MATH102 Exam 2014 PDF

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LANCASTER UNIVERSITY 2014 EXAMINATIONS PART I MATHEMATICS & STATISTICS Math 102: Integration

1 hour

You should answer ALL questions. Please use a separate booklet for each module. There is a total of 100 marks.

1.

(a) Determine all of the roots of the equation (z + 1)6 = 64, and sketch them on an Argand diagram.

[8]

(b) How many of the roots in (a) are purely imaginary, i.e. have ℜ(z) = 0?

2.

[2]

Evaluate the following integrals: (a) Z

I=

∞ 0

3 dx (x + 1)(2x2 + 1)

[18]

(b) J=

Z

2 1

p

[10]

x2 − 1 dx.

3. Determine which of the following statements is true and which is false. Briefly justify your answers. (a) If α is a complex root of the real polynomial f (x), then so is its conjugate α.

[2]

(b) The Laplace transform F (s) of ex is defined for all s > 0.

[2]

2

2

√ 5 . 3

(c) The eccentricity ε of the ellipse x 9 + y4 = 1 is R∞ R (d) If f(x) ≤ g (x) for all x and 0 g(x) dx converges, then so does 0∞ f(x) dx. (e) f(x, y) = e−x cos y is a solution of the differential equation fxx + fyy = 0.

please turn over 1

[2] [2] [2]

MATH102 Integration continued

4.

(a) (i) Show that the point (x, y), where x = 1 − 3t2 and y = t(3t2 − 1), lies on the curve 3y2 = x2 (1 − x).

[4]

(ii) Hence deduce that the length along the curve from (1, 0) to (−2, 2) is 4.

[10]

(b) Determine the equation of the tangent and normal lines to the curve y2 − x3 = 1 at the point (2, 3).

5.

[10]

(a) Find and classify the stationary points of the function f (x, y) =

x3 3

− xy2 + 56 y3 − y.

[16]

(b) Sketch the region T in the plane bounded by the straight lines x = 0, y = 1 − 2x and y = 0, and evaluate the integral Z Z

(xy + x2 ) dx dy.

T

[12]

end of exam

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