Engineering Mathematics-I-Engineering-Amravati-Sum 2015 - Grad Plus

# Engineering Mathematics-I-Engineering-Amravati-Sum 2015

## Engineering Mathematics-I

Marks: 80

[Time : 3 hours]

Section A and B

INSTRUCTIONS TO CANDIDATES
(1) All questions carry marks as indicated.
(2) Answer THREE questions from Section A and THREE questions from Section B.
(3) Assume suitable data wherever necessary.
(4)  Use of calculators is permitted.
(5) Figures to the right indicate full marks.
(6) Use pen of Blue/Black ink/refill only for writing the answer book.
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SECTION—A

1. (a) Find the nth derivative of    \frac{1}{x^{2}+9}    [05M]

(b) Apply Taylor’s- theorem to express (x-2)4-3(x-2)3+4(x-2)2+5 in powers of x. [05M]

(c) Evaluate : \lim_{x\rightarrow 0}\frac{e^{2x}-(1+x)^{2})}{x log (1+x)} [04 M]

OR

2. (a) Expand log sec x in powers of x upto x4. [05M]

(b) If y=(x+\sqrt{x^{2}+a^{2}})^{m}  . Prove that : (x2+a2)yn+2+(2n+1)x yn+1+(n2-m2)yn=0. [05M]

(c) Find : \lim_{x\rightarrow \frac{x}{2}}(sec\; x)^{cot \; x})  [04M]

3. (a) Verify Euler’s theorem for the function:

u=sin^{-1}\frac{x}{y}+tan^{-1}\frac{y}{x} [06M]

(b) If z=f(x,y) where x=uv, y=\frac{u+v}{u-v} . Prove that :

2x\frac{\partial z}{\partial x}=u\frac{\partial z}{\partial u}+v\frac{\partial z}{\partial v}   [07M]

OR

4. (a) If u=log(x^{2}+y^{2})+tan^{-1}\frac{y}{x} , find the value of \frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial y^{2}} [06M]

(b) If z=\frac{(x^{2}+y^{2})}{\sqrt{(x+y)}}. Show that: x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=\frac{3z}{2} [07M]

5. (a) If u=\frac{y-x}{1+xy} \; and \; v=tan^{-1}y-tan^{-1}x, find \frac{\partial (u,v)}{\partial (x,y)}  [06M]

(b) Show that the stationary value of u = xm • yn zp, where x+y+z=a is

m^{m}.n^{n} p^{p}\left ( \frac{a}{m+n+p} \right )^{m+n+p}  [07M]

OR

6. (a) Verify whether the following functions : u=\frac{x-y}{x+y}\; v=\frac{xy}{(x+y)^{2}} are functionally dependent, and if so, find the relation between u and v. [06M]

b. Find the maximum and minimum values of u=2+2x+2y-x2-y2[07M]

SECTION-B

7. (a) If n is positive integer, prove that:   [04M]

(\sqrt{3}+i)^{n}+ (\sqrt{3}-i)^{n}=2^{n+1}\; cos\frac{n\pi }{6}

(b) Separate real and imaginary parts of tan-1 (α+iβ)  [05M]

(c) Prove that the real part of the principal value of ilog(1+i) is :
e^{-\pi ^{2}/8}cos\left ( \frac{\pi }{4} log \; 2\right )  [05M]

OR

8. (a) Prove that : (cosh x + sinh x)n = cosh nx + sinh nx.   [04M]

(b) Prove that : sin h^{-1}x= cosh^{-1} (\sqrt{1+x^{2}}) [05M]

(c) If ii….∞= A+iB, Prove that :     [05M]

(i) tan \frac{A\pi }{2}=\frac{B}{A}  and

(ii) A2+B2=e-Bπ

9. Solve the following differential equations:

(a) \frac{dy}{dx}+2xy=2e^{-x^{2}}     [04M]

(b) cos x. \frac{dy}{dx}+y\; sin x= \sqrt{y \; sec x}    [04M]

(c) y log y dx+ (x-log y)dy=0  [05M]

OR

10. Solve the following differential equations:

(a) y-x\frac{dy}{dx}=a\left ( y^{2}+\frac{dy}{dx} \right )  [04M]

(b) \left [ y\left ( 1+\frac{1}{x}\right )+cos y \right ]dx + [x+log x- sin y]dy=0 [04M]

(c) tan y \frac{dy}{dx}+\; tan x= cos y \; cos^{2} x

11. (a) Solve the following differential equations :

(i) p-\frac{1}{p}=\frac{x}{y}-\frac{y}{x} [04M]

(ii) y+px=x4 p2   [04M]

(b) Find the orthogonal trajectories of x2+y2=2ax. [05M]

OR

12. (a) Solve the following differential equations .

(Ii) y=2px+y2p3     [04M]

(ii) e4x (p-1) +e2yp2=0   [04M]

(b) A resistance of 100 ohms, an inductance henries are connected in series with battery of 20 volts. Find the current in the circuit as a function of time. [05M]

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