Complex Analysis and Calculus Problem Solutions, Exams of Mathematics

Solutions to various problems involving complex analysis and calculus, including finding arguments and loci of complex numbers, expressing complex numbers in rectangular and polar forms, finding derivatives, integrating functions, and determining maximum and minimum turning points of functions.

Typology: Exams

2012/2013

Uploaded on 04/12/2013

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1. (a) What are
z; jzj;arg (z);
when z= 1 j:
[1, 1, 1]
(b) Determine
z1z2;z2
z1
;
when z1= 6 + 8jand z2= 2 j:
[2, 3]
(c) Sketch on separate Argand diagrams the locus of points satisfying
jzj 2;arg (z) = 3
4:
[3, 3]
(d) Express z= 4ej=6in rectangular form.
[2]
(e) If
z1= 1 + j; z2=p3j;
express z1
z2
;
in polar form.
[4]
1
pf3

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  1. (a) What are z;  jzj ; arg (z) ; when z = 1 j: [1, 1, 1] (b) Determine z 1 z 2 ;^ z z^21 ; when z 1 = 6 + 8j and z 2 = 2 j: [2, 3] (c) Sketch on separate Argand diagrams the locus of points satisfying jzj  2 ; arg (z) =^34  : (d) Express z = 4ej=^6 in rectangular form.^ [3, 3] (e) If^ [2] z 1 = 1 + j; z 2 = p 3 j; express (^) z 1 in polar form.^ z^2 ; [4]
  1. (a) Find d dyx in each of the following cases: ii.i.^ yy^ = 4= sinx^3 x ^ xp^5 ;x; iii. iv. yy = e= cosx^ cos x x; x : [2, 2, 2, 2] (b) i. Find the coordinates of the maximum and minimum turning points of thefunction f (x) = x^3 3 x 2 2 6 x + 3: ii. Hence, sketch the graph of the function f (x):^ [4] (c) The distance (measured in metres) that a particle, dropped into a viscous áuid,^ [2] travels in t seconds is given by f (t) = 100^  1 (^) t + 1^1  : Determine i. the distance covered by the particle in the Örst three seconds of travel, [1] ii. the speed of the particle at the instant when t = 4, [3] iii. the maximum distance traversed by the dropping particle, [1] iv. the time when the speed of the particle is 25 m/s. [1]