Download OU B.Sc 2018 Dec 1st Year 3006 Differential Calculus Question Paper

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FACULTY OF SCIENCE

B.Sc. I-Semester (CBCS) Examination, December 2018

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Subject : Mathematics

Paper—I : Differential Calculus

Time : 3 Hours Max. Marks : 80

PART—A (Short Answer Type)

Note : Answer any FIVE of the following questions

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1. Expand y = log (sec x + tan x) as a Maclaurin series.
2. Verify Rolle's mean value theorem for the function f(x) = e^x (sin x - cos x) in [p/4, 5p/4].
3. Evaluate lim_x?0 (cosx)^1/x2.
4. Find the radius of curvature of the curve x = a(? - sin?), y = a(1 - cos?) at ?.
5. If u = sin^-1(xy), x = e^t, y = t³ find du/dt.

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6. If z = f(x-ay) + g(x+ay) then find ?²z/?y².

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Code No. 3006/E

7. If x = r cos ?, y = r sin ?, find ?(x,y)/?(r,?).

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8. Find the envelope of the family of straight lines y = mx + a/m, where m is a parameter.
9. If x^xy^yz^z = C, show that at x = y = z, ?²z/?x?y = -(x log ex)^-1.
10. (a) Show that the evolute of the hyperbola x²/a² - y²/b² = 1 is (ax)^2/3 - (by)^2/3 = (a² + b²)^2/3.
    OR
    (b) Find the circle of curvature at the point P(t², 2t) of the curve y² = 4x.

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11. (a) Expand f(x, y) = x³ + y³ - 2x²y + 1 as a Taylor series around P(1,2).
    (i) If x + y + z - sinxyz = 1 then evaluate ?z/?x at P(0, 1).
    OR
    (b) (i) If u = log(x³ + y³ + z³ - 3xyz) then evaluate x(?u/?x) + y(?u/?y) + z(?u/?z).
    (ii) If f(x, y) = (x³ + y³)/(x-y), x ? y and 0, if x = y ? 0. Show that lim_(x,y)?(0,0) f(x,y) = 0.

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12. (a) Find the minimum value of x² + y² + z² when xyz = a³.
    OR
    (b) Find the extreme values of f(x, y) = x³ + y³ - 3axy.

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