Download PTU B.Tech 2021 Jan CSE 3rd Sem 76438 Mathematics Iii Question Paper

Roll No. Total No. of Pages : 03

Total No. of Questions: 18

B.Tech. (CSE) (2018 Batch) (Sem.-3)

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MATHEMATICS-III

Subject Code: BTAM304-18

M.Code: 76438

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 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.

SECTION-A

Solve the following :

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1. Show that the limit for the function f(x, y) = (x² + y²) / (x² + y²) does not exists as (x, y) ? (0, 0).
2. Evaluate the integral ? x*y*z dx dy dz.
3. Check the convergence of the following sequences whose nth term is given by a_n = ((3n+1)ⁿ) / ((3n-1)ⁿ)
4. State Cauchy Integral test for convergence of a positive term infinite series.
5. Write down the Taylor's series expansion for sin x about x = p/2

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6. Solve by reducing into Clairaut's equation : p = log(px-y), where p= dy/dx.
7. Solve the differential equation dy/dx + ycotx = xcosecx
8. Determine whether the differential equation is exact (x²+y² +2x)dx + 2ydy = 0

9. Solve the differential equation d²y/dx² + dy/dx + y = 0
10. Find Particular integral for d²y/dx² - 2dy/dx + y = e^-x

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SECTION-B

11. Using Method of Lagrange Multipliers, find the maximum and minimum distance of the point (3, 4, 12) from the sphere x² + y² + z² = 1.
12. Solve by changing order of integration: ? a/(x² + y²) dxdy, a is any positive constant.
13. For what value(s) of x does the series converge (i) conditionally (ii) absolutely? x - x²/v2 + x³/v3 - x4/2 + ... to 8 . Also find the interval of convergence.
14. Solve the differential equation: (xy³ + y)dx + 2(x²y² + x + y4)dy = 0

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15. Solve the differential equation d²y/dx² + 2y = xe^3x + sin 2x.

SECTION-C

16. a) Check the convergence of the series S (v(n+1) - vn) / n^3/2 from n=2 to 8.
17. b) Find by double integration, the area lying inside the circle r = a sin ? and outside the cardioid r = a (1 – cos ?).

17. a) Solve the differential equation dy/dx + x/(1-x²) y = xvy.

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18. b) Solve the differential xyp² - (x² + y²) p + xy = 0, where p = dy/dx

18. a) Solve by Method of Variation of parameters d²y/dx² + y = sec x.
19. b) Solve (1+x)² d²y/dx² + (1+x) dy/dx + y= cosln(1+x).

NOTE: Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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