2020 Sydney Technical High School - Adv PDF

Preview

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

Student name: ______________________ Class/Teacher name: ______________________

TRIAL HIGHER SCHOO SCHOOLL CERT CERTIFICATE IFICATE EEXAMINATION XAMINATION

Mathematics Advanced

General Instructions

x x x x x x

Total marks: 100

Section I – 10 marks (pages 1-5) x Attempt Questions 1-10 x Allow about 15 minutes for this section

Reading time – 10 minutes Working time - 180 minutes Write using black pen NESA approved calculators may be used A reference sheet is provided In questions 11-40, show relevant mathematical reasoning and/or calculations

Section II – 90 marks (pages 7-31) x Attempt questions 11-40 x Allow about 2 hours and 45 minutes for this section

Section I

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 indefinite integral of𝑥 3 ?

(A) −

𝑥4

(C) −

2𝑥 2

3

+𝑐

1

(B) − 3𝑥 4 + 𝑐 1

1

+𝑐

(D) − 3𝑥 2 + 𝑐 1

2.

What is the correlation between the variables in this scatterplot? (A) Weak positive (B) Weak negative (C) Moderate positive (D) Moderate negative

1

3. Which diagram could be the graph of the parabola 𝑦 = 2 − (𝑥 + 1)2 ?

4. What is the domain of the function 𝑦 = (A) Domain: [9, ∞) (B) Domain: (9, ∞)

1

√𝑥−9

(C) Domain: (−∞, ∞) (D) Domain: [−3, 3]

2

?

5.

Which of the following equations is likely to be the rule for the graph of the trigonometric function shown above? (A) 𝑦 = 3 + 3sin ( 4 ) π𝑥

(B) 𝑦 = 3 + 3cos ( 4 ) π𝑥

(C) 𝑦 = 3 + 3sin ( 4) 𝑥

(D) 𝑦 = 3 + 3cos (4) 𝑥

6. The probability density function for a continuous random variable 𝑋 is: 𝑓(𝑥) = {

𝑠𝑖𝑛𝑥 0 < 𝑥 < 𝑘 0

otherwise

What is the value of 𝑘? (A)

𝜋

2

(B) 𝜋 (C) 1

(D) 2

3

7. The diagram shows the displacement, 𝑥 metres, of a moving object at time 𝑡 seconds.

Which of the following statements describes the motion of the object at the point 𝐴 (A) Velocity is negative and acceleration is positive (B) Velocity is negative and acceleration is negative (C) Velocity is positive and acceleration is negative (D) Velocity is positive and acceleration is positive

8. The derivative of 𝑒 −4𝑥 𝑐𝑜𝑠2𝑥 with respect to 𝑥 is: (A) 𝑒−4𝑥 (sin2𝑥 − 2cos2𝑥)

(B) 2𝑒−4𝑥 (sin2𝑥 + 2cos2𝑥)

(C) −𝑒−4𝑥 (sin2𝑥 − 2cos2𝑥)

(D) −2𝑒−4𝑥 (sin2𝑥 + 2cos2𝑥) 9. Let 𝑎 = 𝑒 𝑥 . Which expression is equal to 𝑙𝑜𝑔𝑒 (𝑎 2 )? (A) 𝑒 2𝑥 (B) 𝑒 𝑥

2

(C) 2𝑥 (D) 𝑥 2

4

10.

Use the graph above to find the value of k which satisfies ∫ 𝑓(𝑥)𝑑𝑥 = 0 −6 𝑘

(A) 6

(B) 10 (C) 11 (D) 12

5

Mathematics Advanced Section II Answer Booklet

90 marks Attempt Questions 11-40 Allow about 2 hours and 45 minutes for this section

Instructions x

Answer the questions in the spaces provided. These spaces provide guidance for the expected length of response.

x

Your responses should include relevant mathematical reasoning and/or calculations.

Please turn over

7

Question 11 (2 marks) Simplify

𝑥

𝑥 2 −4

−

4 𝑥−2

2

............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ Question 12 (1 mark) Rationalise the denominator of

7

√5 − 2

1

............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ Question 13 (2 marks)

Find the equation of the line that passes through the point (0, −3) and has an angle of inclination of 30°. Leave your answer in gradient-intercept form.

............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ 9

2

Question 14 (2 marks)

Differentiate the following with respect to 𝑥 (a) 𝑓(𝑥) = 𝑡𝑎𝑛7𝑥

1

............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ (b) 𝑓(𝑥) = 𝑙𝑛(𝑥 2 + 2)

1

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

Question 15 (1 mark) Find ∫ (4𝑥 + 3)9 𝑑𝑥

1

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

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

10

Question 16 (3 marks) The number of students absent from Year 12 for the past nine days was as follows: 14, 17, 13, 16, 17, 12, 11, 28, 19 (a) Find the standard deviation. Answer correct to one decimal place.

1

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

(b) Find the lower quartile and the upper quartile

1

............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ (c) Find the interquartile range

1

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

Question 17 (2 marks)

Solve the equation 𝑡𝑎𝑛2 𝑥 = 3 for 0 ≤ 𝑥 ≤ 2𝜋

2

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

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

11

Question 18 (2 marks) There are fifteen marbles in a jar. Five of the marbles are red, five are blue and five are yellow. Ron randomly selects two marbles and puts them in his pocket. (a) What is the probability that the two marbles are red?

1

............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ (b) What is the probability that the two marbles are the same colour?

1

.......................................................................................................................................................................... ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ Question 19 (4 marks) For the following continuous probability distribution

3𝑥 2 for 1 ≤ 𝑥 ≤ 8 𝑓(𝑥) = { 511

(a) Find 𝑃(𝑋 = 5)

0 otherwise

1

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

Question 19 continues on page 13 12

Question 19 continued (b) Find 𝑃(𝑋 ≤ 5)

2

........................................................................................................................................................................... ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ (c) Find 𝑃(𝑋 > 5)

1

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

13

Question 20 (3 marks)

A motorist drives 35 km from Town A to Town B on a bearing of 030°𝑇.

He then drives 55 km to Town C that is on a bearing of 135°𝑇 from Town B.

(a) Find the size of ∠𝐴𝐵𝐶

1

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