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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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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