3U Cambridge 2020 - Extension 1 Mathematics Paper PDF

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Extension 1 Mathematics Paper ...

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CambridgeMATHS Practice HSC Examination

Mathematics Extension 1

General Instructions • Reading time – 10 minutes • Working time – 2 hours • Write using black pen • Draw diagrams in pencil • NESA-approved calculators may be used • The HSC Reference Sheet may be used • In Section II, show relevant mathematical reasoning and/or calculations

Total marks – 70 • Section I is worth 10 marks • Section II is worth 60 marks

Section I Attempt Questions 1–10 Allow about 15 minutes for this section Marks 1.

Which function is a primitive of A. B. C. D.

2.

1

9

1

 ?

1+9฀฀2

tan−1 9฀฀

1 tan−1 3฀฀ 3 1

tan−1

1

tan−1

9

3

฀฀

9

฀฀

3

Which expression is equal to cos ฀ ฀ + sin ฀ ฀ ? ฀฀

A. √2 cos �฀฀ + � 4 ฀฀

B. √2 cos �฀฀ − � 4 ฀฀

C. 2 cos �฀฀ + � 3 ฀฀

D. 2 cos �฀฀ − � 3 3.

Which expression is equal to cos 4฀฀  ? A. 1 − 2 cos 2 8฀฀ B. 1 − 2 cos 2 2฀฀ C. 2 cos 2 8฀฀ − 1 D. 2 cos 2 2฀฀ − 1

2

4.

A slope field is shown in the diagram below. Which differential equation could correspond to this slope field? ฀฀

฀฀

A. B. C. D.

฀฀฀฀

฀฀฀฀

= 2฀ ฀ + ฀฀

฀฀฀฀ ฀฀฀฀

= 2฀฀ − ฀฀

฀฀฀฀ ฀฀฀฀ ฀฀฀฀

฀฀฀฀

= 2฀฀฀฀ =

2฀฀ ฀฀

3

5.

The integral ∫

฀฀2

 ฀฀฀฀ is transformed into which of the following integrals by the

√1−฀฀6

substitution ฀ ฀ = ฀฀ 3 ? 1

A. ∫ 3√1−฀฀2 ฀฀฀฀ 3

B. ∫ √1−฀฀2 ฀฀฀฀ C. ∫

1

3√1−฀฀3

฀฀฀฀

3

D. ∫ √1−฀฀3 ฀฀฀฀ 6.

What is the probability of getting 3 sixes and 2 fours in any order when a standard die is rolled 5 times? A. B. C. D.

7.

1 7776 5

3888 125

3888 625

3888

What is the component of the vector 4฀฀ in the direction of the vector ฀ ฀ + ฀฀? A. 2√2 ฀฀ + 2√2  ฀ ฀ B. 4√2 ฀฀ + 4√ 2 ฀฀ C. 2฀ ฀ + 2฀฀ D. 4฀ ฀ + 4฀฀

8.

Which of the following is a general solution of the differential equation ฀฀฀฀

฀฀฀฀

= 2 sin ฀ ฀ cos ฀ ฀ ? 1

A. ฀ ฀ = ฀฀ − cos 2฀ ฀ for some constant ฀฀ 2 1

B. ฀ ฀ = ฀ ฀ +cos 2฀ ฀ for some constant ฀฀ 2 1

C. ฀ ฀ = ฀฀ − sin 2฀ ฀ for some constant ฀฀ 2 1

sin 2฀ ฀ for some constant ฀฀ D. ฀ ฀ = ฀ ฀ + 2

4

9.

The diagram below shows the velocity at time ฀฀ of a particle that moves over a 10 second time interval. For what percentage of the time is the speed of the particle decreasing? ฀฀ 6

9 −2

0

2

4

10

฀฀

A. 20% B. 30% C. 60% D. 70%

10.

How many solutions does the equation 2฀ ฀ + 3฀ ฀ sin ฀ ฀ = 0 have for 0 ≤ ฀฀ ≤ 2฀฀? A. 1 B. 2 C. 3 D. 4

5

Section II Attempt Questions 11–14 Allow about 1 hour and 45 minutes for this section

Question 11 (15 marks) ฀฀

(a)

By using the substitution ฀ ฀ = sin ฀฀, or otherwise, evaluate04∫  sin4 ฀ ฀ cos ฀฀ ฀฀฀฀ .

(b)

Use a ฀฀-formula to solve the equation sin ฀ ฀ = tan฀฀, for 0 ≤ ฀฀ ≤ 2฀฀.

(c)

The inverse of the function ฀ ฀ = √฀฀ − 1 is ฀ ฀ = �฀฀ − 1.

2

2

1

3

2

฀฀฀฀

Find   and hence find the derivative of the inverse function in terms of ฀฀. ฀฀฀฀

(d)

(e)

A triangle ฀฀฀฀฀฀ has vertices ฀฀(1, 1), ฀฀(5, 2) and ฀฀(4, 6). (i)

����� and �฀฀฀฀ ���� as column vectors. Write ฀฀฀฀

(ii)

Hence show that ∠฀฀฀฀฀฀ = 45°.

(iii)

Find the area of the triangle.

Prove by mathematical induction that for all positive integer values of ฀฀:

1 2 1

4

1 1 × 3 + 2 × 5 + 3 × 7 + ⋯ + ฀฀(2฀ ฀ + 1) = ฀฀(฀ ฀ + 1)(4฀ ฀ + 5) 6

End of Question 11

6

Question 12 (15 marks) (a)

Consider the expression √2 sin ฀฀ − √6 cos ฀฀. (i)

Write the expression in the form ฀ ฀ sin(฀฀ − ฀฀), where ฀ ฀ > 0 and 0 < ฀ ฀ ฀฀. It strikes the inclined plane at ฀฀, which is the vertex of the parabolic path of the particle. You may assume that the parabolic path has parametric equations 1 ฀ ฀ = ฀฀฀฀ cos ฀ ฀ , ฀ ฀ = ฀฀฀฀ sin ฀฀ −2 . ฀฀฀฀ 2 (a)

Find, in terms of ฀฀, ฀฀ and ฀฀, the time it takes for the particle to reach ฀฀.

1

(b)

Show that tan ฀ ฀ = 2 tan ฀฀.

4

End of examination

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