SBHS 2020 MA THSC PDF

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Summary

Due to the implementation of the new syllabus, past paper resources that cover the new content is almost impossible to find. HOWEVER, I have collated 35+ past papers from different schools for their 2020 trials :)...

Description

SYDNEY BOYS HIGH SCHOOL

2020

NESA Number: Name: Class:

YEAR 12 TERM 3 TRIAL HSC

Mathematics Advanced General Instructions

•

Reading time – 10 minutes

•

Working time – 3 hours

•

Write using black pen

•

NESA approved calculators may be used

•

A reference sheet is provided with this paper

•

Marks may NOT be awarded for messy or badly arranged work

•

For questions in Section II, show ALL relevant mathematical reasoning and/or calculations

Total Marks: 100

Section I – 10 marks (pages 2 – 6) •

Attempt Questions 1 – 10

•

Allow about 15 minutes for this section

Section II – 90 marks (pages 7 – 32)

Examiner: BK

Section I

•

Attempt all Questions in Section II

•

Allow about 2 hours and 45 minutes for this section

10 marks Attempt Questions 1–10 Allow about 15 minutes for this section Use the multiple-choice answer sheet for Questions 1–10

1

What is the derivative of x cos x with respect to x ? A.

−x sin x

B.

− sin x

C.

x sin x − cos x

D.

−x sin x + cos x 2

2

3

4

⌠ Evaluate ⎮ e2 x+1 dx. ⌡1 A.

1 2 e 2

B.

1 3 2 e (e − 1) 2

C.

e2

D.

e2 +1

Which of the following is a primitive function of 4 + x ? A.

4x +

C.

4x +

2 x3 3 1

B.

D.

4x +

3 x2 2

1 2 x

2 x

A raffle consists of 20 tickets in which there are two prizes. David buys 5 tickets. First prize is two movie vouchers and second prize is one movie voucher. What is the probability that David wins at least one movie voucher? A.

5 20

B.

27 76

C.

7 16

D.

17 38

–2–

5

The graph of the derivative y = f ′ (x) is drawn below.

At which of the following points is there a maximum turning point on y = f (x) ?

6

A.

x = -1

B.

x = 1.4

C.

x=5

D.

x = 7.2

What is the solution of 5x = 4 ? A.

x=

log e 4 5

B.

x=

4 loge 5

C.

x=

loge 4 loge 5

D.

⎛ 4⎞ x = log ⎜ ⎟ ⎝5⎠

–3–

7

2 from x = 1 to x = d . x What value of d makes the shaded area equal to 2?

The diagram shows the area under the curve y =

NOT TO SCALE

8

A.

e

B.

e +1

C.

2e

D.

e2

What is the domain of the function f (x) =

A.

⎛ 1 1⎞ ⎜⎝ − 2 , 2⎟⎠

B.

⎛ 1⎞ ⎛ 1 ⎞ ⎜⎝ −∞, − 2 ⎟⎠ ∪ ⎜⎝ 2 , ∞⎟⎠

C.

⎛ 1⎤ ⎛1 ⎞ ⎜⎝ −∞, − 2 ⎥ ∪ ⎜⎝ 2 , ∞⎟⎠ ⎦

D.

⎡ 1 1⎤ ⎢ − 2 , 2⎥ ⎦ ⎣

1 4x 2 −1

–4–

?

9

Which one of the following best shows the graph of the function y = 1 - sin x for ⎡⎣ 0, 2π ⎦⎤ ? A.

B.

C.

D.

–5–

10 v 9

4

t

When t = 0 the displacement x is equal to 3 metres. What is the maximum value of the displacement x? A.

9m

B.

15 m

C.

18 m

D.

21 m

–6–

NESA Number

Q11-15

YEAR 12 Mathematics Advanced

TERM 3 Cohort Task #3 (THSC)

Part A

–7–

Section II Part A 14 marks Attempt Questions 11–15 Answer each question in the space provided. A blank page is provided at the end of this question to allow rewriting of a part. Your responses should include relevant mathematical reasoning and/or calculations. Question 11 (1 mark) Factorise 8x 3 +125

1

................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

Question 12 (1 mark) Express 260° as an exact radian value.

1

................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

Question 13 (2 marks) Solve 2x − 1 = 5

2

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. –8–

Question 14 (6 marks) Differentiate the following with respect to x. 2 1 + (a) x 3x

2

................................................................................. ................................................................................. .................................................................................

(b)

8x ex

2

................................................................................. ................................................................................. ................................................................................. .................................................................................

(c)

loge (4 x 2 + 3)

2

................................................................................. .................................................................................

Question 15 (4 marks) Find the following: (a)

⌠ 2 ⎮ 6x − 7x +1 dx ⌡

1

................................................................................. .................................................................................

(b)

⌠ ⎛ 2x 3⎞ ⎮ ⎜⎝ 4e + x ⎟⎠ dx ⌡

2

................................................................................. ................................................................................. .................................................................................

(c)

⌠ ⎮ 6 cos5x dx ⌡

1

................................................................................. ................................................................................. –9–

Use this space to re-write any questions for Part A. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

End of Part A – 10 –

NESA Number

Q16-19

YEAR 12 Mathematics Advanced

TERM 3 Cohort Task #3 (THSC)

Part B

– 11 –

Section II Part B 17 marks Attempt Questions 16–19 Answer each question in the space provided. Your responses should include relevant mathematical reasoning and/or calculations. Question 16 (1 mark) Classify the function y = sin x as one-to-many, many-to-one, many-to-many or one-to-one.

1

................................................................................. .................................................................................

Question 17 (5 marks) Given that f ′′(x) = 6x − 2 and that there is a stationary point on f (x) at (1, 2) , find (a) f (x)

3

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

(b)

The co-ordinates of any point(s) of inflection. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

– 12 –

2

Question 18 (7 marks) (a)

Differentiate y = esin x

1

................................................................................. .................................................................................

(b)

Show y = esin x has 2 stationary points for 0 £ x £ 2p and find the coordinates of these stationary points in simplest exact form.

2

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

(c)

Find the nature of these stationary points.

2

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

(d)

sin x Sketch y = e for 0 ≤ x ≤ 2π .

2

– 13 –

Question 19 (4 marks) Given y = x x + 1 , (a)

Show

dy 3x + 2 = dx 2 x + 1

2

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. 8

(b)

⌠ 3x + 2 dx Hence, or otherwise, evaluate ⎮ ⌡3 x + 1

................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

Use this space to re-write any questions for Part B. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

End of Part B – 14 –

2

NESA Number

Q20-25

YEAR 12 Mathematics Advanced

TERM 3 Cohort Task #3 (THSC)

Part C

– 15 –

Section II Part C 16 marks Attempt Questions 20–25 Answer each question in the space provided. Your responses should include relevant mathematical reasoning and/or calculations. Question 20 (1 mark) 4

⌠ 3 Find ⎮ x dx ⌡−4

1

................................................................................. .................................................................................

Question 21 (1 mark) Given that f (x) = 2x +1 and g(x) = x 2 + 5 , find f (g(−3) ).

1

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

Question 22 (3 marks) Find the value(s) of k for which the equation y = (k +1)x 2 − (2 + k)x + 3 is positive definite. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. – 16 –

3

Question 23 (4 marks) Differentiate the following. (a)

⎛ x + 4⎞ y = loge ⎜ ⎝ x − 3 ⎟⎠

2

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. .................................................................................

(b)

y = 8sin x ln x

2

................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ................................................................................. ............................................................