Download PTU B.Tech 2021 Jan IT 3rd Sem 76393 Mathematics Iii Question Paper

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Roll No. _________________________ Total No. of Pages : 02

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Total No. of Questions : 18

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

MATHEMATICS-III

Subject Code : BTAM-304-18

M.Code : 76393

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Time : 3 Hrs. Max. Marks : 60

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

Write brief:

1. Express ?w/?r and ?w/?s in terms of r and s if w = x + 2y + z², x = r² + s², y = r² - s², z = 2rs
2. Show that the function f(x, y) =

   2xy x2 + y2	(x, y) ? (0, 0)
   0	(x, y) = (0, 0)

   is not continuous at origin.
3. Find the local extreme values of the function f(x, y) = x² + y² + 3x² - 3y² - 8
4. Define convergence of a sequence and give an example of a convergent sequence.

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5. State Leibniz's test for alternating series.
6. Determine for what values of a and b, the following differential equation is exact. (y + x³)dx + (ax + by³)dy = 0
7. Find the integrating factor for the following differential equation (5x³ + 12x² + 6y²)dx + 6xydy = 0

8. Give an example of a fourth order linear differential equation.
9. Find the solution of the differential equation y" - 4y' - 12y = 0

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10. If y₁ and y₂ are two linearly independent solutions of a second order linear differential equation, then what can you say about the general solution of this equation. Justify your answer.

SECTION-B

11. Find the volume of the region in the first octant bounded by the coordinate planes and the planes x + z - 1 and y + 2z = 2.
12. For what values of x does the following power series converge? ?_n=1⁸ (-1)^n-1 x^2n-1}{2n-1}
13. Obtain the general solution and singular solution of the non-linear equation y = xy' - (y')³.
14. Solve the differential equation y" + 16y = 32 sec 2x by using method of variation of parameters.

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15. If ?_n=1⁸ |a_n| converges, then show that ?_n=1⁸ a_n also converges. Is the converse also true? Justify your answer.

SECTION-C

16. Find the extreme values of the function f(x, y, z) = x² + y² + z² subject to the constraints x² + y² - 1 = 0 and x + y + z = 1.
17. Test the convergence of the series (i) ?_n=1⁸ (-1)ⁿ ln n}{n - ln n} (ii) ?_n=1⁸ 8 tan^-1 n}{1 + n²}
18. Find the general solution of the equation y" - 4y' + 13y = 18e^2x sin 3x.

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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