Math Paper
2008
H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N
Mathematics
General Instructions • Reading time – 5 minutes • Working time – 3 hours • Write using black or blue pen • Board-approved calculators may be used • A table of standard integrals is provided at the back of this paper • All necessary working should be shown in every question
Total marks – 120 • Attempt Questions 1–10 • All questions are of equal value
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Total marks – 120 Attempt Questions 1–10 All questions are of equal value Answer each question in the appropriate writing booklet. Extra writing booklets are available.
Marks Question 1 (12 marks) Use the Question 1 Writing Booklet.
(a)
Evaluate 2 cos
π correct to three significant figures. 5
2
(b)
Factorise 3x 2 + x – 2.
2
(c)
Simplify
2 1 . − n n +1
2
(d)
Solve 4 x − 3 = 7.
2
(e)
Expand and simplify
(
3 −1 2 3 + 5 .
)(
)
2
(f)
Find the sum of the first 21 terms of the arithmetic series 3 + 7 + 11 + ··· .
2
– 3 –
Marks Question 2 (12 marks) Use the Question 2 Writing Booklet.
(a)
Differentiate with respect to x : (i) (ii) (iii)
( x 2 + 3) 9 x 2 loge x sin x . x+4
2 2 2
(b)
Let M be the midpoint of (–1, 4) and (5, 8). 1 Find the equation of the line through M with gradient − . 2
2
(c)
(i)
⌠ dx Find ⎮ . x ⌡ + 5 π ⌠ 12
1
(ii)
2 Evaluate ⎮ sec 3x dx . ⌡0
3
– 4 –
Marks Question 3 (12 marks) Use the Question 3 Writing Booklet.
(a)
y C (1, 5) D
O A (0, –1)
2x –y –1 =
B (0, 3)
NOT TO SCALE
0
x
In the diagram, ABCD is a quadrilateral. The equation of the line AD is 2x – y – 1 = 0. (i) (ii) (iii) (iv) (v) Show that ABCD is a trapezium by showing that BC is parallel to AD. The line CD is parallel to the x-axis. Find the coordinates of D. Find the length of BC. Show that the perpendicular distance from B to AD is
H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N
Mathematics
General Instructions • Reading time – 5 minutes • Working time – 3 hours • Write using black or blue pen • Board-approved calculators may be used • A table of standard integrals is provided at the back of this paper • All necessary working should be shown in every question
Total marks – 120 • Attempt Questions 1–10 • All questions are of equal value
212
BLANK PAGE
– 2 –
Total marks – 120 Attempt Questions 1–10 All questions are of equal value Answer each question in the appropriate writing booklet. Extra writing booklets are available.
Marks Question 1 (12 marks) Use the Question 1 Writing Booklet.
(a)
Evaluate 2 cos
π correct to three significant figures. 5
2
(b)
Factorise 3x 2 + x – 2.
2
(c)
Simplify
2 1 . − n n +1
2
(d)
Solve 4 x − 3 = 7.
2
(e)
Expand and simplify
(
3 −1 2 3 + 5 .
)(
)
2
(f)
Find the sum of the first 21 terms of the arithmetic series 3 + 7 + 11 + ··· .
2
– 3 –
Marks Question 2 (12 marks) Use the Question 2 Writing Booklet.
(a)
Differentiate with respect to x : (i) (ii) (iii)
( x 2 + 3) 9 x 2 loge x sin x . x+4
2 2 2
(b)
Let M be the midpoint of (–1, 4) and (5, 8). 1 Find the equation of the line through M with gradient − . 2
2
(c)
(i)
⌠ dx Find ⎮ . x ⌡ + 5 π ⌠ 12
1
(ii)
2 Evaluate ⎮ sec 3x dx . ⌡0
3
– 4 –
Marks Question 3 (12 marks) Use the Question 3 Writing Booklet.
(a)
y C (1, 5) D
O A (0, –1)
2x –y –1 =
B (0, 3)
NOT TO SCALE
0
x
In the diagram, ABCD is a quadrilateral. The equation of the line AD is 2x – y – 1 = 0. (i) (ii) (iii) (iv) (v) Show that ABCD is a trapezium by showing that BC is parallel to AD. The line CD is parallel to the x-axis. Find the coordinates of D. Find the length of BC. Show that the perpendicular distance from B to AD is