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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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16

*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

4

FP2 JUNE 20155

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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

FP2 JUNE 2015 "IAL" PAPER7

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

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