MATH102 Exam 2013 PDF

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LANCASTER UNIVERSITY 2013 EXAMINATIONS PART I MATHEMATICS & STATISTICS Math 102: Integration

1 hour

You should answer ALL questions. Please use a separate booklet for each module. There are 100 marks in total. 1.

(a) Find the roots of the equation (z − 1 + 2i)3 = 1 and sketch them in the Argand plane.

(b) On the same diagram, mark the points 2.

(a) i) Solve the indefinite integral: I=

Z

√

πi

2e− 4 and e

3πi 2

.

[4]

x+2 dx x(x2 + 2)

[12]

ii) Does the following definite integral converge? Justify your answer. Z ∞ x+2 dx √ x(x2 + 2) 2 iii) Does the integral

R∞ 0

x+2 x(x2 +2)

[8]

[6]

dx converge?

[4]

(b) Find the Laplace transform of sin 3x.

[12]

3. Determine which of the following statements is true and which is false. Briefly justify your answers. (a) If f (R) → 0 as R → ∞, then (b) If g(x, y) satisfies

∂g ∂x

R∞ 0

f (x) dx converges.

[2]

= 0 for all x then g(x, y) is constant.

[2]

(c) Two complex numbers z, w satisfy |z − w| ≥ |z| − |w|.

[2]

(d) If z, w are complex numbers and |z| > |w| then ℜ(z) > ℜ(w).

[2] 2

∂ u− (e) u(x, y) = cosh(x + y) is a solution of the differential equation 2∂x 2

∂2u ∂y 2

− u = 0.

please turn over 1

[2]

MATH102 Integration continued

4.

(a) (i) Let T be the region in the plane which is bounded by the lines x = 0, y = 0 and y = 1 − 2x. Sketch T , and calculate the integral Z Z I= xy dx dy.

[10]

(ii) Interpret the integral I as the volume of a body, and describe the body.

[2]

T

(b) Let C be the logarithmic spiral defined in polar coordinates by r = e2θ , 0 ≤ θ ≤

2π. Express the Cartesian coordinates of a point on C in terms of θ, and hence

calculate the length of C.

5.

[10]

(a) Find and classify the stationary points of the function f (x, y) = x3 + y 3 − 2xy.

[14]

(b) Show that the point P = (2, 1) is on the curve C defined by f (x, y) = 5, and find the tangent line to C at P .

[8]

end of exam

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