HSC Papers, Sorted by Topic (1984-1997), Part 1 of 4

4 UNIT MATHEMATICS – GRAPHS – HSC

¤©BOARD OF STUDIES NSW 1984 - 1997

©EDUDATA: DATAVER1.0 1995

Graphs 4U97-3b)!

Let f(x) = 3x5 - 10x

3 + 16x.

i. Show that f (x) 1 for all x.

ii. For what values of x is f (x) positive?

iii. Sketch the graph of y = f(x), indicating any turning points and points of inflection.¤

« i) Proof ii) -1 < x < 0 or x > 1 iii)

-1

-9

9

1

y

x0

Inflectionpoints

»

4U96-4b)!

i. On the same set of axes, sketch and label clearly the graphs of the functions y x1

3 and

y ex .

ii. Hence, on a different set of axes, without using calculus, sketch and label clearly the graph of

the function y x ex1

3 .

iii. Use your sketch to determine for which values of m the equation x e mxx

1

3 1 has exactly

one solution.¤

« i)

y = x1

3

y = ex

0

y

1

x

ii)

(-1, -0.36)

x

y

»

4U95-3a)!

Let f(x) = - x² + 6x - 8. On separate diagrams, and without using calculus, sketch the following

graphs. Indicate clearly any asymptotes and intercepts with the axes.

i. y = f(x)

ii. y = │f(x)│

iii. y² = f(x)

iv. y = 1

f(x)




v. y = ef(x)

. ¤

« (i)

(3, 1)

x

y = f(x)

y

-8

2 4

(ii)

(3, 1)

x

y = f(x)

y

2 4

8

(iii)

y2 = f(x)

x

y

1

-1

42 30

(iv)

18

y1

f(x)y

1

f(x)

y1

f(x)

y

x

(3, 1)

2 4

asymptote

(v)

e

y

x3

(3, e)

y = ef(x)

asymptote »

4U94-5a)!

Let f xx x

x( )

( )( ),

2 1

55 for x .

i. Show that f x xx

( )

418

5.

ii. Explain why the graph of y = f(x) approaches that of y = -x - 4 as x approaches and as x

approaches -.

iii. Find the values of x for which f(x) is positive, and the values of x for which f(x) is negative.

iv. Using part (i), show that the graph of y = f(x) has two stationary points. (There is no need to

find the y coordinates of the stationary points.)

v. Sketch the curve y = f(x). Label all asymptotes, and show the x intercepts. ¤




« (i) Proof (ii) As x approaches , 18

5 x approaches zero and so f(x) approaches -x - 4 (iii) Positive

when x < -1, 2 < x < 5 and negative when -1 < x < 2, x > 5 (iv) Proof (v)

-1 2 5 x

y

y = -x - 4

-4

asymptotes

»

4U93-4a)!

Let f(x)1 x

x

. On separate diagrams sketch the graphs of

the following functions. For each graph label the asymptote.

i. y = f(x)

ii. y = f(│x│)

iii. y = ef(x)

iv. y² = f(x)

Discuss the behaviour of the curve of (iv) at x = 1. ¤

« (i)

y

x

1

-1

asymptotes

(ii)

y

x1

-1

asymptotes

-1




(iii)

1e

y

xasymptotes

(iv)

y

x10

asymptote

NB: there is a

vertical tangent at x = 1 »

4U92-4b)!

Let f(x) = Ln(1 + x) - Ln(1 - x) where -1 < x < 1.

i. Show that f x( ) 0for -1 < x < 1.

ii. On the same diagram, sketch

y = Ln(1 + x) for x > -1,

y = Ln(1 - x) for x < 1 and

y = f(x) for -1 < x < 1.

Clearly label the three graphs.

iii. Find an expression for the inverse function y f x 1( ) . ¤

« (i) Proof (ii)

x

y

y = ln(1 + x)

y = f(x)

y = ln(1 - x)

y = f(x)

-1 10

(iii) ye 1

e 1

x

x

»

4U92-8a)!

Consider the function f x exx( )

1

10

10

.

i. Find the turning points of the graph of y f x ( ) .

ii. Sketch the curve y = f(x) and label the turning points and any asymptotes.

iii. From your graph, deduce that

exx

110

10

for x < 10.

iv. Using (iii), show that

11

10

10

9

10 10

e . ¤




« (i) (0, 1) is a maximum turning point and (10, 0) is a minimum turning point. (ii)

x

y

10

NB: The x-axis is an asymptote. (iii) Proof (iv) Proof »

4U91-4a)!

-1 0-2-3-4 1 2 3 4

1

-1

y

x

The diagram is a sketch of the function y = f(x). On separate diagrams sketch:

i. y = -f(x)

ii. y = |f(x)|

iii. y = f(|x|)

iv. y = sin-1

(f(x)). ¤

« (i)

-3 -2 -1 1 2 3

1

-1

y

x

(ii)

-3 -2 -1 1 2 3

1

y

x

(iii)

-3 -2 -1 1 2 3

1

-1

y

x

(iv)

-2 -1 2 3

2

y

x

2

»

4U91-4b)!

The even function g is defined by

g(x)4e 6e

for x 0

for x 0

x 2x

g x( )

i. Show that the graph of y = g(x) has a maximum turning point at Ln3,2

3

.

ii. Sketch the curve y = g(x) and label the turning points, any points of inflexion, asymptotes,

and points of intersection with the axes.

iii. Discuss the behaviour of the curve y = g(x) at x = 0. ¤




« (i) Proof (ii)

x

(0, -2)

(ln , )32 0( ln , ) 3

2 0

(ln , )6 12( ln , ) 6 1

2

(ln , )3 23( ln , ) 3 2

3

y

(iii) y = g(x) has a cusp

at (0, -2) »

4U91-8a)!

Let f(x)sinx

x

1

for 0 x

2

for x 0

.

i. Find the derivative of f for 02

x

and prove that f ' is negative in this interval.

ii. Sketch the graph of y = f(x) for 02

x

and deduce that sin x >2x

in this interval. ¤

« (i)

f (x)cos x(x tan x)

x2(ii)

2

1

y

x

2

»

4U90-3b)!

Consider the functions f, g defined by

f(x) x 1

x 2, for x 2,

g(x) f(x) 2.

i. Sketch the hyperbola y = f(x), clearly labelling the horizontal and vertical asymptotes and the

points of intersection with the x and y axes.

ii. Find all turning points of y = g(x).

iii. Using the same diagram as used in (i) sketch the curve y = g(x) clearly labelling it.

iv. On a separate diagram sketch the curve given by y = g(-x). ¤

« (i) (iii)

14

y = f(x)

y = g(x)y = g(x)

-2 x

y

1

-0.51

(ii) Min (1, 0) (iv)

14

x

y

y = g(-x)

y = g(-x)

1

2

»




4U90-4a)!

t

f

4

-4

3 6 10 12

(6, -4)

(12, 2)

The diagram shows the graph of the function f, where

f(t)4

43

t , for 0 t 6

t 10, for 6 t 12

.

The function F is defined for 0 x 12 by F(x) f(t) dt0

x

.

i. Calculate F(6) and F(12).

ii. Calculate those values of x for which F(x)=0.

iii. Find all turning points of F. ¤

« (i) F(6) = 0, F(12) = -6 (ii) x = 0, 6 (iii) (3,6) is a relative maximum & (10, -8) is a relative minimum »

4U89-6a)!

x

y

3

y = f '(x)

21-3 -2 -1 0

The function f(x) has derivative f '(x) whose graph appears above. You are given that

f f f x s ( ) ( ) , ( )2 1 0 approache as x approaches and f x( ) approaches 0 as

x approaches .

i. Sketch the graph of f(x) showing its behaviour at its stationary points. You are given that

f(0) = 0 and f(3) > 0.

ii. Describe the behaviour of f(x) as x approaches . ¤




« (i)

-3 -2 -1 1 2 30

y

x

MIN

MAX

(ii) As x approaches , f(x) approaches an

horizontal asymptote. As x approaches -, f(x) approaches -. »

4U89-6b)!

i. Sketch the graph of g(x) x4 4x

3 4x

2

1

2 showing that it has four real zeros.

ii. On different diagrams sketch the curves:

. y = |g(x)|;

. y2 = g(x).

iii. . Indicate the nature of the curve y = |g(x)| at a zero of g(x).

. Calculate the slope of the curve y2 = g(x) at any point x and describe the nature of

the curve at a zero of g(x). ¤

« (i)

y

x1 2

1

2

(ii) ()

y

x

y = g(x)

1 2 3-1

() y

x

y2 = g(x)

-1 1 2 3

(iii) () Since the curve is reflected in the x-axis, the curve has

sharp points at zeros. () dy

dx

g (x)

2 g(x)

. The curve has a vertical tangent. »

4U88-2)!

a. Draw a neat sketch of the function f(x) = (x - 2)(6 - x). State the co-ordinates of its vertex

and of its points of intersection with both co-ordinate axes.

b. Hence or otherwise draw a neat sketch of the function g(x) 16

(x 2)(6 x). Clearly

indicate on your sketch the equations of the vertical asymptotes and the co-ordinates of any

stationary points.

c. The lines with equations x = 3 and x = 5 cut the graph of y = g(x) at P and Q respectively.

Mark on your sketch the co-ordinates of P and Q. Shade the region R bounded by y = g(x)

and the line PQ.

d. Prove that the area of R is 32

38 3

loge square units. ¤




« (a) (b) (c)

P(3, )163 Q(5, )16

3

x

x = 2

43

163

x = 6

y = g(x)

y = f(x)

(4, 4)

2 6y = g(x)

y

y = g(x)

(d) Proof »

4U87-3)

a. A function f(x) is defined by f(x) loge x

x for x > 0.

Prove that the graph of f(x) has a relative maximum turning point at x = e and a point of

inflexion at x = e3

2

b. Discuss the behaviour of f(x) in the neighbourhood of x = 0 and for large values of x.

c. Hence draw a clear sketch of f(x) indicating on it all these features.

d. Draw separate sketches of the graphs of:

. y loge x

x;

. y x

loge x.

(Hint: There is no need to find any further derivatives to answer this part.)

e. What is the range of the function y x

loge x? ¤

« (a) Proof (b) As x approaches 0, f(x) approaches -. As x approaches , f(x) approaches 0. (c)

xf x

x

xe( )

log

e e3

2

3

23

2,

ee

,1

1

y

(d) ()

xf x

x

xe( )

log

ee

,1

1

y

e e3

2

3

23

2,

()

x

f xx

xe

( )log

y

(e, e)

(e) y < 0 and y e »

4U85-2ii)

a. Sketch the function g(x) = xe-x

, for x -1.

b. Given g(x) as in (a) above, the function f(x) is given by the rule




f xg x

g x

x

x( )

( ),

( ),

2 1

1.

Find the zeros of this function, and the maximum and minimum values. Draw a sketch of the

graph of y = f(x). ¤

« (a)

2, 2

e2

1, 1e

(-1, -e)

x

y = xe

y

-x, x -1

(b)

-1, 1e

- 2, 2

e2 4, 2

e2

3, 1e

y

x

(1, -e)

-1 0 1 2

»

4U84-2i)

Sketch the graphs of:

a. (x + 3)(y - 2) = 1;

b. x2 + y

2 + 1 = 2(x + y). ¤

« (a)

x0

y

-3

2

(x + 3)(y - 2) = 1

21

3

31

2 (b)

1

1

C(1, 1)

x2 + y2 + 1 = 2(x + y)y

x0 »

4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC



Complex Numbers 1: 1997 - 1991

4U97-2a)!

i. Express 3 i in modulus-argument form.

ii. Hence evaluate 36

i .¤

« i) 2cis11

6i

ii) -64 »

4U97-2b)!

i. Simplify (2i)3.

ii. Hence find all complex numbers z such that z3 = 8i. Express your answers in the form

x + iy.¤

« i) 8i ii) 2i, 3 i, - 3 i »

4U97-2c)!

Sketch the region where the inequalities z - 3+ i and z +1 z -1 5 both hold.¤

« -5

3

X

Y

(3, -1)

»

4U97-2d)!

Let w =3 + 4i

5and z =

5 +12i

13so that w, z 1.

i. Find wz and wz in the form x + iy.

ii. Hence find two distinct ways of writing 652 as the sum a

2 + b

2, where a and b are integers

and 0 < a < b.¤

« i)

33

65

56

65

16

65i,

63

65i ii) 65 33 56 16 632 2 2 2 2 »

4U97-8b)!

U

S

R

O

T

The diagram shows points O, R, S, T and U in the complex plane. These points correspond to the

complex numbers 0, r, s, t and u respectively. The triangles ORS and OTU are equilateral. Let

cos3

i sin3

.




i. Explain why u = t .

ii. Find the complex number r in terms of s.

iii. Using complex numbers, show that the lengths of RT and SU are equal.¤

« i) Rotation

3 anticlockwise and UOT =

3 ii) r =

s

iii) Proof »

4U96-2a)!

Suppose that c is a real number, and that z c i . Express the following in the form x iy , where x

and y are real numbers:

i. ( )iz ;

ii. 1

z.¤

« i) 1– ci ii) c

c

i

c1 12 2

»

4U96-2b)!

On an Argand diagram, shade the region specified by both the conditions

Re( )z and z i 4 4 5 3 .¤

«

Ox

y

4 – 5i

|z – (4 – 5i)| = 3

Re(z) = 4

»

4U96-2c)!

i. Prove by induction that (cos sin ) cos( ) sin( ) i n i nn for all integers n 1.

ii. Express w i 3 in modulus-argument form.

iii. Hence express w5 in the form x iy , where x and y are real numbers.¤

« i) Proof ii) 2 cis

6 iii) –16 3 – 16i »

4U96-2d)!

3

O-1

z

The diagram shows the locus of points z in the complex plane such that




arg( ) arg( )z z 3 13

.

This locus is part of a circle. The angle between the lines from -1 to z and from 3 to z is , as shown.

Copy this diagram into your Writing Booklet.

i. Explain why

3

.

ii. Find the centre of the circle.¤

« i) The exterior angle of a triangle is equal to the sum of the interior opposite angles. ii) (1, 2 3

3) »

4U96-8a)!

Let w = cos sin2

9

2

9

i .

i. Show that w k is a solution of z 9 - 1 = 0, where k is an integer.

ii. Prove that w + w2 + w

3 + w

4 + w

5 + w

6 + w

7 + w

8 = -1.

iii. Hence show that cos cos cos

9

2

9

4

9

1

8

.¤

« Proof »

4U95-2a)!

Let w1 = 8 - 2i and w2 = - 5 + 3i.

Find w w1 2 .¤

« 3 - 5i »

4U95-2b)!

i. Show that (1 - 2i)² = - 3 - 4i

ii. Hence solve the equation z² - 5z + (7 + i) = 0.¤

« (i) Proof (ii) Z = 3 - i or 2 + i »

4U95-2c)!

Sketch the locus of z satisfying:

i. arg(z - 4) = 34

;

ii. Im z = │z│.¤

« (i)

Im(z)

Re(z)4

4

34

(ii)

Im(z)

z

Re(z) »

4U95-2d)!




Q

P

O

The diagram shows a complex plane with origin 0. The points P and Q represent arbitrary non - zero

complex numbers z and w respectively. Thus the length of PQ is │z - w│.

i. Copy the diagram into your Writing Booklet, and use it to show that │z - w│ │z│ + │w│.

ii. Construct the point R representing z + w. What can be said about the quadrilateral OPRQ?

iii. If │z - w│ = │z + w│, what can be said about the complex number w

z?¤

« (i) (ii)

Q

R

P

O OPRQ is a parallelogram (iii)

w

z is imaginary »

4U95-4a)!

i. Find the least positive integer k such that cos4

7isin

4

7

is a solution of z

k = 1.

ii. Show that if the complex number w is a solution of zn = 1, then so is w

m, where m and n are

arbitrary integers.¤

« (i) k = 7 (ii) Proof »

4U94-2a)!

Let z = a + ib, where a and b are real. Find:

i. Im(4i - z);

ii. (3iz) in the form of x + iy, where x and y are real;

iii. tan , where arg(z )2.¤

« (i) 4 - b (ii) -3b - 3ai (iii) 2ab

a b2 2 »

4U94-2b)!

Express in modulus - argument form:

i. -1 + i;

ii. ( 1 i)n , where n is a positive integer.¤

« (i) 2 cos34

i sin34

(ii) 22

n

cos 3n4 i sin 3n

4 »

4U94-2c)!

i. On the same diagram, draw a neat sketch of the locus specified by each of the following:




. z i ( )3 2 2

. z z 3 5 .

ii. Hence write down all values of z which satisfy simultaneously

z i ( )3 2 2 and z z 3 5 .

iii. Use your diagram in (i) to determine the values of k for which the simultaneous equations

z i ( )3 2 2 and z i k 2

have exactly one solution for z.¤

« (i) Re(z)

Im(z)

(3 + 2i)

1 5

4

(ii) z = 1 + 2i (iii) k = 1, 5 »

4U94-4a)!

Find and , given that z z i z z3 23 2 ( ) ( ) .¤

« = -i, = 2i »

4U94-7a)!

i. It is known that if f x'( ) 0 and f(0) = 0, then f x( ) 0 for x > 0. Show that

sin x xx6

03

for x > 0, and hence show that sin x

x1

x6

2

for x > 0.

Let the points A A A An0 1 2 1, , ,...., represent the nth roots of unity on an Argand diagram, where Ak

represents cos sin2 2k

ni

k

n

. Let P be the regular polygon A A An0 1 1... .

ii. Show that the area of P is n

n2

2sin

.

iii. Using part (i), or otherwise, show that for all n 26 , P covers more than 99% of the unit

circle.¤

« Proof »

4U94-8b)!

Let x = be a root of the quartic polynomial P x x Ax Bx Ax( ) 4 3 2 1, where A and B are

real. Note that may be complex.

i. Show that 0.

ii. Show that x = is also a root of Q x xx

A xx

B( )

2

2

1 1.

iii. With u x1x

, show that Q(x) becomes R u u Au B( ) ( ) 2 2 .

iv. For certain values of A and B, P(x) has no real roots. Let D be the region of the AB plane

where P(x) has no real roots for A 0.




-2

0

L

B

T

c

A

The region D is shaded in the figure. Specify the bounding straight-line segment L and curved

segment c. Determine the coordinates of T.¤

« (i) Proof (ii) Proof (iii) Proof (iv) The straight line segment is B = 2A - 2 and the curved segment is

B A 214

2 . T is the point (4, 6). »

4U93-2a)!

i. On an Argand diagram, shade in the region determined by the inequalities 2 Im z 4 and

6 arg z 4 .

ii. Let z0 be the complex number of maximum modulus satisfying the inequalities of (i).

Express z0 in the form a + ib.¤

« (i)

y

x

4

2

4

6

(ii) z 4 3 4i0 »

4U93-2b)!

Let be a real number and consider (cos + i sin )3.

i. Prove cos 3 cos 3cos sin3 2 .

ii. Find a similar expression for sin 3.¤

« (i) Proof (ii) sin 3 3sin cos sin2 3 »

4U93-2c)!

Find the equation, in Cartesian form, of the locus of the point z if Rez 4

z0

.¤

« x2 + y

2 - 4x = 0, excluding the origin (0, 0) »

4U93-2d)!

By substituting appropriate values of z1 and z2 into the equation argz

zarg z arg z1

2

1 2 show that

4tan 2 tan

1

3

1 1 .¤




« Proof »

4U93-2e)!

Let P, Q, and R represent the complex numbers w1, w2 and w3 respectively. What geometric

properties characterise triangle PQR if w2 - w1 = i(w3 - w1)? Give reasons for your answer.¤

« PQR is a right-angled isosceles triangle, with RPQ

2 and PQ=PR»

4U93-8a)!

Let the points A1, A2, ..., An represent the nth rots of unity, w1, w2, ..., wn, and suppose P represents any

complex number z such that │z│ = 1.

i. Prove that w1 + w2 + ... + wn = 0.

ii. Show that PA (z w )(z w )i2

i i for i = 1, 2, ..., n.

iii. Prove that PA 2ni2

z 1

n

.¤

« Proof »

4U92-2a)!

The points A and B represent the complex numbers 3 - 2i and 1 + i respectively.

i. Plot the points A and B on an Argand diagram and mark the point P such that OAPB is a

parallelogram.

ii. What complex number does P represent?¤

« (i)

1 2 3 4 x0

1

-2

2

-1 P(4, -1)

A

B

y

(ii) 4 - i »

4U92-2b)!

Let z = a + ib where a b 0.2 2

i Show that if Im(z) > 0 then Im1z 0 .

ii. Prove that 1 1

z z .¤

« Proof »

4U92-2c)!

Describe and sketch the locus of those points z such that:

i. z i z i

ii. z i z i 2 .¤




« (i) The locus is the real axis.

x

-i

i

y

(ii) The locus is a circle, centre (0, -3) and

radius 2 2 .

x

-3

y

»

4U92-2d)!

It is given that 1 + i is a root of P z z z rz s( ) 2 33 2 where r and s are real numbers.

i. Explain why 1 - i is also a root of P(z).

ii. Factorise P(z) over the real numbers.¤

« (i) If z1 is a root of P(z) = 0 then z1 is also a root. Thus, if (1 + i) is a root then 1 i 1 i is also a

root. (ii) P(z) (2z 1)(z 2z 2)2 »

4U92-7b)!

Suppose that z7 1 where z 1.

i. Deduce that z z zz z z

3 22 31

1 1 10 .

ii. By letting x zz

1

reduce the equation in (i) to a cubic equation in x.

iii. Hence deduce that cos cos cos 7

27

37

18

.¤

« (i) Proof (ii) x3 + x

2 - 2x - 1 = 0 (iii) Proof »

4U91-2a)!

Plot on an Argand diagram the points P, Q, and R which correspond to the complex numbers 2i, 3 -

i, and - 3 - i, respectively.

Prove that P, Q, and R are the vertices of an equilateral triangle.¤




«

-2

Q( 3, 1)R( 3, 1)

-1 1 2 3

1

2

3

-1

y

x

P(0, 2)

»

4U91-2b)!

Let z1 = cos1 + isin1 and z2 = cos2 + isin2, where 1 and 2 are real. Show that:

i. 1

z1

= cos1 - isin1

ii. z1z2 = cos(1 + 2) + isin(1 + 2).¤

« Proof »

4U91-2c)!

i. Find all pairs of integers x and y such that (x + iy)2 = -3 - 4i.

ii. Using (i), or otherwise, solve the quadratic equation z 3z (3 i) 02 .¤

« (i) x = 1, y = -2 and x = -1, y = 2 (ii) z = 2 - i or 1 i »

4U91-2d)!

A

C

F

D

B

0

E

In the Argand diagram, ABCD is a square, and OE and OF are parallel and equal in length to AB and

AD respectively. The vertices A and B correspond to the complex numbers w1 and w2 respectively.

i. Explain why the point E corresponds to w2 - w1.

ii. What complex number corresponds to the point F?

iii. What complex number corresponds to the vertex D?¤

« (i) Proof (ii) i(w2 - w1) (iii) w1(1 - i) + iw2 »




Complex Numbers 2: 1990 - 1984

4U90-1a)!

Let z = a+ib, where a and b are real numbers and a 0 .

i. Express |z| and tan(arg z) in terms of a and b.

ii. Express z

3 5i in the form x+iy, where x and y are real. ¤

« (i) z a b2 2 , tan(arg z)b

a (ii)

(3a 5b)

34

i( 5a 3b)

34

»

4U90-1b)!

i. If wi

1 3

2 show that w

3 1.

ii. Hence calculate w10

. ¤

« (i) Proof (ii) 12

i3

2 »

4U90-1c)!

If z = 5 - 5i write z, z2 and

1

z in modulus argument form. ¤

« z 5 2 cos4

i sin4

, z cos

2i sin

2

2

50

,

1

zcos

4isin

4

1

5 2

»

4U90-1d)!

Let u and v be two complex numbers, where u = -2+i, and v is defined by |v| = 3 and argv

3.

i. On an Argand diagram plot the points A and B representing the complex numbers u and v

respectively.

ii. Plot the points C and D represented by the complex numbers u-v and iu, respectively.

Indicate any geometric relationships between the four points A, B, C, and D. ¤

«

A

B

C D

O x

3

y

3

»

4U89-1a)!

Evaluate |2 + 3i|. ¤

« 13 »




4U89-1b)!

Given that a and b are real numbers, express in the form x + iy, where x and y are real:

i. (a bi)(5 i) ;

ii. a bi

3 4i. ¤

« (i) (5a + b) + (5b - a)i (ii) 3a 4b

25

3b 4a

25i

»

4U89-1c)!

Find the complex square roots of 10 - 24i, giving your answers in the form x + iy, where x and y are

real. ¤

« (3 2 2 2i) »

4U89-1d)!

On and Argand diagram shade in the region containing all points representing complex numbers z

such that 2 Rez4 and 1 Imz3. ¤

«

3

-1

2 4

y

x

»

4U89-1e)!

Find in modulus-argument form all complex numbers z such that z3 = -1 and plot them on an Argand

diagram. ¤

« z cos3

isin31

, z cos isin2 , z cos

53

isin533

;

y

x

z1

z3

z2

3

»

4U89-1f)!

On separate diagrams draw a neat sketch of the locus specified by:

i. arg ( ( ))z z i

1 33

;

ii. z z 16i2 2 . ¤




« (i)

3

1

y

x

z

3

(ii)

y

x

xy = 4

»

4U88-4a)!

i. Express z = 2 2 i in modulus-argument form.

ii. Hence write z22

in the form a + ib, where a and b are real. ¤

« (i) z 2 cos4

i sin4

(ii) 2

22i »

4U88-4b)!

i. On an Argand diagram shade in the region R containing all points representing complex

numbers z such that 1 < |z| < 2 and

4 argz

2.

ii. In R mark with a dot the point K representing a complex number z. Clearly indicate on your

diagram the points M, N, P and Q representing the complex numbers z , -z, 1

z and 2z

respectively. ¤

«

y

x

N M

P

K

21

2

1

Q

4

»

4U88-4c)!

Show that the locus specified by 3|z - (4 + 4i)| = |z - (12 + 12i)| is a circle. Write down its radius and

the co-ordinates of its centre. Draw a neat sketch of the circle. ¤

« Centre is (3, 3) and radius is 3 2 .

(3, 3)

R(z)

Im(z)

6

6

O »




4U87-4i)

Find the complex square roots of 7 6 2i giving your answers in the form x + iy, where x and y are

real. ¤

« 3 i 2 and (3 i 2) »

4U87-4ii)

Let z1 = 4 + 8i and z2 = -4 - 8i.

a. Draw a neat sketch of the locus specified by |z - z1| = |z - z2|.

b. Show that the locus specified by |z - z1| = 3|z - z2| is a circle. Give its centre and radius. ¤

« (a)

z1

z2

P(z)

Im(z)

Re(z)

(b) Proof, centre is (-5 - 10i) and radius is 3 5 units. »

4U87-4iii)

a. Let OABC be a square on an Argand diagram where O is the origin. The points A and C

represent the complex numbers z and iz respectively. Find the complex number representing

B.

b. The square is now rotated about O through 45° in an anticlockwise direction to OABC.

Find the complex numbers representing A, B and C. ¤

« (a) (1 + i)z (b) 1

2(1 i)z , i 2z and

1

2(-1 i)z represent A, B and C respectively. »

4U86-4i)

Given that z1 = 3 - i, z2 = 2 + 5i, express in the form a + ib, where a, b are real,

a. (z1 )2;

b. z

z1

2

;

c. z

z1

2

. ¤

« (a) 8 + 6i (b) 1

29i17

29 (c)

10

29 »




4U86-4ii)

Given that for the complex number z, |z| = 2, arg z = 2

5

write in the form a + ib, where a, b are real:

a. z;

b. z7. ¤

« (a) z 2 cos 2

5i 2sin

2

5

(b) z 2 (-cos

5i sin

57 7

) »

4U86-4iii)

a. Draw a sketch of the region of the Argand diagram consisting of the set of all values of z for

which 1 |z| 4 and

4arg z

3

4 .

b. . The curve in the Argand diagram for which

|z - 2| + |z - 4| = 10 is an ellipse. Write down the coordinates of the centre, and the

lengths of the major and minor axes of this ellipse.

. On a separate Argand diagram, show the region for which z satisfies the inequalities

z + z 6 or |z - 2| + |z - 4| 10. ¤

« (a)

1 4 x

y

4

(b) () Centre is (3, 0). Major axis is 10 units and Minor axis

is 4 6 units. ()

y

-2 3 x8

»

4U85-3i)

Reduce the complex expression (2 i)(8 3i)

(3 i) to the form a + ib, where a, b are real numbers. ¤

« 11

2

5

2 i »

4U85-3ii)

The complex number z is given by z 3 i .

a. Write down the values of arg z and |z|.

b. Hence, or otherwise, show that z7 + 64z = 0. ¤

« (a) arg z5

6

, z = 2 (b) Proof »

4U85-3iii)

On the Argand diagram, let A = 3 + 4i, B = 9 + 4i.




a. Draw a clear sketch to show the important features of the curve defined by |z - A| = 5. Also,

for z on this curve, find the maximum value of |z|.

b. On a separate diagram, draw a clear sketch to show the important features of the curve

defined by |z - A| + |z - B| = 12. For z on this curve, find the greatest value of arg z. ¤

« (a)

y

x

4

3

5

z - a = 5

A(3 + 4i)

Maximum value of z = 10. (b)

A(3 + 4i) B(9 + 4i)

6 + 4i

y

xz - A + z - B = 12

Greatest value of arg z2

. »

4U84-3i)

Calculate the modulus and argument of the product of the roots of the equation

(5 + 3i)z2 - (1 - 4i)z + (8 - 2i) = 0. ¤

« P 2 , arg P4

»

4U84-3ii)

Let A = 1 + i, B = 2 - i. Draw sketches to show the loci specified on the Argand diagram by

a. arg(z - A) =

4

b. |z - A| = |z - B|. ¤

« (a)

A(1 + i)

arg(z - A)

Locus of z

P(z)

1

10 x

y

4

(b)

A(1 + i)

x

y

0

Locus of zB(2 - i)

P(z)

»

4U84-3iii)

Show that the point representing cos sin

3 3i on the Argand diagram lies on the circle of radius one

with centre at the point which represents 1. ¤

« Proof »

4U84-3iv)




R is a positive real number and z1, z2 are complex numbers. Show that the points on the Argand

diagram which represent respectively the numbers z1, z2, z1 iRz2

1 iR, form the vertices of a right

angled triangle. ¤

« Proof »

4 UNIT MATHEMATICS – CONICS – CSSA

†©CSSA OF NSW 1984 - 1996


Conics 4U96-4)!

a. The diameter of the ellipse x

a

y

b

2

2

2

2 1 (where a > b > 0) through the point P(a cos ,

b sin ) meets the circle x2 + y

2 = a

2 at the point R(a cos , a sin ).

i. Show this information on a sketch.

ii. Show that tan = b

atan .

iii. Prove that the tangent to the ellipse at P has equation bx + ay tan = ab sec .

iv. Show that the tangent to the circle at R has equation ax + by tan = a2sec .

v. If the tangent to the ellipse at P and the tangent to the circle at R are concurrent with

the right hand directrix of the ellipse, show that sec = 2

e, where e is the

eccentricity of the ellipse.†

b. The diameter of the ellipse x y2 2

25 91 through the point P on the ellipse meets the circle

x2 + y

2 = 25 at R. Tangents to the ellipse at P and the circle at R are concurrent with a

directrix of the ellipse. Using the results from part (a):

If P lies in the first quadrant,

i. find the coordinates of P and R.

ii. Sketch the ellipse and the circle, showing the coordinates of P and R, and the point

of intersection of the tangents and the appropriate directrix.†

« a) i)

x

R

P

a-a

-a

-b

b

a

ii) iii) iv) v) Proof b) i) P 23 21

5,

, R

50

17

15 21

17,

ii)

Y

x

x 25

4

25

4

5 21

14,

R

P

5-5

-5

-3

3

5

»

4U95-4)!

The hyperbola xy = c2 meets the ellipse

x

a

y

b

2

2

2

21 at P ct

c

t11

,

and Q ct

c

t22

,

where t1 > t2 > 0.

Tangents to the hyperbola at P and Q meet in T, while tangents to the ellipse at P and Q meet in V.

i. Show this information on a sketch.

ii. Show that the parameter t of a point ctc

t,

where xy = c

2 and

x

a

y

b

2

2

2

21 intersect

satisfies the equation b2c

2t4 - a

2b

2t2 + a

2c

2=0.


†©CSSA OF NSW 1984 - 1996


iii. Using without proof the result that the tangent to hyperbola xy = c2 at the point ct

c

t,

has

equation x + t2y = 2ct, show that T has coordinates

2 21 2

1 2 1 2

ct t

t t

c

t t

, .

iv. Using without proof the result that the tangent to x

a

y

b

2

2

2

21 at the point (x1, y1) has

equation b2x1x + a

2y1y = a

2b

2, , show that V has co-ordinates

a

c t t

b t t

t t

2

1 2

21 2

1 2( ),

( )

v. Show that TV passes through the origin.

vi. Show that if V lies at a focus of the hyperbola, then the ellipse is in fact a circle and find the

radius of this circle in terms of c.†

« i)

Q V

PT

x

a

y

b

2

2

2

2 1

xy = c2y

x

ii) iii) iv) v) Proof vi) Proof

Radius = c 1 5 »

4U94-4)!

P(20cos , 12sin ) is a point on the ellipse x y2

2

2

220 121 .

P lies in the first quadrant, and the tangent to the ellipse at P meets the directrices in Q and R where Q

is nearer the focus S and R is nearer the focus S. Q and R each lie above the x axis, and QS meets

RS in K where K lies in the third quadrant.

i. Sketch the ellipse showing it’s directrices and foci and the points P, Q, R and K.

ii. Show that the tangent at P has equation 3x cos + 5y sin = 60.

iii. Show that K has co-ordinates

20

4 9 25

3

2

cos ,( sin )

sin

.

iv. If K lies on the ellipse, find the co-ordinates of P and show that PSKS is a rectangle.†

« i)

R

Q

P(20cos, 12sin)

x

y

x = -25 x = 25-12

12

2016O-20 -16

ii) iii) Proof iv) k(-5 7 , -9), Proof »

4U93-4a)!


†©CSSA OF NSW 1984 - 1996


-a a ae0 x

y

x

a

y

b1

2

2

2

2

(x - ae)2 + y2 = a2(e2 + 1)

i. Show that the tangent at P(a sec , b tan ) on the hyperbola x

a

y

b

2

2

2

21 has equation

x

a

y

b

sec tan 1 0 .

ii. Show that if the tangent at P is also a tangent to the circle with centre (ae, 0) and radius

a e2 1 , then sec = -e.

iii. Deduce that the points of contact P, Q on the hyperbola of the common tangents to the circle

and hyperbola are the extremities of a latus rectum of the hyperbola, and state the co-

ordinates of P and Q.

iv. Find the equations of the common tangents to the circle and hyperbola, and find the

coordinates of their points of contact with the circle.†

« i) ii) Proof iii) P(-ae, b e2 1 ), Q(-ae, -b e2 1 ) iv) xe y + a = 0, (0, a) »

4U92-4b)!

Consider the ellipse x

a

y

b

2

2

2

21 , where a > b > 0.

i. Show that the tangent to the ellipse at the point P(a cos , b sin ) has equation

bx cos + ay sin - ab = 0.

ii. R and R are the feet of the perpendiculars from the foci S and S on to the tangent at P. Show

that SR.SR = b2.†

« Proof »


†©CSSA OF NSW 1984 - 1996


4U91-4)!

a. P(a cos , b sin ), Q(a sec , b tan ) lie on the ellipse x

a

y

b

2

2

2

21 and the hyperbola

x

a

y

b

2

2

2

21 respectively. M and N are the feet of the perpendiculars from P, Q respectively

to the x axis. O < <

2, and QP produced meets the x axis in K. A is the point with

coordinates (a, 0).

i. Using without proof the similarity of KPM and KQN, show that KM

KN cos ,

and hence show that K has coordinates (-a, 0).

ii. Sketch the ellipse and hyperbola showing the positions of P, Q, M, N, A and K.

iii. Show that the tangent to the ellipse at P has equation x

a

y

b

cos sin 1 and

deduce that this tangent passes through N.

iv. Given that the tangent to the hyperbola at Q has equation x

a

y

b

sec tan 1,

show that this tangent passes through M.

v. Show that the tangents PN and QM and the common tangent at A are concurrent,

and show that the point of concurrence is T(a, b tan

2).

vi. If the common tangent at A meets QP in V, show that T is the midpoint of AV.†

b. The result in (a) (i) provides a method of constructing the hyperbola x

y2

2

41 from the

auxiliary circle x2 + y

2 = 4, and the ellipse

xy

22

41 . Indicate why this is so on a new

sketch by using the auxiliary circle to construct one such pair of points P, Q each with

parameter , 0 < <

2.†


†©CSSA OF NSW 1984 - 1996


« a) i) Proof ii)

PQ

A NM

-b

x

y

b

Ka-a

iii) iv) v) vi)

Proof b)

y

x

Q

NM 2-2

-2

-1

2

1 P

»

4U90-4)!

a. Show that the tangent to the ellipse x y2 2

9 41 at the point P(x1, y1) has cartesian equation

xx yy1 1

9 41 .

b. Show that if tangents are drawn from a point W(x0, y0) external to the ellipse x y2 2

9 41 ,

touching the ellipse at P, Q respectively, then the equation of the chord of contact PQ is

xx yy0 0

9 41 .


†©CSSA OF NSW 1984 - 1996


c.

y = mx + k

-2

-3 0 3 5 7 x9

P(x1, y1)

R(x2, y2)

y = 2

y = -2

(x - 7)2 + y2 = 4

2

y

x y2

9

2

41

The above diagram shows the ellipse x y2 2

9 41 and the circle (x - 7)

2 + y

2= 4. Clearly

y = 2 and y = -2 are common tangents to the ellipse and the circle. Suppose the line

y = mx + k, m 0, is also a common tangent, touching the ellipse at P(x1, y1) and the circle at

R(x2, y2) as shown.

i. Copy the diagram and use symmetry to draw a fourth common tangent, touching the

ellipse at Q and the circle at T, and write down the coordinates of Q and T, on your

diagram. Deduce that the equation of QT is y = -mx - k.

ii. PR and QT intersect at V. Show V has coordinates

k

m, 0 .

iii. Use the fact that PQ is the chord of contact of tangents from V to the ellipse to show

that x1 = 9m

k.

iv. Deduce that x1 = 9m

k is a double root of the equation

x mx k2 2

9 41

( ), and

hence show that 9m2 - k

2 + 4 = 0.

v. Show that if y = mx + k is a tangent to the circle (x - 7)2 + y

2 = 4, then

45m2 + k

2 + 14mk - 4 = 0.

vi. Show that m

k7

27, and find the coordinates of P, Q and V, and the equations of

the two oblique common tangents.†


†©CSSA OF NSW 1984 - 1996


« a) b) Proof c) i)

-2

-3 0 3 5 7 9

P(x1, y1)

Q(x1, y1)

T(x2, -y2)

R(x2, -y2)

y = 2

y = -2

2

y

ii) iii) iv)

v) Proof vi) P7

3

8 2

9,

, Q

7

3

8 2

9,

, V

27

70,

,

7

27

2 2

91

x y ,

7

27

2 2

91

x y »

4U90-5b)!

A parabola passes through the points (-a, 0), (0, h) and (a, 0) where a > 0, h > 0. Show that the area

enclosed by this parabola and the x axis is 4

3ah.†

« Proof »

4U89-4a)!

i. Show that the ellipse 4x2 + 9y

2 = 36 and the hyperbola 4x

2 - y

2 = 4 intersect at right angles.

ii. Find the equation of the circle through the points of intersection of the two conics.†

« i) Proof ii) x2 + y

2 = 5 »

4U89-4b)!

i. Show that the tangent to the hyperbola x

a

y

b

2

2

2

21 (where a > b > 0) at the point P(a sec,

b tan) has equation bx sec - ay tan = ab.

ii. If this tangent passes through a focus of the ellipse x

a

y

b

2

2

2

2 1 (where a > b > 0) show that

it is parallel to one of the lines y = x, y = -x and that its point of contact with the hyperbola

lies on a directrix of the ellipse.†

« Proof »

4U88-4)!

P(2Ap, Ap2) is a point on the parabola x

2 = 4Ay. Q(a cos, b sin) is a point on the ellipse

x

a

y

b

2

2

2

21 .

In what follows you may use without proof the results that the tangent to x2 = 4Ay at P and the tangent

to x

a

y

b

2

2

2

21 at Q have equations px - y = Ap

2 and

x

a

y

b

cos sin 1 respectively.

a. Using the fact that two lines are coincident if the corresponding coefficients are in proportion

deduce that the tangent to x2 = 4Ay at P is also the tangent to

x

a

y

b

2

2

2

21 at Q if

cos = a

Ap and sin =

b

Ap2.


†©CSSA OF NSW 1984 - 1996


b. Hence show that PQ is a common tangent to x2 = 4Ay and

x

a

y

b

2

2

2

21 if

A2p

4 - a

2p

2 - b

2 = 0, and deduce that there are exactly two such common tangents.

c. Let p0 > 0 be the parameter of the point of contact of one of these common tangents with the

parabola and let A > 0.

Sketch the parabola, the ellipse and both common tangents showing, in terms of p0 the

coordinates of the points of contact of the tangents with both curves and the intercepts of the

tangents on the coordinate axes.

d. Using symmetry sketch on the same diagram the parabola x2 = -4Ay and the two common

tangents to x2 = -4Ay and

x

a

y

b

2

2

2

21 .

What is the nature of the quadrilateral formed by the four tangents on this diagram? Deduce

that this quadrilateral is a square if A2 = a

2 + b

2.

e. Find the equation of the circle with centre (0, 0) for which the quadrilateral formed by the

four tangents common to the circle and the curve x2 = 8y is a square.†

« a) Proof b) Proof c) d)

y

Ox-Ap0 Ap0

-Ap02

x

a

y

b

2

2

2

2 1

(2Ap0, Ap02)(-2Ap0,

Ap02)

a

Ap

b

Ap

2

0

2

02,

a

Ap

b

Ap

2

0

2

02,

p0x - y = Ap02 -p0x - y = Ap0

2

e) x2 + y

2 = 2 »

4U87-4i)!

a. Show that the normal to the ellipse x

a

y

b

2

2

2

21 (a

2 > b

2) at the point P(x1, y1) has equation

a2y1x - b

2x1y = (a

2 - b

2)x1y1.

b. This normal meets the major axis of the ellipse at G. S is one focus of the ellipse. Show that

GS = e.PS where e is the eccentricity of the ellipse.†

« Proof »

4U87-4ii)!

a. By using the result in part (i)(a) above, or otherwise, show that the normal to the ellipse

x y2 2

25 91 at the point P(5 cos, 3 sin) has equation 5xsin - 3ysin = 16sin cos.

b. This normal cuts the major and minor axes of the ellipse at G and H respectively. Show that

as P moves on the ellipse the mid point of GH describes another ellipse with the same

eccentricity as the first.

c. On the same axes sketch the two ellipses showing clearly the coordinates of the intercepts on

the coordinate axes.†


†©CSSA OF NSW 1984 - 1996


« a) Proof b) Proof c)

x

y

5-5

3

-3-2.6

2.6

1.6-1.6

»

4U86-4i)!

Show that the curves x2 - y

2 = c

2 and xy = c

2 cross at right angles.†

« Proof »

4U86-4ii)!

Show that the tangent to the hyperbola x

a

y

b

2

2

2

21 at the point P(a sec, b tan) has equation

bx sec - ay tan = ab, and deduce that the normal there has equation by

sec + ax tan = (a2 + b

2) sectan. The tangent and the normal cut the y-axis at A and B

respectively. Show that the circle on AB as diameter passes through the foci of the hyperbola. (It is

enough to show that the circle passes through one focus and then to use symmetry).†

« Proof »

4U85-4)!

i. Show that the point P (a sec , b tan ) lies on the hyperbola x

a

y

b

2

2

2

2 1 for all values of .

If Q is the point (a sec , b tan ) where + =

2 show that the locus of the midpoint of PQ

is x

a

y

b

y

b

2

2

2

2 .†

ii. Show that the equation of the normal to the hyperbola x

a

y

b

2

2

2

2 1 at the point (a sec , b tan

) is ax tan + by sec = (a2 + b

2) sec tan .

The ordinate at P meets an asymptote of the hyperbola at Q. The normal at P meets the x

axis at G. Show that GQ is at right angles to the asymptote.†

« Proof »

4U84-4i)!

Show that the condition for the line y = mx + c to be tangent to the ellipse x

a

y

b

2

2

2

21 is

c2 = a

2m

2 + b

2. Show that the pair of tangents drawn from the point (3, 4) to the ellipse

x y2 2

16 91

are at right angles to one another.†

« Proof »

4U84-4ii)!

Show that the equation of the normal at the point P(a sec, b tan) on the hyperbola x

a

y

b

2

2

2

21 is

ax sin + by = (a2 + b

2)tan. The normal at the point P(a sec, b tan) on the hyperbola

x

a

y

b

2

2

2

21

meets the x axis at G and PN is the perpendicular from P to the x axis. Prove that OG = e2.ON (where

O is the origin).†


†©CSSA OF NSW 1984 - 1996


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Documents

HSC Papers, Sorted by Topic (1984-1997), Part 1 of 4