Applied Mathematics 1 - May 2012

First Year Engineering (Semester 1)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) If $$Args\left(z+1\right)=\frac{\pi{}}{6}\ \&\ Args\left(z-1\right)=\frac{2\pi{}}{3}\ find\ z.$$(5 marks) 1 (b) Find n^th derivative of 2^x⋅ sin²x ⋅cos³x.(5 marks) 1 (c) Expand $$ f(x)=x^4-3x^2+2x^2-x+1 $$ in power of (x-3).(5 marks) 1 (d) Show that $$ \lim_{x\rightarrow\infty}\left( \frac{1^{1/x}+2^{1/x}+3^{1/x}+4^{1/x} }{4}\right)^{4x}=24 $$(5 marks) 2 (a) $$ If\ z=\log{\left(x^2+y^2\right)}+\frac{x^2+y^2}{x+y}-2\log{\left(x+y\right)},\\ prove\ that\ x\frac{\partial{}z}{\partial{}x}+y\frac{\partial{}z}{\partial{}y}=\frac{x^2+y^2}{x+y}$$(7 marks) 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 $$\left\vert{}\begin{array}{ccc}f(a) & \varnothing{}(a) & \varphi{}(a) \\f(b) & \varnothing{}(b) & \varphi{}(b) \\f^{'}\left(c\right) & {\varnothing{}}^{'}\left(c\right) & \varphi{}'(c)\end{array}\right\vert{}=0$$(6 marks) 2 (c) $$If\ \tan\left(\alpha +i\beta{}\right)=e^{i\theta},\ prove\ that\ \\ 1)\ \alpha{}=\frac{n\pi{}}{2}+\frac{\pi{}}{4}\ ,\ and\ \\ 2)\ \beta{}=\frac{1}{2}\log{\tan{\left(\frac{\pi{}}{4}+\frac{\theta{}}{2}\right)}.}$$(7 marks) 3 (a) $$ show \ that \left(1-e^{i\theta}\right)^{-\frac{1}{2}} \left(1-e^{-i\theta}\right)^{-\frac{1}{2}}=\left(1 \mathrm{cosec} \left(\frac{\theta}{2}\right)\right)^{\frac{1}{2}} $$(7 marks) 3 (b) $$ If\ f(x)=\frac{1}{x^2}\ and\ g(x)=\frac{1}{x}\ $$ then show that c of C.M.V.T. is H.M of a & b where a>0, b>0.(6 marks) 3 (c) $$if\ y=\sin \left[\log\left(x^2+2x+1\right) \right] \ then\ prove\ that\ \\\left(x+1\right)^2y_{n+2}\left(2n+1\right)\left(x+1\right)y_{n+1}+\left(n^2+4\right)y_n=0.$$(7 marks) 4 (a) $$Solve\ x^6-x^5+x^4-x^3+x^2-x+1=0.$$(7 marks) 4 (b) If u=1x+my, v=mx-1y, prove that $$ 1]\ \left(\frac{\partial u}{\partial x}\right)_y\cdot \left(\frac{\partial x}{\partial u}\right)_V=\frac{I^2}{I^2+m^2} $$
$$ 2]\ \ \left(\frac{\partial y}{\partial v}\right)_x\cdot \left(\frac{\partial v}{\partial y}\right)_u=\frac{I^2+m^2}{I^2} $$
(6 marks) 4 (c) Show that$$ \bar{F}=\left(y^2-z^2+3yz-2x\right)i+\left(3xz+2xy\right)i+\left(3xy-2xz+2z\right)k$$ is both solenoidal and irrotational.(7 marks) 5 (a) $$prove\ that\ \tan 7\theta =\frac{7\tan\theta -35\tan^3 \theta +21\tan^5 \theta -\tan^7\theta}{1-21\tan^2\theta+35\tan^4\theta -7\tan^6 \theta} $$(7 marks) 5 (b) Test the convergence of $$\sum\frac{2^n+1}{3^n+n}$$(6 marks) 5 (c) $$If\ x=\sqrt{vw\ \ },\ y=\sqrt{wu},\ z=\sqrt{uv},\ and\ \varphi{}\ is\ a\ function\ of\ x,y,z\ \\ prove\ that\ u\frac{\partial{}\varphi{}}{\partial{}u}+v\frac{\partial{}\varphi{}}{\partial{}v}+w\frac{\partial{}\varphi{}}{\partial{}w}=x\frac{\partial{}\varphi{}}{\partial{}x}+y\frac{\partial{}\varphi{}}{\partial{}y}+z\frac{\partial{}\varphi{}}{\partial{}z}$$(7 marks) 6 (a) $$ Prove\ that \ \nabla\cdot\left\{ \frac{f(r)}{r}\bar{r} \right \}= \frac{1}{r^2} \frac{d}{dr}\ [r^2f(r)] $$
hence or otherwise prove that div $$\left(r^n\bar{r}\right)=\left(n+3\right)r^n$$(7 marks) 6 (b) $$ Show\ that\frac{b-a}{\sqrt{1-a^2}}< \sin^{-1}b-\sin^{-1}a< \frac{b-a}{\sqrt{1-b^2}}\ hence\ show\ that\ $$
$$ 1]\ \frac{\pi}{6}+\frac{\sqrt{3}}{15}<\sin^{-1}\left(\frac{3}{5}\right)<\frac{\pi}{6}+\frac{1}{8} $$
$$ 2]\ \frac{\pi}{6}+\frac{1}{2\sqrt{3}}< \sin^{-1}\left(\frac{1}{4}\right)<\frac{\pi}{6}-\frac{1}{\sqrt{15}} $$(6 marks) 6 (c) $$ If\ u= \frac{x^3y^3z^3}{x^3+y^3+z^3}+\log \left(\frac{xy+yz+zx}{x^2+y^2+z^2}\right) \ \ then\ prove\ that\\ \ X\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=6\frac{x^3y^3z^3}{x^3+y^3+z^3}$$(7 marks) 7 (a) Find the extreme values of the function $$x^3+3xy^2+72x-15x^2-15y^2$$(7 marks) 7 (b) Examine the convergence of $${\left(\frac{2^2}{1^2}-\frac{2}{1}\right)}^{-1}+{\left(\frac{3^3}{2^3}-\frac{3}{2}\right)}^{-2}+{\left(\frac{4^4}{3^4}-\frac{4}{3}\right)}^{-3}+?\ ...$$(6 marks) 7 (c) $$If\frac{{\left(a+ib\right)}^{x+1y}\ }{{\left(a-ib\right)}^{x-1y}}=a+i\beta \ then\ find\ \alpha{}\ \&\ \beta{}$$(7 marks)