{1}{8} \%$

(d) none of these

Q 10 | Page 13

If y = xn, then the ratio of relative errors in y and x is
(a) 1 : 1
(b) 2 : 1
(c) 1 : n
(d) n : 1

Q 11 | Page 13

The approximate value of (33)1/5 is
(a) 2.0125
(b) 2.1
(c) 2.01
(d) none of these

Q 12 | Page 13

The circumference of a circle is measured as 28 cm with an error of 0.01 cm. The percentage error in the area is

(a) $\frac{1}{14}$

(b) 0.01

(c) $\frac{1}{7}$

(d) none of these

#### Pages 8 - 9

Q 1.1 | Page 8

f(x) = 3 + (x − 2)2/3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Q 1.2 | Page 8

f (x) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Q 1.3 | Page 8

f (x) = sin $\frac{1}{x}$ for −1 ≤ x ≤ 1 Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Q 1.4 | Page 8

f (x) = 2x2 − 5x + 3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Q 1.5 | Page 8

f (x) = x2/3 on [−1, 1] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Q 1.6 | Page 8

$f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}$ Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Q 2.1 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 8x + 12 on [2, 6] ?

Q 2.2 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 4x + 3 on [1, 3] ?

Q 2.3 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = (x − 1) (x − 2)2 on [1, 2] ?

Q 2.4 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f (x) = x(x − 1)2 on [0, 1] ?

Q 2.5 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f (x) = (x2 − 1) (x − 2) on [−1, 2] ?

Q 2.6 | Page 9

Verify Rolle's theorem for the following function on the indicated interval   f (x) = x(x − 4)2 on the interval [0, 4] ?

Q 2.7 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f(x) = x(x −2)2 on the interval [0, 2] ?

Q 2.8 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = x2 + 5x + 6 on the interval [−3, −2]  ?

Q 3.01 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = cos 2 (x − π/4) on [0, π/2] ?

Q 3.02 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f(x) = sin 2x on [0, π/2] ?

Q 3.03 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [−π/4, π/4] ?

Q 3.04 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = ex sin x on [0, π] ?

Q 3.05 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = ecos x on [−π/2, π/2] ?

Q 3.06 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f(x) = cos 2x on [0, π] ?

Q 3.07 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = $\frac{\sin x}{e^x}$ on 0 ≤ x ≤ π ?

Q 3.08 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 3x on [0, π] ?

Q 3.09 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = ${e^{1 - x}}^2$ on [−1, 1] ?

Q 3.1 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x2 + 2) − log 3 on [−1, 1] ?

Q 3.11 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?

Q 3.12 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = 2 sin x + sin 2x on [0, π] ?

Q 3.13 | Page 9

Verify Rolle's theorem for the following function on the indicated interval $f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]$?

Q 3.14 | Page 9

Verify Rolle's theorem for the following function on the indicated interval $f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]$ ?

Q 3.15 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?

Q 3.16 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 5x + 4 on [1, 4] ?

Q 3.17 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on $\left[ 0, \frac{\pi}{2} \right]$ ?

Q 3.18 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f(x) = sin x − sin 2x on [0, π] ?

Q 7 | Page 9

Using Rolle's theorem, find points on the curve y = 16 − x2x ∈ [−1, 1], where tangent is parallel to x-axis.

Q 8.1 | Page 9

At what point  on the following curve, is the tangent parallel to x-axis y = x2 on [−2, 2]
?

Q 8.2 | Page 9

At what point  on the following curve, is the tangent parallel to x-axis y = $e^{1 - x^2}$ on [−1, 1] ?

Q 8.3 | Page 9

At what point  on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?

Q 9 | Page 9

If f : [−5, 5] → is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?

Q 10 | Page 9

Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?

Q 11 | Page 9

It is given that the Rolle's theorem holds for the function f(x) = x3 + bx2 + cx, x  $\in$ at the point x = $\frac{4}{3}$ , Find the values of b and c ?

#### Pages 17 - 18

Q 1.01 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x2 − 1 on [2, 3] ?

Q 1.02 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = x3 − 2x2 − x + 3 on [0, 1] ?

Q 1.03 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x(x −1) on [1, 2] ?

Q 1.04 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = x2 − 3x + 2 on [−1, 2] ?

Q 1.05 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = 2x2 − 3x + 1 on [1, 3] ?

Q 1.06 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = x2 − 2x + 4 on [1, 5] ?

Q 1.07 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = 2x − x2 on [0, 1] ?

Q 1.08 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore f(x) = (x − 1)(x − 2)(x − 3) on [0, 4] ?

Q 1.09 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore $f\left( x \right) = \sqrt{25 - x^2}$ on [−3, 4] ?

Q 1.1 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore  f(x) = tan1 x on [0, 1] ?

Q 1.11 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem $f\left( x \right) = x + \frac{1}{x} \text { on }[1, 3]$ ?

Q 1.12 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x(x + 4)2 on [0, 4] ?

Q 1.13 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem $f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]$ ?

Q 1.14 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = x2 + x − 1 on [0, 4] ?

Q 1.15 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = sin x − sin 2x − x on [0, π] ?

Q 1.16 | Page 17

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x3 − 5x2 − 3x on [1, 3] ?

Q 2 | Page 18

Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?

Q 3 | Page 18

Show that the lagrange's mean value theorem is not applicable to the function
f(x) = $\frac{1}{x}$ on [−1, 1] ?

Q 4 | Page 18

Verify the  hypothesis and conclusion of Lagrange's man value theorem for the function
f(x) = $\frac{1}{4x - 1},$ 1≤ x ≤ 4 ?

Q 5 | Page 18

Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?

Q 6 | Page 18

Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?

Q 7 | Page 18

Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?

Q 8 | Page 18

Find the points on the curve y = x3 − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?

Q 9 | Page 18

Find a point on the curve y = x3 + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?

Q 10 | Page 18

Let C be a curve defined parametrically as $x = a \cos^3 \theta, y = a \sin^3 \theta, 0 \leq \theta \leq \frac{\pi}{2}$ . Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).

Q 11 | Page 18

Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < $\frac{\pi}{2}$ ?

#### Page 19

Q 1 | Page 19

If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?

Q 2 | Page 19

State Rolle's theorem ?

Q 3 | Page 19

State Lagrange's mean value theorem ?

Q 4 | Page 19

If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)n on the interval $[0, 2\sqrt{3}] \text { is } \frac{3}{4},$ write the value of n (a positive integer) ?

Q 5 | Page 19

Find the value of c prescribed by Lagrange's mean value theorem for the function $f\left( x \right) = \sqrt{x^2 - 4}$ defined on [2, 3] ?

#### Pages 19 - 20

Q 1 | Page 19

If the polynomial equation $a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0$ n positive integer, has two different real roots α and β, then between α and β, the equation $n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }$.

(a) exactly one root
(b) almost one root
(c) at least one root
(d) no root

Q 2 | Page 19

If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
(a) (0, 1)
(b) (1, 2)
(c) (0, 2)
(d) none of these

Q 3 | Page 19

For the function f (x) = x + $\frac{1}{x}$ ∈ [1, 3], the value of c for the Lagrange's mean value theorem is

(a) 1
(b) $\sqrt{3}$

(c) 2
(d) none of these

Q 4 | Page 19

If from Lagrange's mean value theorem, we have $f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }$

(a) a < x1 ≤ b
(b) a ≤ x1 < b
(c) a < x1 < b
(d) a ≤ x1 ≤ b

Q 5 | Page 19

Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
(a) any interval
(b) the interval [0, π]
(c) the interval (0, π/2)
(d) none of these

Q 6 | Page 20

The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is

(a) 2

(b)$- \frac{1}{3}$

(c) −2

(d) $\frac{2}{3}$

Q 7 | Page 20

When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (ee), the value of x is

(a) e1/1−e

(b) e(e−1)(2e−1)

(c) $e^\frac{2e - 1}{e - 1}$

(d) $\frac{e - 1}{e}$

Q 8 | Page 20

The value of c in Rolle's theorem for the function $f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}$ defined on [−1, 0] is
(a) 0.5
(b) $\frac{1 + \sqrt{5}}{2}$

(c)$\frac{1 - \sqrt{5}}{2}$

(d) −0.5

Q 9 | Page 20

The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
(a) 1
(b) 1/2
(c) 2/3
(d) 3/2

Q 10 | Page 20

$c = \frac{3}{2} \in \left( 1, 2 \right)$The value of c in Rolle's theorem for the function f (x) = x3 − 3x in the interval [0,$\sqrt{3}$] is

(a) 1
(b) −1
(c) 3/2
(d) 1/3

Q 11 | Page 20

If f (x) = ex sin x in [0, π], then c in Rolle's theorem is

(a) π/6

(b) π/4

(c) π/2

(d) 3π/4

#### Pages 10 - 11

Q 1.01 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point $y = \sqrt{x^3} \text { at } x = 4$ ?

Q 1.02 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point $y = \sqrt{x} \text { at }x = 9$ ?

Q 1.03 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  y = x3 − x at x = 2 ?

Q 1.04 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 0 ?

Q 1.05 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point y =  x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?

Q 1.06 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  x = a cos3 θ, y = a sin3 θ at θ = π/4 ?

Q 1.07 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?

Q 1.08 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  y = (sin 2x + cot x + 2)2 at x = π/2 ?

Q 1.09 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  x2 + 3y + y2 = 5 at (1, 1)  ?

Q 1.1 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  xy = 6 at (1, 6) ?

Q 2 | Page 10

Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?

Q 3 | Page 10

If the tangent to the curve y = x3 + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?

Q 4 | Page 10

Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?

Q 5 | Page 10

Find the points on the curve y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?

Q 6 | Page 10

Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?

Q 7 | Page 10

Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?

Q 8 | Page 10

Find the point on the curvey = x2 where the slope of the tangent is equal to the x-coordinate of the point ?

Q 9 | Page 10

At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?

Q 10 | Page 10

At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?

Q 11 | Page 10

Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?

Q 12 | Page 10

At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?

Q 13 | Page 10

Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is $- \frac{1}{6}$  ?

Q 14 | Page 11

Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?

Q 15 | Page 11

Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?

Q 16 | Page 11

At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?

Q 17.1 | Page 11

Find the points on the curve $\frac{x^2}{4} + \frac{y^2}{25} = 1$ at which the tangent is parallel to the x-axis ?

Q 17.2 | Page 11

Find the points on the curve$\frac{x^2}{4} + \frac{y^2}{25} = 1$ at which the tangent is  parallel to the y-axis ?

Q 18 | Page 11

Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

Q 19.1 | Page 11

Find the points on the curve $\frac{x^2}{9} + \frac{y^2}{16} = 1$ at which the tangent is  parallel to x-axis ?

Q 19.2 | Page 11

Find the points on the curve $\frac{x^2}{9} + \frac{y^2}{16} = 1$ at which the tangent is  parallel to y-axis ?

Q 20 | Page 11

Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?

Q 21 | Page 11

Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?

#### Pages 27 - 29

Q 1 | Page 27

Find the equation of the tangent to the curve $\sqrt{x} + \sqrt{y} = a$ at the point $\left( \frac{a^2}{4}, \frac{a^2}{4} \right)$ ?

Q 2 | Page 27

Find the equation of the normal to y = 2x3 − x2 + 3 at (1, 4) ?

Q 3.01 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point x4 − bx3 + 13x2 − 10x + 5 at (0, 5)  ?

Q 3.02 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1 ?

Q 3.03 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  y = x2 at (0, 0) ?

Q 3.04 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y = 2x2 − 3x − 1 at (1, −2) ?

Q 3.05 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)$ ?

Q 3.06 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3  ?

Q 3.07 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 at \left( a\cos\theta, b\sin\theta \right)$ ?

Q 3.08 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text {at} \left( a\sec\theta, b\tan\theta \right)$ ?

Q 3.09 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at $\left( \frac{a}{m^2}, \frac{2a}{m} \right)$ ?

Q 3.1 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)$ ?

Q 3.11 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at $\left( ct, \frac{c}{t} \right)$ ?

Q 3.12 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)$ ?

Q 3.13 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)$ ?

Q 3.14 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  $x^\frac{2}{3} + y^\frac{2}{3}$ = 2 at (1, 1) ?

Q 3.15 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  x2 = 4y at (2, 1) ?

Q 3.16 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  y2 = 4x at (1, 2)  ?

Q 3.17 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?

Q 3.18 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  y2 = 4ax at (x1, y1)?

Q 3.19 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)$ ?

Q 4 | Page 27

Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?

Q 5.1 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = $\frac{\pi}{2}$ ?

Q 5.2 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points $x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}$ ?

Q 5.3 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?

Q 5.4 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points  x = asect, y = btant at t ?

Q 5.5 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?

Q 5.6 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points  x = 3cosθ − cos3θ, y = 3sinθ − sin3θ

Q 6 | Page 28

Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?

Q 7 | Page 28

Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?

Q 8 | Page 28

The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?

Q 9 | Page 28

Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?

Q 10 | Page 28

Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?

Q 11 | Page 28

Determine the equation(s) of tangent (s) line to the curve y = 4x3 − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?

Q 12 | Page 28

Find the equation of a normal to the curvey = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?

Q 13.1 | Page 28

Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?

Q 13.2 | Page 28

Find the equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?

Q 14 | Page 28

Find the equations of all lines having slope 2 and that are tangent to the curve $y = \frac{1}{x - 3}, x \neq 3$ ?

Q 15 | Page 28

Find the equations of all lines of slope zero and that are tangent to the curve $y = \frac{1}{x^2 - 2x + 3}$ ?

Q 16 | Page 28

Find the equation of the tangent to the curve  $y = \sqrt{3x - 2}$ which is parallel to the 4x − 2y + 5 = 0 ?

Q 17 | Page 28

Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?

Q 18 | Page 29

Prove that $\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2$ touches the straight line $\frac{x}{a} + \frac{y}{b} = 2$ for all n ∈ N, at the point (a, b) ?

Q 19 | Page 29

Find the equation of the tangent to the curve x = sin 3ty = cos 2t at

$t = \frac{\pi}{4}$ ?

Q 20 | Page 29

At what points will be tangents to the curvey = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?

Q 21 | Page 29

Find the equation of  the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?

#### Pages 40 - 41

Q 1.1 | Page 40

Find the angle of intersection of the following curve y2 = x and x2 = y  ?

Q 1.2 | Page 40

Find the angle of intersection of the following curve  y = x2 and x2 + y2 = 20  ?

Q 1.3 | Page 40

Find the angle of intersection of the following curve  2y2 = x3 and y2 = 32x ?

Q 1.4 | Page 40

Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?

Q 1.5 | Page 40

Find the angle of intersection of the following curve $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and x2 + y2 = ab ?

Q 1.6 | Page 40

Find the angle of intersection of the following curve  x2 + 4y2 = 8 and x2 − 2y2 = 2 ?

Q 1.7 | Page 40

Find the angle of intersection of the following curve  x2 = 27y and y2 = 8x ?

Q 1.8 | Page 40

Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?

Q 1.9 | Page 40

Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?

Q 2.1 | Page 40

Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x?

Q 2.2 | Page 40

Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?

Q 2.3 | Page 40

Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?

Q 3.1 | Page 40

Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?

Q 3.2 | Page 40

Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?

Q 3.3 | Page 40

Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 +  y2 = 10 at  $\left( 1, 2\sqrt{2} \right)$ ?

Q 4 | Page 40

Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?

Q 5 | Page 40

Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?

Q 6 | Page 40

Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?

Q 7 | Page 40

Prove that the curves y2 = 4x and x2 + y2

$-$ 6x + 1 = 0 touch each other at the point (1, 2) ?
Q 8.1 | Page 41

Find the condition for the following set of curve to intersect orthogonally $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2$ ?

Q 8.2 | Page 41

Find the condition for the following set of curve to intersect orthogonally $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ ?

Q 9 | Page 41

Show that the curves $\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1$ intersect at right angles ?

Q 10 | Page 41

If the straight line xcos $\alpha$ +y sin $\alpha$ = p touches the curve  $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ then prove that a2cos2 $\alpha$ $-$ b2sin$\alpha$ = p?

#### Pages 41 - 42

Q 1 | Page 41

Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?

Q 2 | Page 41

Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?

Q 3 | Page 41

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of $\frac{dy}{dx}$ ?

Q 4 | Page 41

Write the value of $\frac{dy}{dx}$ , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?

Q 5 | Page 41

If the tangent to a curve at a point (xy) is equally inclined to the coordinates axes then write the value of $\frac{dy}{dx}$ ?

Q 6 | Page 41

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of $\frac{dx}{dy}$ ?

Q 7 | Page 41

Find the slope of the normal at the point 't' on the curve $x = \frac{1}{t}, y = t$ ?

Q 8 | Page 41

Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle $\frac{\pi}{4}$ with x-axis  ?

Q 9 | Page 41

Write the angle made by the tangent to the curve x = et cos t, y = et sin t at $t = \frac{\pi}{4}$ with the x-axis ?

Q 10 | Page 42

Write the equation of the normal to the curve y = x + sin x cos x at $x = \frac{\pi}{2}$ ?

Q 11 | Page 42

Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?

Q 12 | Page 42

Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?

Q 13 | Page 42

Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?

Q 14 | Page 42

Write the angle between the curves y = e−x and y = ex at their point of intersections ?

Q 15 | Page 42

Write the slope of the normal to the curve $y = \frac{1}{x}$  at the point $\left( 3, \frac{1}{3} \right)$ ?

Q 16 | Page 42

Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?

Q 17 | Page 42

Write the equation of the normal to the curve y = cos x at (0, 1) ?

Q 18 | Page 42

Write the equation of the tangent drawn to the curve $y = \sin x$ at the point (0,0) ?

#### Pages 42 - 44

Q 1 | Page 42

The equation to the normal to the curve y = sin x at (0, 0) is

(a) x = 0
(b) y = 0
(c) x + y = 0
(d) x − y = 0

Q 2 | Page 42

The equation of the normal to the curve y = x + sin x cos x at x = π/2 is
(a) = 2
(b) x = π
(c) x + π = 0
(d) 2x = π

Q 3 | Page 42

The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is
(a) x − 2y = 2
(b) x − 2y + 2 = 0
(c) 2x +  y = 4
(d) 2x + y − 4 = 0

Q 4 | Page 42

The point on the curve y2 = x where tangent makes 45° angle with x-axis is
(a) (1/2, 1/4)
(b) (1/4, 1/2)
(c) (4, 2)
(d) (1, 1)

Q 5 | Page 42

If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is
(a) (a, a)
(b) (0, a)
(c) (0, 0)
(d) (a, 0)

Q 6 | Page 42

The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is
(a) (0, 2)
(b) (1, 0)
(c) (−1, 6)
(d) (2, −2)

Q 7 | Page 42

The point on the curve y2 = x where tangent makes 45° angle with x-axis is
(a) (1/2, 1/4)
(b) (1/4, 1/2)
(c) (4, 2)(d) (1, 1)

Q 8 | Page 42

The point at the curve y = 12x − x2 where the slope of the tangent is zero will be
(a) (0, 0)
(b) (2, 16)
(c) (3, 9)
(d) none of these

Q 9 | Page 42

The angle between the curves y2 = x and x2 = y at (1, 1) is
(a) $\tan^{- 1} \frac{4}{3}$

(b)$\tan^{- 1} \frac{3}{4}$

(c) 90°
(d) 45°

Q 10 | Page 43

The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is
(a) x + 3y = 8
(b) x + 3y + 8 = 0
(c) x + 3y ± 8 = 0
(d) x + 3y = 0

Q 11 | Page 43

The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are
(a) x − y + 2 = 0 = x − y − 1
(b) x + y − 1 = 0 = x − y − 2
(c) x − y − 1 = 0 = x − y
(d) x − y = 0 = x + y

Q 12 | Page 43

The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is
(a) 22/7
(b) 6/7
(c) −6
(d) none of these

Q 13 | Page 43

At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
(a) (3, 0), (−1, 0)
(b) (3, 0), (1, 2)
(c) (−1, 0), (1, 2)
(d) (1, 2), (1, −2)

Q 14 | Page 43

The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is
(a) 0°
(b) 45°
(c) 90°
(d) none of these

Q 15 | Page 43

If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to
(a) 1
(b) −6
(c) 6
(d) 0

Q 16 | Page 43

If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then
(a) b = 1, c = 2
(b) b = −1, c = 1
(c) b = 2, c = 1
(d) b = −2, c = 1

Q 17 | Page 43

The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is
(a) 1/2
(b) 0
(c) −2
(d) ∞

Q 18 | Page 43

The curves y = aex and y = be−x cut orthogonally, if
(a) a = b
(b) a = −b
(c) ab = 1
(d) ab = 2

Q 19 | Page 43

The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is
(a) x = 0
(b) y = 0
(c) c = y
(d) x + y = a

Q 20 | Page 43

If the curves y = 2 ex and y = ae−x intersect orthogonally, then a =
(a) 1/2
(b) −1/2
(c) 2
(d) 2e2

Q 21 | Page 43

The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is
(a) (−3, −27)
(b) (3, 9)
(c) (7/2, 35/4)
(d) (0, 0)

Q 22 | Page 43

The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is
(a) π/6
(b) π/3
(c) π/2
(d) π/4

Q 23 | Page 43

The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at $x = \frac{\pi}{6}$ is

(a) π/4
(b) π/2
(c) π/3
(d) none of these

Q 24 | Page 43

Any tangent to the curve y = 2x7 + 3x + 5
(a) is parallel to x-axis
(b) is parallel to y-axis
(c) makes an acute angle with x-axis
(d) makes an obtuse angle with x-axis

Q 25 | Page 43

The point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes is

(a) $\left( 4, \frac{8}{3} \right)$

(b) $\left( - 4, \frac{8}{3} \right)$

(c) $\left( 4, - \frac{8}{3} \right)$

(d) none of these

Q 26 | Page 43

The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is

(a) $\frac{22}{7}$

(b) $\frac{6}{7}$

(c) $\frac{7}{6}$

(d) $- \frac{6}{7}$

Q 27 | Page 43

The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is
(a) 1
(b) 2
(c) 3
(d)$\frac{1}{2}$

Q 28 | Page 44

The normal at the point (1, 1) on the curve 2y + x2 = 3 is
(a) x + y = 0
(b) x − y = 0
(c) x + y + 1 = 0
(d) x − y = 1

Q 29 | Page 44

The normal to the curve x2 = 4y passing through (1, 2) is
(a) x + y = 3
(b) x − y = 3
(c) x + y = 1
(d) x − y = 1

#### Page 10

Q 1 | Page 10

Prove that the function f(x) = loge x is increasing on (0, ∞) ?

Q 2 | Page 10

Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?

Q 3 | Page 10

Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?

Q 4 | Page 10

Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?

Q 5 | Page 10

Show that f(x) = $\frac{1}{x}$ is a decreasing function on (0, ∞) ?

Q 6 | Page 10

Show that f(x) = $\frac{1}{1 + x^2}$ decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?

Q 7 | Page 10

Show that f(x) = $\frac{1}{1 + x^2}$ is neither increasing nor decreasing on R ?

Q 8 | Page 10

Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .

Q 9 | Page 10

Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?

#### Pages 33 - 35

Q 1.01 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2  ?

Q 1.02 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = x2 + 2x − 5  ?

Q 1.03 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 6 − 9x − x2  ?

Q 1.04 | Page 33

Find the interval in which the following function are increasing or decreasing   f(x) = 2x3 − 12x2 + 18x + 15 ?

Q 1.05 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x?

Q 1.06 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x?

Q 1.07 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 5x3 − 15x2 − 120x + 3 ?

Q 1.08 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?

Q 1.09 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?

Q 1.1 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20  ?

Q 1.11 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?

Q 1.12 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?

Q 1.13 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 2x3 − 24x + 107  ?

Q 1.14 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1  ?

Q 1.15 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)?

Q 1.16 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?

Q 1.17 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = 2x3 − 24x + 7 ?

Q 1.18 | Page 33

Find the interval in which the following function are increasing or decreasing $f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11$ ?

Q 1.19 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?

Q 1.2 | Page 33

Find the interval in which the following function are increasing or decreasing $f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7$ ?

Q 1.21 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) = x4 − 4x3 + 4x2 + 15 ?

Q 1.22 | Page 33

Find the interval in which the following function are increasing or decreasing  f(x) =  $5 x^\frac{3}{2} - 3 x^\frac{5}{2}$  x > 0 ?

Q 1.23 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2  ?

Q 1.24 | Page 33

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?

Q 1.25 | Page 33

Find the interval in which the following function are increasing or decreasing $f\left( x \right) = \left\{ x(x - 2) \right\}^2$ ?

Q 1.26 | Page 33

Find the interval in which the following function are increasing or decreasing $f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5$ ?

Q 1.27 | Page 33

Find the interval in which the following function are increasing or decreasing $f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51$ ?

Q 1.28 | Page 33

Find the interval in which the following function are increasing or decreasing $f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R$ ?

Q 2 | Page 34

Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?

Q 3 | Page 34

Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?

Q 4 | Page 34

Show that f(x) = e2x is increasing on R ?

Q 5 | Page 34

Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?

Q 6 | Page 34

Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?

Q 7 | Page 34

Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?

Q 8 | Page 34

Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?

Q 9 | Page 34

Show that f(x) = x − sin x is increasing for all x ∈ R ?

Q 10 | Page 34

Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?

Q 11 | Page 34

Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?

Q 12 | Page 34

Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?

Q 13 | Page 34

Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?

Q 14 | Page 34

Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?

Q 15 | Page 34

Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?

Q 16 | Page 34

Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?

Q 17 | Page 34

Show that the function f(x) = cot $-$ l(sinx + cosx) is decreasing on $\left( 0, \frac{\pi}{4} \right)$ and increasing on $\left( 0, \frac{\pi}{4} \right)$ ?

Q 18 | Page 34

Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?

Q 19 | Page 34

Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?

Q 20 | Page 34

Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?

Q 21 | Page 35

Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?

Q 22 | Page 35

State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?

Q 23 | Page 35

Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?

Q 24 | Page 35

Show that f(x) = tan−1 x − x is a decreasing function on R ?

Q 25 | Page 35

Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?

Q 26 | Page 35

Find the intervals in which f(x) = log (1 + x) −$\frac{x}{1 + x}$ is increasing or decreasing ?

Q 27 | Page 35

Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?

Q 28 | Page 35

Show that the function f given by f(x) = 10x is increasing for all x ?

Q 29 | Page 35

Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?

Q 30.1 | Page 35

Prove that the following function is increasing on R f $(x) =$3 $x^5$ + 40 $x^3$ + 240$x$ ?

Q 30.2 | Page 35

Prove that the following function is increasing on R f $f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27$ ?

Q 31 | Page 35

Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?

Q 32 | Page 35

Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?

Q 33 | Page 35

Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).

Q 34 | Page 35

Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?

Q 35 | Page 35

Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?

Q 36 | Page 35

Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?

Q 37 | Page 35

Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?

Q 38 | Page 35

Let f defined on [0, 1] be twice differentiable such that | f"(x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?

Q 39.1 | Page 35

Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x $\in$ R ?

Q 39.2 | Page 35

Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x $\leq 2\pi$ ?

Q 39.3 | Page 35

Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < $\frac{\pi}{2}$ ?

#### Pages 39 - 40

Q 1 | Page 39

What are the values of 'a' for which f(x) = ax is increasing on R ?

Q 2 | Page 39

What are the values of 'a' for which f(x) = ax is decreasing on R ?

Q 3 | Page 39

Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?

Q 4 | Page 39

Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?

Q 5 | Page 39

Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?

Q 6 | Page 39

Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?

Q 7 | Page 39

Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?

Q 8 | Page 40

Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?

Q 9 | Page 40

Write the set of values of k for which f(x) = kx − sin x is increasing on R ?

Q 10 | Page 40

If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?

Q 11 | Page 40

Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?

Q 12 | Page 40

Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?

Q 13 | Page 40

State whether f(x) = tan x − x is increasing or decreasing its domain ?

Q 14 | Page 40

Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?

#### Pages 40 - 42

Q 1 | Page 40

The interval of increase of the function f(x) = x − ex + tan (2π/7) is
(a) (0, ∞)
(b) (−∞, 0)
(c) (1, ∞)
(d) (−∞, 1)

Q 2 | Page 40

The function f(x) = cot−1 x + x increases in the interval
(a) (1, ∞)
(b) (−1, ∞)
(c) (−∞, ∞)
(d) (0, ∞)

Q 3 | Page 40

The function f(x) = xx decreases on the interval
(a) (0, e)
(b) (0, 1)
(c) (0, 1/e)
(d) none of these

Q 4 | Page 40

The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
(a) (1, 2)
(b) (2, 3)
(c) (1, 3)
(d) (2, 4)

Q 5 | Page 40

If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
(a) (−∞, 4)
(b) (4, ∞)
(c) (−∞, 8)
(d) (8, ∞)

Q 6 | Page 40

Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and bsatisfy
(a) a2 − 3b − 15 > 0
(b) a2 − 3b + 15 > 0
(c) a2 − 3b + 15 < 0
(d) a > 0 and b > 0

Q 7 | Page 40

The function $f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)$ is of the following types:
(a) even and increasing
(b) odd and increasing
(c) even and decreasing
(d) odd and decreasing

Q 8 | Page 40

If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
(a) a ∈ (1/2, ∞)
(b) a ∈ (−1/2, 1/2)
(c) a = 1/2
(d) a ∈ R

Q 9 | Page 40

Let $f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),$,where g (x) is monotonically increasing for 0 < x < $\frac{\pi}{2} .$ Then, f(x) is
(a) increasing on (0, π/2)
(b) decreasing on (0, π/2)
(c) increasing on (0, π/4) and decreasing on (π/4, π/2)
(d) none of these

Q 10 | Page 40

Let f(x) = x3 − 6x2 + 15x + 3. Then,
(a) f(x) > 0 for all x ∈ R
(b) f(x) > f(x + 1) for all x ∈ R
(c) f(x) is invertible
(d) none of these

Q 11 | Page 41

The function f(x) = x2 e−x is monotonic increasing when
(a) x ∈ R − [0, 2]
(b) 0 < x < 2
(c) 2 < x < ∞
(d) x < 0

Q 12 | Page 41

Function f(x) = cos x − 2 λ x is monotonic decreasing when
(a) λ > 1/2
(b) λ < 1/2
(c) λ < 2
(d) λ > 2

Q 13 | Page 41

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) monotonically increasing
(b) monotonically decreasing
(c) not monotonic
(d) constant

Q 14 | Page 41

Function f(x) = x3 − 27x + 5 is monotonically increasing when
(a) x < −3
(b) | x | > 3
(c) x ≤ −3
(d) | x | ≥ 3

Q 15 | Page 41

Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
(a) x < 2
(b) x > 2
(c) x > 3
(d) 1 < x < 2

Q 16 | Page 41

If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
(a) k < 3
(b) k ≤ 3
(c) k > 3
(d) k ≥ 3

Q 17 | Page 41

f(x) = 2x − tan−1 x − log $\left\{ x + \sqrt{x^2 + 1} \right\}$ is monotonically increasing when

(a) x > 0
(b) x < 0
(c) x ∈ R
(d) x ∈ R − {0}

Q 18 | Page 41

Function f(x) = | x | − | x − 1 | is monotonically increasing when
(a) x < 0
(b) x > 1
(c) x < 1
(d) 0 < x < 1

Q 19 | Page 41

Every invertible function is
(a) monotonic function
(b) constant function
(c) identity function
(d) not necessarily monotonic function

Q 20 | Page 41

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) increasing
(b) decreasing
(c) constant
(d) none of these

Q 21 | Page 41

If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then

(a) a = b
(b) $a = \frac{1}{2}b$

(c) $a \leq - \frac{1}{2}$

(d)  $a > - \frac{3}{2}$

Q 22 | Page 41

The function $f\left( x \right) = \frac{x}{1 + \left| x \right|}$ is

(a) strictly increasing
(b) strictly decreasing
(c) neither increasing nor decreasing
(d) none of these

Q 23 | Page 41

The function $f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}$ is increasing, if

(a) λ < 1
(b) λ > 1
(c) λ < 2
(d) λ > 2

Q 24 | Page 41

Function f(x) = ax is increasing on R, if
(a) a > 0
(b) a < 0
(c) 0 < a < 1
(d) a > 1

Q 25 | Page 41

Function f(x) = loga x is increasing on R, if
(a) 0 < a < 1
(b) a > 1
(c) a < 1
(d) a > 0

Q 26 | Page 41

Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
(a) increases on [0, a]
(b) decreases on [0, a]
(c) increases on [−a, 0]
(d) decreases on [a, 2a]

Q 27 | Page 41

If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
(a) k ∈ (2, ∞)
(b) k ∈ (−∞, 2)
(c) k ∈ (4, ∞)
(d) k ∈ (−∞, 4).

Q 28 | Page 41

The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
(a) increasing
(b) decreasing
(c) constant
(d) none of these

Q 29 | Page 42

If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
(a) −1 ≤ k < 1
(b) k < −1 or k > 1
(c) 0 < k < 1
(d) −1 < k < 0

Q 30 | Page 42

The function f(x) = x9 + 3x7 + 64 is increasing on
(a) R
(b) (−∞, 0)
(c) (0, ∞)
(d) R0

#### Page 7

Q 1 | Page 7

f ( $x$)=4$x^2$-4  $x$ + 4 on R .

Q 2 | Page 7

f(x)=(x-1)2+2 on R ?

Q 3 | Page 7

f(x)=| x+2 | on R .

Q 4 | Page 7

f(x)=sin 2x+5 on R .

Q 5 | Page 7

f(x) = | sin 4x+3 | on R ?

Q 6 | Page 7

f(x)=2x3 +5 on R .

Q 7 | Page 7

f (x) = $-$ | x + 1 | + 3 on R .

Q 8 | Page 7

f(x) = 16x2 $-$ 16x + 28 on R ?

Q 9 | Page 7

f(x) = x$-$ 1 on R .

#### Page 16

Q 1 | Page 16

f(x) = (x $-$ 5)4.

Q 2 | Page 16

f(x) = x$-$ 3x .

Q 3 | Page 16

f(x) = x3  (x $-$ 1).

Q 4 | Page 16

f(x) =  (x $-$ 1) (x+2)2

Q 5 | Page 16

f(x) = $\frac{1}{x^2 + 2}$ .

Q 6 | Page 16

f(x) =  x$-$ 6x2 + 9x + 15 .

Q 7 | Page 16

f(x) = sin 2x, 0<x< $\pi$ .

Q 8 | Page 16

f(x) =  sin x $-$ cos x, 0 < x<2 $\pi$ .

Q 9 | Page 16

f(x) =  cos x, 0<x< $\pi$ .

Q 10 | Page 16

f(x) = sin 2x $-$ x, $- \frac{\pi}{2} < \frac{<}{}x\frac{<}{}\frac{\pi}{2}$ .

Q 11 | Page 16

f(x) = 2sin x$-$ x, $- \frac{\pi}{2} < \frac{<}{}x\frac{<}{}\frac{\pi}{2}$ .

Q 12 | Page 16

f(x) =$x\sqrt{1 - x} , x > 0$.

Q 13 | Page 16

f(x) = x3 (2x $-$ 1)3.

Q 14 | Page 16

f(x) =$\frac{x}{2} + \frac{2}{x} , x > 0$ .

#### Page 31

Q 1.01 | Page 31

f(x) = x4 $-$ 62x2 + 120x + 9.

Q 1.02 | Page 31

f(x) = x3$-$ 6x2 + 9x + 15

Q 1.03 | Page 31

f(x) = (x $-$ 1) (x+2)2.

Q 1.04 | Page 31

f(x) = 2/x $-$ 2/x2 , x>0 .

Q 1.05 | Page 31

f(x) = xex.

Q 1.06 | Page 31

f(x) = x/2+2/x, x>0 .

Q 1.07 | Page 31

f(x) = (x+1) (x+2)1/3, $x\frac{>}{} - 2$ .

Q 1.08 | Page 31

f(x) = $x\sqrt{32 - x^2}, - 5\frac{<}{}x\frac{<}{}5$ .

Q 1.09 | Page 31

f(x) = $x^3 - 2a x^2 + a^2 x, a > 0, x \in R$ .

Q 1.1 | Page 31

f(x) = $x + \frac{a2}{x}, a > 0,$ , x ≠ 0 .

Q 1.11 | Page 31

f(x) = $x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}$ .

Q 1.12 | Page 31

f(x) = $x + \sqrt{1 - x}, x \leq 1$ .

Q 2.1 | Page 31

f(x) = (x $-$ 1) (x $-$ 2)2.

Q 2.2 | Page 31

f(x) = $x\sqrt{1 - x} , x\frac{<}{}1$ .

Q 2.3 | Page 31

f(x) = $- (x - 1 )^3 (x + 1 )^2$ .

Q 3 | Page 31

The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?

Q 4 | Page 31

Show that $\frac{\log x}{x}$ has a maximum value at x = e ?

Q 5 | Page 31

Find the maximum and minimum values of the function f(x) = $\frac{4}{x + 2} + x .$

Q 6 | Page 31

Find the maximum and minimum values of y = tan $x - 2x$ .

Q 7 | Page 31

If f(x) = x3 + ax2 + bx + c has a maximum at x = $-$ 1 and minimum at x = 3. Determine a, b and c ?

Q 8 | Page 31

Prove that f(x) = sinx + $\sqrt{3}$ cosx has maximum value at x = $\frac{\pi}{6}$ ?

#### Page 37

Q 1.1 | Page 37

f(x) = 4x $-$ $\frac{x^2}{2}$ in [ $-$ 2,4,5] .

Q 1.2 | Page 37

f(x) = (x $-$ 1)2 + 3 in [ $-$ 3,1] ?

Q 1.3 | Page 37

f(x) = 3x4 $-$ 8x3 + 12x2$-$ 48x + 25 in [0,3] .

Q 1.4 | Page 37

f(x) = (x $-$ 2) $\sqrt{x - 1} \text { in }[1, 9]$ .

Q 2 | Page 37

Find the maximum value of 2x3$-$ 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ $-$ 3, $-$ 1].

Q 3 | Page 37

Find the absolute maximum and minimum values of the function of given by $f(x) = \cos^2 x + \sin x, x \in [0, \pi]$ .

Q 4 | Page 37

Find the absolute maximum and minimum values of a function f given by $f(x) = 12 x^{4/3} - 6 x^{1/3} , x \in [ - 1, 1]$ .

Q 5 | Page 37

Find the absolute maximum and minimum values of a function f given by $f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]$ ?

#### Pages 72 - 74

Q 1 | Page 72

Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.

Q 2 | Page 72

Divide 64 into two parts such that the sum of the cubes of two parts is minimum.

Q 3 | Page 72

How should we choose two numbers, each greater than or equal to $-$ 2, whose sum______________ so that the sum of the first and the cube of the second is minimum?

Q 4 | Page 72

Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.

Q 5 | Page 72

Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?

Q 6.1 | Page 72

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by $M = \frac{WL}{2}x - \frac{W}{2} x^2$ .

Find the point at which M is maximum in case.

Q 6.2 | Page 72

A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by $M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}$ .

Find the point at which M is maximum in case.

Q 7 | Page 72

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?

Q 8 | Page 72

A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?

Q 9 | Page 72

Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.

Q 10 | Page 73

Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.

Q 11 | Page 73

Two sides of a triangle have lengths 'a' and 'b' and the angle between them is $\theta$ What value of $\theta$ will maximize the area of the triangle? Find the maximum area of the triangle also.

Q 12 | Page 73

A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.

Q 13 | Page 73

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?

Q 14 | Page 73

A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

Q 15 | Page 73

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

Q 16 | Page 73

A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.

Q 17 | Page 73

Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is $\frac{2R}{\sqrt{3}} .$

Q 18 | Page 73

A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?

Q 19 | Page 73

Prove that a conical tent of given capacity will require the least amount of  canavas when the height is $\sqrt{2}$ times the radius of the base.

Q 20 | Page 73

Show that the cone of the greatest volume which can be inscribed in a given spher has an altitude equal to $\frac{2}{3}$ of the diameter of the sphere.

Q 21 | Page 73

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is  $\cot^{- 1} \left( \sqrt{2} \right)$ .

Q 22 | Page 73

An isosceles triangle of vertical angle 2 $\theta$ is inscribed in a circle of radius a. Show that the area of the triangle is maximum when $\theta$ = $\frac{\pi}{6}$ .

Q 23 | Page 73

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}$r.

Q 24 | Page 73

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?

Q 25 | Page 73

Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?

Q 26 | Page 73

A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?

Q 27 | Page 73

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .$

Q 28 | Page 74

Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r $\sqrt{2}$ .

Q 29 | Page 74

Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?

Q 30 | Page 74

Find the point on the curve y2=4x which is nearest to the point (2,$-$ 8).

Q 31 | Page 74

Find the point on the curve x2=8y which is nearest to the point (2,4) ?

Q 32 | Page 74

Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?

Q 33 | Page 74

Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x $-$ 3 ?

Q 34 | Page 74

Find the point on the curvey y2=2x which is at a minimum distance from the point (1,4).

Q 35 | Page 74

Find the maximum slope of the curve y= $- x^3 + 3 x^2 + 2x - 27 .$

Q 36 | Page 74

The total cost of producing x radio sets per  day is Rs $\left( \frac{x^2}{4} + 35x + 25 \right)$ and the price per set  at which they may be sold is Rs. $\left( 50 - \frac{x}{2} \right) .$ ind the daily output to maximum the total profit.

Q 37 | Page 74

Manufacturer can sell x items at a price of rupees $\left( 5 - \frac{x}{100} \right)$ each. The cost price is Rs  $\left( \frac{x}{5} + 500 \right) .$ Find the number of items he should sell to earn maximum profit.

Q 38 | Page 74

An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.

Q 39 | Page 74

A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

Q 40 | Page 74

The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.

Q 41 | Page 74

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is $\pi : (\pi + 2)$ .

Q 42 | Page 74

The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?

Q 43 | Page 74

A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?

Q 44 | Page 74

The total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?

Q 45 | Page 74

The space s described in time by a particle moving in a straight line is given by S = $t5 - 40 t^3 + 30 t^2 + 80t - 250 .$ Find the minimum value of acceleration.

Q 46 | Page 74

A particle is moving in a straight line such that its distance at any time t is given by  S = $\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .$  Find when its velocity is maximum and acceleration minimum.

#### Page 80

Q 1 | Page 80

Write necessary condition for a point x = c to be an extreme point of the function f(x).

Q 2 | Page 80

Write sufficient conditions for a point x=c to be a point of local maximum.

Q 3 | Page 80

If f(x) attains a local minimum at x=c, then write the values off' (c) and f'' (c).

Q 4 | Page 80

Write the minimum value of f(x) = $x + \frac{1}{x}, x > 0 .$

Q 5 | Page 80

Write the maximum value of f(x) = $x + \frac{1}{x}, x > 0 .$

Q 6 | Page 80

Write the point where f(x) = x log, x attains minimum value.

Q 7 | Page 80

Find the least value of f(x) =$ax + \frac{b}{x}$, where a>0, b>0 and x>0 .

Q 8 | Page 80

Write the minimum value of f(x) = xx .

Q 9 | Page 80

Write the maximum value of f(x) = x1/x.

Q 10 | Page 80

Write the maximum value of f(x) = $\frac{\log x}{x}$, if it exists .

#### Pages 80 - 82

Q 1 | Page 80

The maximum value of x1/x, x>0 is

(a) e1/e

(b) $\left( \frac{1}{e} \right)^e$

(c) 1

(d) none of these

Q 2 | Page 81

If $ax + \frac{b}{x} \frac{>}{} c$ for all positive x where a,b,>0, then

(a) $ab < \frac{c^2}{4}$

(b) $ab \frac{>}{} \frac{c^2}{4}$

(c) $ab \frac{>}{} \frac{c^{}}{4}$

Q 3 | Page 81

The minimum value of $\frac{x}{\log_e x}$ is

(a) e
(b) 1/e
(c) 1
(d) none of these

Q 4 | Page 81

For the function f(x) = $x + \frac{1}{x}$

(a) x = 1 is a point of maximum
(b) x = $-$ 1 is a point of minimum
(c) maximum value > minimum value
(d) maximum value< minimum value

Q 5 | Page 81

Let f(x) = x3+3x$-$ 9x+2. Then, f(x) has
(a) a maximum at x = 1
(b) a minimum at x = 1
(c) neither a maximum nor a minimum at x = $-$ 3
(d) none of these

Q 6 | Page 81

The minimum value of f(x) = $x4 - x2 - 2x + 6$ is
(a) 6
(b) 4
(c) 8
(d) none of these

Q 7 | Page 81

The number which exceeds its square by the greatest possible quantity is

(a) $\frac{1}{2}$

(b) $\frac{1}{4}$

(c) $\frac{3}{4}$

(d) none of these

Q 8 | Page 81

Let f(x) = (x $-$ a)2 + (x $-$ b)2 + (x $-$ c)2. Then, f(x) has a minimum at x =

(a) $\frac{a + b + c}{3}$

(b) $\sqrt[3]{abc}$

(c) $\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$

(d) none of these

Q 9 | Page 81

The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is

(a) $\frac{1}{4}$

(b) $\frac{1}{2}$

(c) $\frac{1}{8}$

(d) none of these

Q 10 | Page 81

The function f(x) = $\sum^5_{r = 1}$ (x $-$ r)2 assumes minimum value at x =

(a) 5
(b) $\frac{5}{2}$

(c) 3
(d) 2

Q 11 | Page 81

At x= $\frac{5\pi}{6}$ f(x) = 2 sin 3x + 3 cos 3x is

(a) 0
(b) maximum
(c) minimum
(d) none of these

Q 12 | Page 81

If x lies in the interval [0,1], then the least value of x2 + x + 1 is

(a) 3
(b) $\frac{3}{4}$

(c) 1
(d) none of these

Q 13 | Page 81

The least value of the function f(x) = $x3 - 18x2 + 96x$ in the interval [0,9] is

(a) 126
(b) 135
(c) 160
(d) 0

Q 14 | Page 81

The maximum value of f(x) = $\frac{x}{4 - x + x^2}$ on [ $-$ 1, 1] is

(a) $- \frac{1}{4}$

(b) $- \frac{1}{3}$

(c) $\frac{1}{6}$

(d) $\frac{1}{5}$

Q 15 | Page 81

The point on the curve y2 = 4x which is nearest to, the point (2,1) is

(a) $1, 2\sqrt{2}$

(b) (1,2)
(c) (1,$-$ 2)

(d) ( $-$ 2,1)

Q 16 | Page 82

If x+y=8, then the maximum value of xy is
(a) 8
(b) 16
(c) 20
(d) 24

Q 17 | Page 82

The least and greatest values of f(x) = x3$-$ 6x2+9x in [0,6], are

(a) 3,4
(b) 0,6
(c) 0,3
(d) 3,6

Q 18 | Page 82

f(x) = $\sin + \sqrt{3} \cos x$ is maximum when x =

(a) $\frac{\pi}{3}$

(b) $\frac{\pi}{4}$

(c) $\frac{\pi}{6}$

(d) 0

Q 19 | Page 82

If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is

(a) $\frac{3}{4}$

(b) $\frac{1}{3}$

(c) $\frac{1}{4}$

(d) $\frac{2}{3}$

Q 20 | Page 82

The minimum value of $\left( x^2 + \frac{250}{x} \right)$ is

(a) 75
(b) 50
(c) 25
(d) 55

Q 21 | Page 82

If(x) = x+$\frac{1}{x}$,x > 0, then its greatest value is

(a) $-$ 2

(b) 0
(c) 3
(d) none of these

Q 22 | Page 82

If(x) = $\frac{1}{4x2 + 2x + 1}$ then its maximum value is

(a) $\frac{4}{3}$

(b) $\frac{2}{3}$

(c) 1

(d) $\frac{3}{4}$

Q 23 | Page 82

Let x, y be two variables and x>0, xy=1, then minimum value of x+y is
(a) 1
(b) 2
(c) $2\frac{1}{2}$

(d) $3\frac{1}{3}$

Q 24 | Page 82

f(x) = 1+2 sin x+3 cos2x, $0\frac{<}{}x\frac{<}{}\frac{2\pi}{3}$ is

(a) Minimum at x =$\frac{\pi}{2}$

(b) Maximum at x = sin $- 1$ ( $\frac{1}{\sqrt{3}}$)

(c) Minimum at x = $\frac{\pi}{6}$

(d) Maximum at sin $- 1$ ($\frac{1}{6})$

Q 25 | Page 82

The function f(x) = $2 x^3 - 15 x^2 + 36x + 4$ is maximum at x =
(a) 3
(b) 0
(c) 4
(d) 2

Q 26 | Page 82

The maximum value of f(x) = $\frac{x}{4 + x + x^2}$ on [ $-$ 1,1] is

(a) $- \frac{1}{4}$

(b) $- \frac{1}{3}$

(c) $\frac{1}{6}$

(d) $\frac{1}{5}$

Q 27 | Page 82

Let f(x) = 2x3$-$ 3x2$-$ 12x + 5 on [ 2, 4]. The relative maximum occurs at x =

(a) $-$ 2

(b) $-$1
(c) 2
(d) 4

Q 28 | Page 82

The minimum value of x loge x is equal to
(a) e
(b) 1/e
(c) $-$ 1/e

(d) 2/e

(e) $-$ e

## R.D. Sharma solutions for Class 12 Mathematics chapter 15 - Mean Value Theorems

R.D. Sharma solutions for Class 12 Mathematics chapter 15 (Mean Value Theorems) include all questions with solution and detail explanation from Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session). This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 15 Mean Value Theorems are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

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