Download OU B.Sc 2019 Dec 1st Sem (1st Year ) 9007 Mathematics Question Paper

Time : 3 Hours

Code No. 9007 / E

B.Sc.

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Semester (CBCS) Examination, November | December 20

Subject : Mathematics (Differential and Integral Calculus)

Paper - 1

Max. Marks: 80

PART — A (8 x 4 = 32 Marks)

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(Short Answer Type)

Note : Answer any EIGHT of the following questions.

1. If f(x, y) = y cos xy then evaluate ?f/?x
2. If f(x,y)= xⁿ+yⁿ/x+y then evaluate x?f/?x + y?f/?y
3. If u=sin^-1[x²+y²/x+y] then show that x?u/?x + y?u/?y = tan u.

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4. If H=(y-2z, z-x, x-y) then show that ?H/?x + ?H/?y + ?H/?z = 0
5. If z=x²+y², x=at², y² = 2at then evaluate dz/dt
6. Expand f(x, y) = x²+ 2xy - y² as a Taylor’s series in powers of (x—1) and (y - 2)
7. Find the radius of curvature for the curve y² = 2x at P(2, 2).
8. Find the envelope of the family of curves y = mx + a/m².

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9. Using Newton’s method, find the radius of curvature for the curve x³ + y³ — 2x² + 3y = 0 at the origin O(0, 0).
10. Find the length of the curve y = x^3/2 from x = 0 to x =4.
11. Find the length of the curve x = a sin t, y = a cos t from t = 0 to t = p/2
12. Find the volume of the region generated by revolving the curve y = cosx , y = 0 from x=0 to x= p/2 about x-axis.
13. (a) If u=tan^-1[x³+y³/x-y], x=r cos ?, y=r sin ? then show that (i) x ?u/?x + y ?u/?y = sin 2u (ii) ?²u/?r² + 1/r ?u/?r + 1/r² ?²u/?² = (1=4sin² u)sin 2u

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PART - B (4 x 12 = 48 Marks)

(Essay Answer Type)

Note: Answer ALL from the questions.

13. (a) If u=tan^-1[x³+y³/x-y], x=r cos ?, y=r sin ? then show that (i) x ?u/?x + y ?u/?y = sin 2u (ii) ?²u/?r² + 1/r ?u/?r + 1/r² ?²u/?² = (1=4sin² u)sin 2u FirstRanker.com
14. (B) If f(x, y) = x^1/4+y^1/4/x^1/5+y^1/5 then using Eulers theorem show that x?f/?x + y?f/?y =f/20 FirstRanker.com

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15. (a) If f(x, y) possesses continuous second order partial derivatives f_xy and f_yx, then show that f_xy = f_yx

    OR

    (b) Show that the minimum value of u(x,y)=xy+a³/x+a³/y is 3a².
16. (a) Find the evolute of the hyperbola x²/a² - y²/b² = 1

    OR

    (b) Find the envelope of the curve lx/a + my/b = 1 where a²l² +b²m²=c².
17. (a) Show that the length of the curve x=t (1-t²/3), y=t²/2 measured form O(0, 0) to P(x, y) is s=t²+x²

    OR

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    (b) Find the volume of the solid obtained by revolving one arc of the cycloid x=a(?+sin?), y=a(1+cos?) about X — axis