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

# Engineering Mathematics-II-Engineering-Amravati-Sum 2015

## Engineering Mathematics-II

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.

SECTION-A

1.(a)  Find inverse of a matrix by partitioning method : [06M]

\begin{bmatrix}1 & 2&3 & 1\\1 & 3 & 3 &2 \\2& 4 & 3& 3\\1& 1& 1 & 1\end{bmatrix}

(b) Find eigen values and eigen vectors of : [07M]

\begin{bmatrix}6 & -2 & 2\\ -2 & 3& -1\\ 2& -1& 3\end{bmatrix}

OR

2. (a) Find the values of a and b for which the equations :  [06M]
x + ay + z = 3
x 2y + 2z = b
x + 5y + 3z = 9 are consistent.

(b) Verify Cayley-Hamilton theorem and find inverse of the matrix : [07M]

A=\begin{bmatrix}1 & 3 & 7\\ 4& 2 &3 \\ 1 & 2 & 1\end{bmatrix}

3. (a) Find the Fourier series for the function : [06M]
f(x) = -1      for -π < x < -π/2
= 0       for -π/2 < x < π/2
= 1       for π/2 < x < π

(b) Find the half range cosine series expansion of the function : [07M]

f(x)= 0, 0 ≤ x ≤ l/2
= l,   l/2 ≤ x ≤ l

OR

4. (a) Expand f(x) =2x -x2 in (0, 3); in terms of Fourier series and hence show that : [06M]

\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}....=\frac{\pi ^{2}}{12}

(b) Compute the first two harmonics of the Fourier series of f(x) given in the following table :  [07M]
x             f(x)
0            1.0
π/3        1.4
2π/3      1.9
π         1.7
4π/3      1.5
5π/3      1.2
2π         1.0

5. (a) Find a such that the vectors 2i – j + k ; i + 2j – 3k and 3i + aj + 5k are coplanar. [04M]

(b) Use rule of differentiation under integral sign : [06M]

\int_{0}^{a}\frac{log(1+ax)}{1+x^{2}}dx and hence show that

\int_{0}^{1}\frac{log(1+x)}{1+x^{2}}dx=\frac{\pi }{8}.\; log2

(c) Trace the curve r = a(1 – cos θ). [04M]

OR

6. (a)  Prove that :

\overline{a}\times [\overline{b}\times (\overline{c}\times\overline{d})]\; = \overline{b}.\overline{d}(\overline{a}\times\overline{c})-\overline{b}.\overline{c}(\overline{a}\times\overline{d}) [04M]

(b) Prove that :  \int_{0}^{1}\frac{x^{a}-1}{log x}dx=log (1+a)

(a>1). [05M]

(c) Trace the curve : [05M]
9ay2 = x(x – 3a)2 .

SECTION—B

7. (a) Find a reduction formula for   \int e^{ax}\; sin^{n}x dx . Hence evaluate \int e^{x}\; sin^{3}x dx  [05M]

(b) Evaluate : \int_{0}^{\infty }\frac{x^{m-1}}{(a+bx)^{m+n}}dx [04M]

(c) Prove that the length of the curve ay2 = x3 from the vertex to the point whose abscissa is ‘b’ is

\frac{1}{27\sqrt{a}}(9b-4a)3/2-\frac{8a}{27}  [05M]

OR

8. (a) If   I_{n}=\int x^{n}e^{x}dx  show that : I_{n}+nI_{n-1}=x^{n}e^{x}  Hence find I4. [04M]

(b) Evaluate \int_{0}^{\pi /2}\frac{sin^{2m-1}\theta.Cos^{2n-1}\theta}{(a cos^{2}\theta +bsin^{2}\theta)^{m+n}}d\theta [05M]

(c) Find the length of the cycloid :
x = a(θ+sinθ), y=a(1-cosθ) between two cusp.  [05M]

9. (a) Change the order of integration : [06M]

\int_{0}^{1}\int_{x^{2}}^{\sqrt{2-x^{2}}}f(x,y)dxdy

(b) Find by double integration the area lying inside the circle r = a sin θ and outside the cardiod r = a(1- cos θ). [07M]

OR

10. (a) Evaluate by transforming to polar co-ordinates :

\int_{0}^{2a}\int_{0}^{\sqrt{2ax-x^{2}}}(x^{2}+ y^{2})dxdy  [07M]

(b) Evaluate : \int \int \sqrt{xy(1-x-y)}dxdy  in the area bounded by x = 0, y = 0 and x+y = 1. [06M]

11. (a) Prove that : [06M]

\int_{0}^{\pi /2}\int_{0}^{asin\theta }\int_{0}^{\frac{a^{2}-r^{2}}{a}}rdrd\theta d\phi = \frac{5\pi a^{3}}{64}

(b) Find the volume bounded by cylinder x2 + y2 = 4 and the plane y + z = 3 and z = 0. [07M]

OR

12. (a) Change to spherical polar co-ordinates and evaluate :
\int_{0}^{a}\int_{0}^{\sqrt{a^{2}-x^{2}}}\int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}}}(x^{2}+y^{2}+z^{2})dxdydz [07M]

(b) Obtain the root mean square value of f(t)=3sin 2t+4cos2t over the range 0≤t≤π. [06M]

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