Download PTU B.Tech 2020 March ME 1st and 2nd Sem Mathematics II Question Paper

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Roll No. [ ] Total No. of Pages : 02
Total No. of Questions : 18
B.Tech. (Mechnical Engg/Automobile Engg./
Civil Engg./CSE/ECE/Electrical & Electronics Engg.) (2018 & onwards)
(Sem.-2)

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MATHEMATICS-II
Subject Code : BTAM-203-18
M.Code : 76256
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :

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1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION - B & C have FOUR questions each.
Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.
SECTION-A

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Answer briefly :
1) Define Bernoulli’s equation with an example.
2) Solve: p²—7p+12=0.
3) Solve: (ycosx + 1) dx + sin xdy=0.
4) Write Clairaut’s equation with example.

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5) What is the significance of integrating factor.
6) Check the analyticity of log z, where z =x + iy.
7) Define conformal mapping.
8) Expand f(z)= 1/z about z=-2.
9) State Cauchy Integral formula.

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10) Evaluate, ? e^z/(z+1)² dz along the circle C: |z—3 |=3.
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SECTION-B

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11) a) Find the power series solution about the origin of the equation (1-x²)y'' -2xy' +6y=0
b) Solve (2x log x —xy) dy + 2ydx = 0.
12) a) Solve y e^x/ydx = (y³ + 2xe^x/y) dy.
b) Solve : (x² + 2x²y²) dx + (y² — x³y³)dy = 0.
13) Solve by method of variation of parameters : (D²+2D+ 1)y = e^-x log x.

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14) Solve : x² d²y/dx² +3x dy/dx + y = x² log x SECTION-C
15) a) Show that function f (z) defined by f (z) = x³/(x²+y²) z?0, f(0) =0, is not analytic at the origin even though it satisfies C-R equations.
b) Find the bilinear transformation that map the points z =1, i, —1 into the points w =i, 0, —i.
16) a) Determine the analytic function whose real part is e^x (x cos 2y — y sin 2y).
b) Prove that u = e^x sin (x cos y) is harmonic. Find a function v such that f (z) = u + iv is analytic. Also express f(z) in terms of z.

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17) a) Use the concept of residues to evaluate ?₀^2p dx/(5—4sinx)
b) Evaluate ?_C z² dz/(z⁵+2z+1) along the circle C:|z+1—i|=2.
18) Expand f(z) = 1/((z+1)(z+4)) in the following given regions :
a) |z|<1, b) 1<|z|<4, c) |z|>4.
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