2008

HSC Preliminary Course

FINAL EXAMINATION

Mathematics – Extension 1

General Instructions

Reading Time – 5 minutes

Working Time – 2 hours

Write using a blue or black pen

Board approved calculators may be use o All necessary working should be shown for every question o Begin each question on a fresh sheet of paper o o o o Total marks (87) o Attempt Questions 1 – 6

Preliminary Course 2008

Question 1

Final Examination

(15 Marks)

Mathematics – Extension 1

Use a Separate Sheet of paper

3x

≤3

Marks

(a)

Solve the inequality

(b)

Solve a – b + c = 7 and a + 2b – c = –4 and

3a – b – c = 3 simultaneously.

3

(c)

Show that the line x + y + 4 = 0 is a tangent to the circle x2 + y2 = 8.

3

(d)

Find the acute angle between the lines x + 2y = 0 and x – 3y = 0.

3

(e)

For the points A(–3, –7) and B(–1, –4). Find the coordinates of the point P(x , y) which divides the interval AB externally in the ratio 4:3.

3

3x – 4

3

End of Question 1

–2–

Preliminary Course 2008

Question 2

(a)

Mathematics – Extension 1

Final Examination

(12 Marks)

Use a Separate Sheet of paper

In the triangle below, AB || FE and FCE = DEB = 90.

Marks

4

A

D

F

C

(b)

E

B

i)

Prove that ∆ABC ||| ∆DBE and ∆ABC ||| ∆FCE

ii)

If DE : FC = 5 : 2, FE = 3.2cm and CE = 2.4cm, find the length of AB

Find the values of w, x, y and z, giving reasons.

P

4

Q

z

T

70

O

•

x

w

50

y

S

R

Question 2 continues on page 4

–3–

Preliminary Course 2008

Mathematics – Extension 1

Final Examination

Question 2 (continued)

(c)

Marks

The point O is the centre of the circle, TU is a tangent to the circle, contacting the circle at P.

T

R

P

O •

U

S

Q

i)

Show that ROP = 2RPT

ii)

Show that RPT and QPU are complementary

iii)

Show that RP || SQ

End of Question 2

–4–

4

Preliminary Course 2008

Question 3

Mathematics – Extension 1

Final Examination

(15 Marks)

Use a Separate Sheet of paper

Marks

θ

(a)

Write sinθ + cosθ in…