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*P42955A0232*
1. A transformation T from the z-plane to the w-plane is given by
w zz
= + 2ii
z����
The transformation maps points on the real axis in the z-plane onto a line in the w-plane.
Find an equation of this line.(4)
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PhysicsAndMathsTutor.com June 2013 (R)FP2 COMPLEX-NUMBER LOCI PAST-PAPER QUESTIONS
1
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*P44517A0832*
4. A transformation from the z-plane to the w-plane is given by
w zz
z=+
≠ −3
3,
Under this transformation, the circle z = 2 in the z-plane is mapped onto a circle C in the w-plane.
Determine the centre and the radius of the circle C.(7)
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FP2 JUNE 2014 "IAL" PAPER2
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*M35144A01628*
6. A transformation T from the z-plane to the w-plane is given by
wz
zz=
+≠ −
ii,
The circle with equation |z | = 3 is mapped by T onto the curve C.
(a) Show that C is a circle and find its centre and radius.(8)
The region |z | < 3 in the z-plane is mapped by T onto the region R in the w-plane.
(b) Shade the region R on an Argand diagram.(2)
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PhysicsAndMathsTutor.com June 2009
June 2009
FP2 COMPLEX LOCI PAST-PAPER QUESTIONS
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*P35413A01228*
5. The point P represents the complex number z on an Argand diagram, where
The locus of P as z varies is the curve C.
(a) Find a cartesian equation of C.(2)
(b) Sketch the curve C. (2)
A transformation T from the z-plane to the w-plane is given by
The point Q is mapped by T onto the point R. Given that R lies on the real axis,
(c) show that Q lies on C.(5)
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PhysicsAndMathsTutor.com June 2011
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*P44517A02032*
7. The point P represents a complex number z on an Argand diagram, where
z z+ = −1 2 1
and the point Q represents a complex number w on the Argand diagram, where
w w= − +1 i
Find the exact coordinates of the points where the locus of P intersects the locus of Q.(7)
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June 2015
6
4. A complex number z is represented by the point P in the Argand diagram. Given that
⎢z – 3i ⎢ = 3,
(a) sketch the locus of P. (2)
(b) Find the complex number z which satisfies both ⎢z – 3i ⎢ = 3 and arg (z – 3i) = (4)
The transformation T from the z-plane to the w-plane is given by
w = zi2 .
(c) Show that T maps ⎢z – 3i ⎢ = 3 to a line in the w-plane, and give the cartesian equation of this line.
(5)
(Total 11 marks)
5. (1 + 2x)
xy
dd = x + 4y2.
(a) Show that
(1 + 2x) 2
2
dd
xy = 1 + 2(4y – 1)
xy
dd . (1)
(2)
(b) Differentiate equation (1) with respect to x to obtain an equation involving
3
3
dd
xy , 2
2
dd
xy ,
xy
dd , x and y.
(3)
Given that y = 21 at x = 0,
(c) find a series solution for y, in ascending powers of x, up to and including the term in x3. (6)
(Total 11 marks)
π43 .
FP2 "PRACTICE PAPER" ISSUED BY EDEXCEL8
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*P44512A01628*
6. The transformation T from the z-plane, where z = x + iy, to the w-plane, where w = u + iv, is given by
w = 4 1 82 1( )( )
− −− + −i ii iz
z, z
14
– 14
i
The transformation T maps the points on the line l with equation y = x in the z-plane to a circle C in the w-plane.
(a) Show that
w ax bx cx
= + ++
2
216 1i
where a, b and c are real constants to be found.(6)
(b) Hence show that the circle C has equation
(u – 3)2 + v2 = k2
where k is a constant to be found.(4)
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PMT
FP2 JUNE 20149
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*P43150A01628*
6. The transformation T maps points from the z-plane, where z = x + iy, to the w-plane, where w = u + iv.
The transformation T is given by
w = z
zi + 1, z i
The transformation T maps the line l in the z-plane onto the line with equation v = –1 in the w-plane.
(a) Find a cartesian equation of l in terms of x and y.(5)
The transformation T maps the line with equation y = 12
in the z-plane onto the curve C in
the w-plane.
(b) (i) Show that C is a circle with centre the origin.
(ii) Write down a cartesian equation of C in terms of u and v.(6)
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PMT
FP2 JUNE 2014 (R)1110
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*N35388A01424*
6. A complex number z is represented by the point P in the Argand diagram.
(a) Given that 6z z− = , sketch the locus of P.(2)
(b) Find the complex numbers z which satisfy both 6z z− = and 3 4i 5z − − = .(3)
The transformation T from the z-plane to the w-plane is given by 30w
z= .
(c) Show that T maps 6z z− = onto a circle in the w-plane and give the cartesian
equation of this circle.(5)
PhysicsAndMathsTutor.com June 2010
21011
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*P40104A02428*
8. The point P represents a complex number z on an Argand diagram such that
z z− = −6 2 3i
(a) Show that, as z varies, the locus of P is a circle, stating the radius and the coordinates of the centre of this circle.
(6)
The point Q represents a complex number z on an Argand diagram such that
arg ( )z − = −6 34π
(b) Sketch, on the same Argand diagram, the locus of P and the locus of Q as z varies. (4)
(c) Find the complex number for which both z z− = −6 2 3i and arg ( )z − = −6 34π
(4)
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PhysicsAndMathsTutor.com June 2012FP2 COMPLEX-NUMBER LOCI PAST-PAPER QUESTIONS
112
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