MU Applied Mathematics 1 - May 2012 Exam Question Paper

MU First Year Engineering (Semester 1)
Applied Mathematics 1
May 2012

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) If

A r g s (z + 1) = \frac{π}{6} & A r g s (z - 1) = \frac{2 π}{3} f i n d z .

$Args\left(z+1\right)=\frac{\pi{}}{6}\ \&\ Args\left(z-1\right)=\frac{2\pi{}}{3}\ find\ z.$

5 M

1 (b) Find n^th derivative of 2^x⋅ sin²x ⋅cos³x.

5 M

1 (c) Expand

f (x) = x^{4} - 3 x^{2} + 2 x^{2} - x + 1

$f(x)=x^4-3x^2+2x^2-x+1$ in power of (x-3).

5 M

1 (d) Show that

lim_{x \to \infty} {(\frac{1^{1 / x} + 2^{1 / x} + 3^{1 / x} + 4^{1 / x}}{4})}^{4 x} = 24

$\lim_{x\rightarrow\infty}\left( \frac{1^{1/x}+2^{1/x}+3^{1/x}+4^{1/x} }{4}\right)^{4x}=24$

5 M

2 (a)

I f z = \log (x^{2} + y^{2}) + \frac{x^{2} + y^{2}}{x + y} - 2 \log (x + y), p r o v e t h a t x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = \frac{x^{2} + y^{2}}{x + y}

$If\ z=\log{\left(x^2+y^2\right)}+\frac{x^2+y^2}{x+y}-2\log{\left(x+y\right)},\\ prove\ that\ x\frac{\partial{}z}{\partial{}x}+y\frac{\partial{}z}{\partial{}y}=\frac{x^2+y^2}{x+y}$

7 M

2 (b) If f(x), ∅(x), φ(x) are differentiable in [a,b], show that there exist a value c in (a,b) such that

| \begin{array}{ccc} f (a) & \emptyset (a) & φ (a) \\ f (b) & \emptyset (b) & φ (b) \\ f^{^{'}} (c) & \emptyset^{^{'}} (c) & φ^{'} (c) \end{array} | = 0

$\left\vert{}\begin{array}{ccc}f(a) & \varnothing{}(a) & \varphi{}(a) \\f(b) & \varnothing{}(b) & \varphi{}(b) \\f^{'}\left(c\right) & {\varnothing{}}^{'}\left(c\right) & \varphi{}'(c)\end{array}\right\vert{}=0$

6 M

2 (c)

I f \tan (α + i β) = e^{i θ}, p r o v e t h a t 1) α = \frac{n π}{2} + \frac{π}{4}, a n d 2) β = \frac{1}{2} \log \tan (\frac{π}{4} + \frac{θ}{2}) .

$If\ \tan\left(\alpha +i\beta{}\right)=e^{i\theta},\ prove\ that\ \\ 1)\ \alpha{}=\frac{n\pi{}}{2}+\frac{\pi{}}{4}\ ,\ and\ \\ 2)\ \beta{}=\frac{1}{2}\log{\tan{\left(\frac{\pi{}}{4}+\frac{\theta{}}{2}\right)}.}$

7 M

3 (a)

s h o w t h a t {(1 - e^{i θ})}^{- \frac{1}{2}} {(1 - e^{- i θ})}^{- \frac{1}{2}} = {(1 c o s e c (\frac{θ}{2}))}^{\frac{1}{2}}

$show \ that \left(1-e^{i\theta}\right)^{-\frac{1}{2}} \left(1-e^{-i\theta}\right)^{-\frac{1}{2}}=\left(1 \mathrm{cosec} \left(\frac{\theta}{2}\right)\right)^{\frac{1}{2}}$

7 M

3 (b)

I f f (x) = \frac{1}{x^{2}} a n d g (x) = \frac{1}{x}

$If\ f(x)=\frac{1}{x^2}\ and\ g(x)=\frac{1}{x}\$ then show that c of C.M.V.T. is H.M of a & b where a>0, b>0.

6 M

3 (c)

i f y = \sin [\log (x^{2} + 2 x + 1)] t h e n p r o v e t h a t {(x + 1)}^{2} y_{n + 2} (2 n + 1) (x + 1) y_{n + 1} + (n^{2} + 4) y_{n} = 0.

$if\ y=\sin \left[\log\left(x^2+2x+1\right) \right] \ then\ prove\ that\ \\\left(x+1\right)^2y_{n+2}\left(2n+1\right)\left(x+1\right)y_{n+1}+\left(n^2+4\right)y_n=0.$

7 M

4 (a)

S o l v e x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 = 0.

$Solve\ x^6-x^5+x^4-x^3+x^2-x+1=0.$

7 M

4 (b) If u=1x+my, v=mx-1y, prove that

1] {(\frac{\partial u}{\partial x})}_{y} \cdot {(\frac{\partial x}{\partial u})}_{V} = \frac{I^{2}}{I^{2} + m^{2}}

$1]\ \left(\frac{\partial u}{\partial x}\right)_y\cdot \left(\frac{\partial x}{\partial u}\right)_V=\frac{I^2}{I^2+m^2}$

2] {(\frac{\partial y}{\partial v})}_{x} \cdot {(\frac{\partial v}{\partial y})}_{u} = \frac{I^{2} + m^{2}}{I^{2}}

$2]\ \ \left(\frac{\partial y}{\partial v}\right)_x\cdot \left(\frac{\partial v}{\partial y}\right)_u=\frac{I^2+m^2}{I^2}$

6 M

4 (c) Show that

\bar{F} = (y^{2} - z^{2} + 3 y z - 2 x) i + (3 x z + 2 x y) i + (3 x y - 2 x z + 2 z) k

$\bar{F}=\left(y^2-z^2+3yz-2x\right)i+\left(3xz+2xy\right)i+\left(3xy-2xz+2z\right)k$ is both solenoidal and irrotational.

7 M

5 (a)

p r o v e t h a t \tan 7 θ = \frac{7 \tan θ - 35 \tan^{3} θ + 21 \tan^{5} θ - \tan^{7} θ}{1 - 21 \tan^{2} θ + 35 \tan^{4} θ - 7 \tan^{6} θ}

$prove\ that\ \tan 7\theta =\frac{7\tan\theta -35\tan^3 \theta +21\tan^5 \theta -\tan^7\theta}{1-21\tan^2\theta+35\tan^4\theta -7\tan^6 \theta}$

7 M

5 (b) Test the convergence of

\sum \frac{2^{n} + 1}{3^{n} + n}

$\sum\frac{2^n+1}{3^n+n}$

6 M

5 (c)

I f x = \sqrt{v w}, y = \sqrt{w u}, z = \sqrt{u v}, a n d φ i s a f u n c t i o n o f x, y, z p r o v e t h a t u \frac{\partial φ}{\partial u} + v \frac{\partial φ}{\partial v} + w \frac{\partial φ}{\partial w} = x \frac{\partial φ}{\partial x} + y \frac{\partial φ}{\partial y} + z \frac{\partial φ}{\partial z}

$If\ x=\sqrt{vw\ \ },\ y=\sqrt{wu},\ z=\sqrt{uv},\ and\ \varphi{}\ is\ a\ function\ of\ x,y,z\ \\ prove\ that\ u\frac{\partial{}\varphi{}}{\partial{}u}+v\frac{\partial{}\varphi{}}{\partial{}v}+w\frac{\partial{}\varphi{}}{\partial{}w}=x\frac{\partial{}\varphi{}}{\partial{}x}+y\frac{\partial{}\varphi{}}{\partial{}y}+z\frac{\partial{}\varphi{}}{\partial{}z}$

7 M

6 (a)

P r o v e t h a t \nabla \cdot {\frac{f (r)}{r} \bar{r}} = \frac{1}{r^{2}} \frac{d}{d r} [r^{2} f (r)]

$Prove\ that \ \nabla\cdot\left\{ \frac{f(r)}{r}\bar{r} \right \}= \frac{1}{r^2} \frac{d}{dr}\ [r^2f(r)]$
hence or otherwise prove that div

(r^{n} \bar{r}) = (n + 3) r^{n}

$\left(r^n\bar{r}\right)=\left(n+3\right)r^n$

7 M

6 (b)

S h o w t h a t \frac{b - a}{\sqrt{1 - a^{2}}} < \sin^{- 1} b - \sin^{- 1} a < \frac{b - a}{\sqrt{1 - b^{2}}} h e n c e s h o w t h a t

$Show\ that\frac{b-a}{\sqrt{1-a^2}}< \sin^{-1}b-\sin^{-1}a< \frac{b-a}{\sqrt{1-b^2}}\ hence\ show\ that\$

1] \frac{π}{6} + \frac{\sqrt{3}}{15} < \sin^{- 1} (\frac{3}{5}) < \frac{π}{6} + \frac{1}{8}

$1]\ \frac{\pi}{6}+\frac{\sqrt{3}}{15}<\sin^{-1}\left(\frac{3}{5}\right)<\frac{\pi}{6}+\frac{1}{8}$

2] \frac{π}{6} + \frac{1}{2 \sqrt{3}} < \sin^{- 1} (\frac{1}{4}) < \frac{π}{6} - \frac{1}{\sqrt{15}}

$2]\ \frac{\pi}{6}+\frac{1}{2\sqrt{3}}< \sin^{-1}\left(\frac{1}{4}\right)<\frac{\pi}{6}-\frac{1}{\sqrt{15}}$

6 M

6 (c)

I f u = \frac{x^{3} y^{3} z^{3}}{x^{3} + y^{3} + z^{3}} + \log (\frac{x y + y z + z x}{x^{2} + y^{2} + z^{2}}) t h e n p r o v e t h a t X \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 6 \frac{x^{3} y^{3} z^{3}}{x^{3} + y^{3} + z^{3}}

$If\ u= \frac{x^3y^3z^3}{x^3+y^3+z^3}+\log \left(\frac{xy+yz+zx}{x^2+y^2+z^2}\right) \ \ then\ prove\ that\\ \ X\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=6\frac{x^3y^3z^3}{x^3+y^3+z^3}$

7 M

7 (a) Find the extreme values of the function

x^{3} + 3 x y^{2} + 72 x - 15 x^{2} - 15 y^{2}

$x^3+3xy^2+72x-15x^2-15y^2$

7 M

7 (b) Examine the convergence of

{(\frac{2^{2}}{1^{2}} - \frac{2}{1})}^{- 1} + {(\frac{3^{3}}{2^{3}} - \frac{3}{2})}^{- 2} + {(\frac{4^{4}}{3^{4}} - \frac{4}{3})}^{- 3} + ? . . .

${\left(\frac{2^2}{1^2}-\frac{2}{1}\right)}^{-1}+{\left(\frac{3^3}{2^3}-\frac{3}{2}\right)}^{-2}+{\left(\frac{4^4}{3^4}-\frac{4}{3}\right)}^{-3}+?\ ...$

6 M

7 (c)

I f \frac{{(a + i b)}^{x + 1 y}}{{(a - i b)}^{x - 1 y}} = a + i β t h e n f i n d α & β

$If\frac{{\left(a+ib\right)}^{x+1y}\ }{{\left(a-ib\right)}^{x-1y}}=a+i\beta \ then\ find\ \alpha{}\ \&\ \beta{}$

7 M

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