B.E. APPLIED MATHEMATICS- III-Mumbai-December 2017 - Grad Plus

B.E. APPLIED MATHEMATICS- III-Mumbai-December 2017

Semester: 3

[Total Time: 3hours]
[Total marks: 80]
N.B. 1) Question No. 1 is compulsory.
2) Answer any Three from remaining
3) Figures to the right indicate full marks.
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Q. 1) a) Find Laplace transform of f(t) = te-3t sint. (5M)

b) Obtain Complex form of Fourier series of f (x)= ex , -1< x< 1 in (-1, 1).  (5M)

c) Does there exist an analytic function whose real part is u = k (1+ cos Θ )? Give justification. (5M)

d) The equations of lines of regression are 3x + 2y = 26 and 6x + y = 31. Find i) means of x and y, ii) coefficient of correlation between x and y. (5M)

Q. 2) a) Evaluate \int_0^\infty e^t\sin\;2t\;\cos\;\;3t\;dt. (6M)

b Find the image of the square bounded by lines x = 0,x=2, y = 0, y = 2 in the z-plane under the transformation w= (1 + I) z +2 -i. (6M)

c) Obtain Fourier series of f(x) = |x|  in (-π, π). Hence, deduce that –
\frac{\pi^2}8=\frac1{1^2}+\frac1{3^2}+\frac1{5^2}+........

Q. 3 a) Find the inverse Laplace transform of F(s)=\frac s{\left(s^2+9\right)\left(s^2+4\right)}  (6M)

b) Solve \frac{d^2u}{dx^2}-100\;\frac{du}{dt}=0  with u(0, t) =0 u(1, t) = 0, u(x, 0) = x (1 – x) taking h = 0.1 for three time steps up to t = 1.5 by Bender-Schmidt method. (6M)

c) Using Residue theorem, evaluate  (8M)

i) \int_0^{2\mathrm\pi}\frac{d\theta}{5+4\;\cos\theta}

ii) \int_{-\infty}^\infty\frac{dx}{\left(x^2+1\right)^2}

Q. 4) a) Solve hy Crank Nicholson simplified formula \frac{d^2u}{dx^2}-\frac{du}{dt}=0 u(0.1) = 0, u(5,t) = 100, u(x, 0) = 20 taking h= 1 for one time step. (6M)

b) Obtain the Taylor’s and Laurent series which represent the function. (6M)

f(z)=\frac z{\left(z-1\right))z-2)}  in the region, I |z| < 1 ii 1< |z| , 2

c) Solve (D2 – 3D + 2)y = 4e2t with y) = -3.y’ (0) = 5 where D\equiv\;\frac d{dt} (8M)

Q. 5) a) Find an analytic function f(x) = u + iv, if u= e-x{(x2-y2) cos y + 2 xy sin y } . (6M)

b) Find the Laplace transform of f(t)=t\;\sqrt{1+\;\sin\;t} (6M)

c) Obtain half range Fourier cosine series of f(x)=x 0< x < 2. Using Parseval’s identity, deduce that –

\frac{\pi^4}{96}=\frac1{1^4}+\frac1{3^4}+\frac1{5\left(4\right)}+....... (8M)

Q. 6) a) If f(a)=\oint\frac{3z^2\;+\;7z\;+\;1}{z-a}dz (6M)

C : x2 + y2 = 4 find the values of f (3).f’ (1-i) and f”(1-i)

b) Find the coefficient of correlation between height of father and height of son from the following data, (6M)

 Height of father 65 66 67 68 69 71 73 Height of Son 67 68 64 68 72 69 702

c) A lightly stretched string with fixed endpoints x = 0 and x = 1, in the shape defined by y = kx (1-x) where k is a constant is released from this position of test Find y(x, t), the vertical displacement if \frac{d^2y}{dt^2}=\;C^{2\;}\frac{d^2y}{dx2}   (8M)
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