2010 HSC Mathematics Sample Answers This document contains ‘sample answers’, or, in the case of some questions, ‘answers could include’. These are developed by the examination committee for two purposes. The committee does this: (a) as part of the development of the examination paper to ensure the questions will effectively assess students’ knowledge and skills, and (b) in order to provide some advice to the Supervisor of Marking about the nature and scope of the responses expected of students. The ‘sample answers’ or similar advice are not intended to be exemplary or even complete answers or responses. As they are part of the examination committee’s ‘working document’, they may contain typographical errors, omissions, or only some of the possible correct answers. – 1 –
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2010 HSC Mathematics Sample Answers
This document contains ‘sample answers’, or, in the case of some questions, ‘answers could include’. These are developed by the examination committee for two purposes. The committee does this:
(a) as part of the development of the examination paper to ensure the questions will effectively assess students’ knowledge and skills, and
(b) in order to provide some advice to the Supervisor of Marking about the nature and scope of the responses expected of students.
The ‘sample answers’ or similar advice are not intended to be exemplary or even complete answers or responses. As they are part of the examination committee’s ‘working document’, they may contain typographical errors, omissions, or only some of the possible correct answers.
– 1 –
2010 HSC Mathematics Sample Answers
Question 1 (a)
2x = 4x
x2 − 4x = 0
x(x − 4) = 0 x = 0 or x = 4
Question 1 (b)
1 5 + 2 5 + 2× = = 5 + 2 5 − 2 5 + 2 5 − 4
∴ a = 2 b = 1
Question 1 (c)
(x + 1)2 + ( y − 2)2 = 25
Question 1 (d)
2x + 3 = ±9
2x + 3 = 9 or 2x + 3 = −9
2x = 6 2x = −12
x = 3 or x = −6
Question 1 (e)
Let y = x2 tan x
y′ = x2 sec2 x + 2x tan x
Question 1 (f)
1 a 1 3 a = 1 r = − S = = =
3 1 − r 1 4 1 − −
3
Question 1 (g)
x ≥ 8
– 2 –
2010 HSC Mathematics Sample Answers
Question 2 (a)
cos xLet y =
x
x(– sin x) − cos x −x sin x − cos x y′ = =
x2 x2
Question 2 (b)
(x − 4)(x + 3) < 0
−3 < x < 4
Question 2 (c)
y = ln3x
1 y′ =
x 1
When x = 2 y′ = 2
Question 2 (d) (i)
1 3⌠ 1 2 2⎮ (5x + 1)2 dx = . (5x + 1)5 3⌡
3
= 2 (5x + 1)2 + C
15
Question 2 (d) (ii)
⌠ x 1dx = ln(4 + x2) + C⎮ 2⌡ 4 + x 2
– 3 –
2010 HSC Mathematics Sample Answers
Question 2 (e)
⌠ 6 ⎡ 6
x2 ⎤ ⎮ (x + k) dx =⎢ + kx⎥ = 18 + 6k⎮⌡ 2
0 ⎣ ⎦0
18 + 6k = 30
6k = 12
k = 2
Question 3 (a) (i)
⎛ −2 + 12 −4 + 6⎞ , = (5, 1)⎝⎜ 2 2 ⎠⎟
Question 3 (a) (ii)
8 − 6 1 m = = −
BC 6 − 12 3
Question 3 (a) (iii)
2 − 1 1 m = = −
MN 2 − 5 3 ∴ BC � MN
∴∠ANM = ∠ACB (corresponding angles on parallel lines equal)
and ∠AMN = ∠ABC (corresponding angles on parallel lines equal)
n=� 174.143� ∴There will be money in the account for 175 months.
– 16 –
2010 HSC Mathematics Sample Answers
Question 9 (b) (i)
0 ≤ x < 2
Question 9 (b) (ii) 4
Question 9 (b) (iii) −6
Question 9 (b) (iv)
1 2 3 4 5 6
(2, 4)
– 17 –
2010 HSC Mathematics Sample Answers
Question 10 (a) (i)
Let ∠CAD = α Then ΔACD has two angles equal to α (since it is isosceles) and ΔABC has two angles equal to α (since it is isosceles). Hence (since 3rd angles must be equal) ΔABC ||| ΔACD (equiangular).
Question 10 (a) (ii)
Since corresponding sides are in proportion, AB AC = AC AD