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1 Lesson 2: Graphing And Loci Complex Loci: 1. Sketch the locus of the points in an Argand diagram representing the complex number z where 1 2 1 2 z i i −+ = . [2] 2. [Assignment] Sketch the locus of the points in an Argand diagram representing the complex number z where 2 arg 0 1 iz i = + . [3]

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Lesson 2: Graphing And Loci Complex Loci:

1. Sketch the locus of the points in an Argand diagram representing the complex number z where 1 2 1 2z i i− + = − . [2]

2. [Assignment] Sketch the locus of the points in an Argand diagram representing the complex

number z where 2

arg 01iz

i⎛ ⎞

=⎜ ⎟+⎝ ⎠. [3]

2

3. Shade, on a single Argand diagram, the region representing the complex number w for which 1w w i≥ + − and |w + 1 − i| ≤ 2 . [3]

Hence find (a) the maximum value of 3w+ in exact form, [2] (b) the maximum value of arg( 3)w+ , giving your answer to 3 significant figures. [2]

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4. Find the complex number w such that arg( )3

w π= and |w −2 −2 i| = 5. [2]

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5. [2008 (b)] The complex number z satisfies the relation |z| ≤ 6 and |z| = |z – 8 – 6i|. (i) Illustrate both of these relations on a single Argand diagram. [3] (ii) Find the greatest and least possible values of arg z, giving your answers in

radians correct to 3 decimal places. [4]

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6. [2009 modified] (i) Solve the equation 5 1 0z i− + = , giving the roots in the form of ire α , where 0r > and π α π− < ≤ . [3]

(ii) Show all the roots on an Argand diagram. [3]

(iii) The roots represented by 1z and 2z are such that ( ) ( )1 2arg arg 02

z zπ− < < < . Explain

why the locus of all points z such that |z−z1| = |z−z2| passes through the origin. Draw this locus on your Argand diagram and find its exact Cartesian equation. [5]

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Graphing NOTE: For any graphs drawn, you must label the axial intercepts and asymptotes, if any.  1 Sketch the curves on separate diagrams:

(a) 2

45

xyx+

=−

[3]

(b) 2

1x xy

xλ−

=+

(Sketch 3 distinct members of the family) [4]

By differentiation, find the range of values of λ such that the graph 2

1x xy

xλ−

=+

has no

stationary points. [4]

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2. Sketch (a) 2 2 2 2 212 13 13x y+ = (b) 2 2 4y x− = (c) 2 24 16 20x y y− − = .

Label all intersections with axes and asymptotes if any.

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3. Prove, using an algebraic method, that the curve 3 6( 6)xy

x x−

=+

cannot lie between two

certain values of y (which is to be determined). [4]

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4. The graph, 3 22xy y x+ = undergoes transformations in the following sequence:

(A) Reflect graph in the y-axis, (B) Scaling of graph by a factor of 3 parallel to the y-axis.

State the equation of the resulting curve. [2] 5. Describe a sequence of transformations geometrically to obtain y = f(2x −1) from

y = f(x). [2]

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6. Two different graphs of y = h(x) is as shown below.

0

A(2,4)

y = 2

1

y

x x

(−4, ½)

y = − ¼

x = −2

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7(a) Sketch the graph given by the parametric equations 3cosx t= and 3siny t= . You should label the points of intersection with the axes. [2] Find the exact y-ordinate of the point where x = 1 / 8. [2]

(b) Find the Cartesian equation of the curve given by 13 tan and cos2

x t y t= = . [2]