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

TitleHSC Papers, Sorted by Topic (1984-1997), Part 1 of 4
TagsEllipse Circle Complex Number Tangent Asymptote
File Size527.7 KB
Total Pages36
Document Text Contents
Page 1

4 UNIT MATHEMATICS – GRAPHS – HSC

¤©BOARD OF STUDIES NSW 1984 - 1997

©EDUDATA: DATAVER1.0 1995

Graphs
4U97-3b)!

Let f(x) = 3x
5
- 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

Inflection
points

»

4U96-4b)!

i. On the same set of axes, sketch and label clearly the graphs of the functions y x
1

3 and

y e
x

 .

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

the function y x e
x


1

3 .

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

1

3 1  has exactly
one solution.¤

« i)

y = x
1

3

y = e
x

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)

Page 2

4 UNIT MATHEMATICS GRAPHS HSC

¤©BOARD OF STUDIES NSW 1984 - 1997

©EDUDATA: DATAVER1.0 1995

v. y = e
f(x)

. ¤

« (i)

(3, 1)

x

y = f(x)

y

-8

2 4

(ii)

(3, 1)

x

y = f(x)

y

2 4

8



(iii)

y
2
= f(x)

x

y

1

-1

42 30

(iv)

 18
y

1

f(x)
y

1

f(x)


y
1

f(x)


y

x

(3, 1)

2 4

asymptote



(v)

e

y

x3

(3, e)

y = e
f(x)

asymptote
»

4U94-5a)!

Let f x
x x

x
( )

( )( )
,

 




2 1

5
5 for x .

i. Show that f x x
x

( )    


4
18

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. ¤

Page 18

4 UNIT MATHEMATICS COMPLEX NUMBERS 1 HSC

¤©BOARD OF STUDIES NSW 1984 - 1997

©EDUDATA: DATAVER1.0 1995

« (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 3
3 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 z
7

1 where z 1.

i. Deduce that z z z
z z z

3 2
2 31

1 1 1
0       .

ii. By letting x z
z

 
1

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

iii. Hence deduce that cos cos cos
  
7

2
7

3
7

1
8

 .¤

« (i) Proof (ii) x
3
+ 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.¤

Page 19

4 UNIT MATHEMATICS COMPLEX NUMBERS 1 HSC

¤©BOARD OF STUDIES NSW 1984 - 1997

©EDUDATA: DATAVER1.0 1995

«

-2

Q( 3, 1)R( 3, 1) 

-1 1 2 3

1

2

3

-1

y

x

P(0, 2)

»

4U91-2b)!

Let z1 = cos1 + isin1 and z2 = cos2 + isin2, where 1 and 2 are real. Show that:



i.
1

z1
= cos1 - isin1

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) 0
2
    .¤

« (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 »

Page 35

4 UNIT MATHEMATICS CONICS CSSA

OF NSW 1984 - 1996

©EDUDATA: DATAVER1.0 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 x
2
- y

2
= c

2
and xy = c

2


« Proof »

4U86-4ii)!

Show that the tangent to the hyperbola
x

a

y

b

2

2

2

2
1  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 = (a
2
+ b

2
) sectan. 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



« 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  = (a
2
+ b

2
) sec  tan .

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



« 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

2
1  is

c
2
= a

2
m

2
+ b

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

x y
2 2

16 9
1 



« 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

2
1  is

ax sin + by = (a
2
+ b

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

x

a

y

b

2

2

2

2
1 

meets the x axis at G and PN is the perpendicular from P to the x axis. Prove that OG = e
2
.ON (where

Page 36

4 UNIT MATHEMATICS – CONICS – CSSA

†©CSSA OF NSW 1984 - 1996

©EDUDATA: DATAVER1.0 1996

« Proof »

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