Download PTU B-Tech CSE-IT 2020 Dec 3rd Sem 76438 Mathematics Iii Question Paper

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

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SECTION-A

Solve the following :

1. Show that the limit for the function f(x, y) = ^2x-y/_x+y does not exist as (x, y) ? (0,0).
2. Evaluate the integral ?₀¹ ?₀^x dydx
3. Check the convergence of the following sequences whose nth term is given by a_n= n/(n²+1)

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4. State Leibnitz test for convergence of an alternating series S (-1)ⁿ⁺¹ a_n
5. Write down the Taylor's series expansion for cos x about x=p/2.
6. Solve by reducing into Clairaut's equation: y = px + p², where p = dy/dx
7. Solve the differential equation dy/dx + y=x
8. Determine whether the differential equation is exact, if found exact solve it. (x² + y²) dx + 2xydy =0

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9. Solve the differential equation (16d²y/dx²) —8(dy/dx) +5y=0
10. Find Particular solution of the differential equation : (d²y/dx²) - 3(dy/dx) + 2y = x²

SECTION-B

1. Find the maximum and minimum distance of the point (1, 2, —1) from the sphere X²+ y² + z²=24.
2. Evaluate ?_D e^(x2+y2) dydx where D is the region bounded x²+ y² = 1

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3. For what value(s) of x does the series converge (i) conditionally (ii) absolutely? x - x²/2 + x³/3 - x⁴/4 + ... Also find the interval of convergence.
4. Solve the differential equation by finding integrating factor (xy+1) ydx+x(1+xy+ x²y²)dy =0
5. Solve the differential equation (d²y/dx²) - 3(dy/dx) +2y= xe^2x +sin 2x

SECTION-C

1. a) Show that the series S 1/n^p converges for p > 1 and diverges for 0 < p = 1.
   

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   b) Using double integration, find the area bounded between the parabolas y² = 4ax and x²=4ay.
2. a) Solve the Bernoulli’s equation dy/dx + y/x = x²y⁶
   b) Solve the differential equation xp²— 2yp +x =0, where p = dy/dx
3. a) Solve by Method of Variation of parameters (d²y/dx²) - 4(dy/dx)+4y= e^2x/x²
   b) Find the complete solution of (d²y/dx²) - 5(dy/dx)+ 6y = e^x sin2x

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