4U Cambridge 2020 - Extension 2 Mathematics Paper PDF

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Extension 2 Mathematics Paper ...

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Practice HSC Examination

Mathematics Extension 2

General Instructions • Reading time – 10 minutes • Working time – 3 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 – 100 • Section I is worth 10 marks • Section II is worth 90 marks

Practice HSC Examination

Extension 2

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

What is the negation of the statement: If a triangle has two equal sides, then it has two equal angles. A. If a triangle does not have two equal sides, then it does not have two equal angles. B. If a triangle does not have two equal angles, then it does not have two equal sides. C. A triangle can have two equal sides and it not have two equal angles. D. A triangle can have two equal angles and it not have two equal sides.

2.

Which function is a primitive of A. ln|2฀฀ − 1|

2฀฀+1 ? 2฀฀−1

B. ฀ ฀ + ln|2฀฀ − 1|

C. ฀ ฀ + 2 ln|2฀฀ − 1| D. 2฀ ฀ + ln|2฀฀ − 1|

3.

Which number is a fourth root of ฀฀? ⁄ A. ฀฀ ฀฀฀฀

8

⁄ C. ฀฀ ฀฀฀฀

4

⁄ B. ฀฀ 3฀฀฀฀

⁄ D. ฀฀ 3฀฀฀฀

8 4

Page 2 of 11

Practice HSC Examination

4.

Extension 2

2 A line has gradient − 3 and passes through the point (3, 2). Which of the following is

a vector equation of the line?

฀฀ −2 3 A. �฀฀� = � � + ฀฀ � � , where ฀฀ ∈ ℝ 3 2 ฀฀ 3 3 B. �฀฀� = � � + ฀฀ � � , where ฀฀ ∈ ℝ −2 2 ฀฀ −2 0 C. �฀฀� = � � + ฀฀ � � , where ฀฀ ∈ ℝ 4 3 ฀฀ 3 0 D. �฀฀� = � � + ฀฀ � � , where ฀฀ ∈ ℝ 4 −2

5.

The acceleration of a particle moving in a straight line with velocity ฀฀ is given by ฀฀󰇘 = ฀฀ 2 . Initially ฀ ฀ = 1. What is ฀฀ as a function of ฀฀? A. ฀ ฀ = 1 − ฀฀

B. ฀ ฀ = ln|1 − ฀฀| ฀฀

C. ฀ ฀ =1−฀฀ 1

D. ฀ ฀ =1−฀฀

6.

Suppose that ฀฀ is an integer and ฀ ฀ > 2. What is the largest number that is always a divisor of ฀฀3 − ฀฀? A. 2 B. 3 C. 6 D. 12

Page 3 of 11

Practice HSC Examination

7.

Extension 2

In the diagram below, the points ฀฀ and ฀฀ represent the complex numbers ฀฀1 and ฀฀2 respectively. In which quadrant is the point ฀฀ representing ฀฀2 − ฀฀1 ? Im

฀฀(฀฀1 ) ฀฀(฀฀2 ) Re

A. 1st quadrant B. 2nd quadrant C. 3rd quadrant D. 4th quadrant

8.

8

Into which definite integral is ∫0 8 2฀฀

A. ∫0

1+฀฀

4 ฀฀

B. ∫0

1+฀฀

8 2

C. ∫0

4

D. ∫0

9.

1+฀฀ 1

1+฀฀

฀฀฀฀

1

1+√2฀฀

฀฀฀฀transformed by the substitution ฀ ฀ = √2฀฀?

฀฀฀฀

฀฀฀฀

฀฀฀฀

The line ℓ1 has vector equation ฀฀฀ ฀ = ฀ ฀ + ฀฀(฀฀ − ฀฀) and the line ℓ2 has vector equation ฀฀฀ ฀ = (3฀ ฀ + 2฀฀ − ฀฀) + ฀฀(2฀ ฀ + ฀฀).

Which of the following statements is correct? A. ℓ1 and ℓ2 are parallel

B. ℓ1 and ℓ2 are perpendicular

C. ℓ1 and ℓ2 intersect at a point D. ℓ1 and ℓ2 are skew

Page 4 of 11

Practice HSC Examination

10.

Extension 2

Which of these inequalities is false? (Do not attempt to evaluate the integrals.) A. B. C. D.

2 1

∫1

฀฀

∫฀฀3   6

2

1+฀฀

21

sin ฀฀ ฀

฀฀฀฀< ∫1

฀

฀฀

฀฀

฀฀฀฀

1

< ∫฀฀3   ฀฀฀฀ ฀฀฀฀ ฀฀ 6

1

∫1 ฀฀ −฀฀ ฀฀฀฀ < ∫0 ฀฀ −฀฀ ฀฀฀฀ ฀฀

2

2

฀฀

∫04  tan2 ฀฀ ฀฀฀฀< ∫04  tan3 ฀฀ ฀฀฀฀

Page 5 of 11

Practice HSC Examination

Extension 2

Section II Attempt Questions 11–16 Allow about 2 hours and 45 minutes for this section

Question 11 (15 marks) 23−14฀฀ 3−4฀฀

in the form ฀ ฀ + ฀฀฀฀, where ฀฀, ฀฀ ∈ ℝ.

(a)

Express

(b)

Find the square roots of −16 + 30฀฀.

(c)

Let ฀ ฀ = −√3 + ฀฀. (i) (ii)

(d)

Find ∫

(e)

(i) (ii)

(f)

2

2

Write ฀฀ in modulus-argument form.

1

Hence find ฀฀ 9 in Cartesian form. 1

√10฀฀−฀฀2

2

฀฀฀฀.

2

Find the values of ฀฀ and ฀฀ such that Hence find ∫

2฀฀2 +3 ฀฀2 +฀฀

฀฀฀฀.

2฀฀2 +3 ฀฀2 +฀฀

฀฀

฀฀

= 2 +฀ ฀ +฀฀+1.

2 1

฀฀

Find the exact value of ∫1 ฀฀ 4 ln ฀฀ ฀฀฀฀.

End of Question 11

Page 6 of 11

3

Practice HSC Examination

Extension 2

Question 12 (15 marks) (a)

The points ฀฀(1, −2, 3) and ฀฀(−5, 4, −1) lie on the line ℓ. (i)

(ii)

(b)

Does the point ฀฀(43, −44, 29) lie on ℓ ?

1

where ฀฀ is real.

(ii)

Find the other two zeroes of ฀฀(฀฀).

2

Find the value of ฀฀.

2

(i)

Use the result cos 3฀ ฀ + ฀ ฀ sin 3฀ ฀ = (cos ฀ ฀ + ฀ ฀ sin ฀฀)3 to find

(ii)

Hence show that tan 3฀ ฀ =

(iii)

(d)

2

It is known that 5 + 6฀฀ is a zero of the polynomial ฀฀(฀฀) = 2฀฀ 3 − 19฀฀ 2 + 112฀ ฀ + ฀฀, (i)

(c)

Find a vector equation for ℓ.

2

expressions for sin 3฀฀ and cos 3฀฀ in terms of sin ฀฀ and cos ฀฀. ฀฀

3 tan ฀฀−tan3 ฀฀ 1−3 tan2 ฀ ฀

.

1

Deduce that ฀ ฀ = tan12 is a root of the equation ฀฀ 3 − 3฀฀ 2 − 3฀ ฀ + 1 = 0. 1

Use the substitution ฀ ฀ = √2 sin ฀฀ to show that0∫

End of Question 12

Page 7 of 11

฀฀2

√2−฀฀2

฀฀฀฀ =4 (฀฀ − 2). 1

2

3

Practice HSC Examination

Extension 2

Question 13 (15 marks)

(a)

−3 4−9�� = √14. Two spheres have vector equations �฀฀ − �−5 �� = 2√14 and �฀฀ − �7 10 Prove that the spheres touch each other at a single point.

3

(b)

Prove by contradiction that log 5 13 is an irrational number.

3

(c)

Suppose that ฀฀ and ฀ ฀ + 1 are positive integers, neither of which is divisible by 3.

(d)

A particle is projected from the origin with positive velocity and moves in simple

3

Prove that ฀฀3 + (฀ ฀ + 1)3 is divisible by 9.

harmonic motion about the origin. The motion has amplitude 0.2m   and period

4 seconds. Find, leaving answers in exact form: (i)

the initial speed of the particle,

2

(ii)

the speed of the particle after 1.5 seconds,

2

(iii)

when the particle first reaches a point 0.1m   from the origin.

End of Question 13

Page 8 of 11

2

Practice HSC Examination

Extension 2

Question 14 (15 marks) (a)

Sketch the subset of the complex plane determined by the relation

3

1 11 . += 2 ฀ ฀  ฀  ฀

(b)

3 2 Let ฀฀ and ฀฀ be the points on the line ฀ ฀ = �1 � + ฀฀ �−2� corresponding to ฀ ฀ = 1 and −1 1 ฀ ฀ = −1 respectively, and let ฀฀ be the point (1, 0, 2). (i) (ii)

(c)

(d)

����� onto �� ��� . ฀฀฀฀ Find the projection of ฀฀฀฀

Hence find the perpendicular distance ฀฀ from ฀฀ to the line ฀฀฀฀. ฀฀ − 1

For a complex number z, arg �

� = ฀฀ + 1

2 2

฀฀ 4

(i)

Find the cartesian equation of the locus of z.

3

(ii)

Sketch the locus of z.

2

Find

�

1

฀฀ �฀฀ (1 + ฀฀ )

฀฀฀฀

3

End of Question 14

Page 9 of 11

Practice HSC Examination

Extension 2

Question 15 (15 marks) (a)

Consider the polynomial ฀฀(฀฀) = ฀฀ 4 + 9. (i) (ii)

(b)

Find the zeroes of ฀฀(฀฀) in modulus-argument form and Cartesian form. Hence write ฀฀(฀฀) as a product of two real quadratic factors. 1

Let ฀฀฀ ฀ =0∫ (i) (ii)

4 2

฀฀฀฀

฀฀2 +1

฀฀฀฀, for ฀฀ ≥ 2.

1

Show that ฀฀฀ ฀ + ฀฀฀฀−2 = . ฀฀−1

1 ฀฀2 (฀฀−1)2

Hence show that ∫0  

฀฀2 +1

2 2

฀฀฀฀ = ln 2− 3.

฀฀(฀฀)

฀฀฀฀

(c)

3

฀฀(฀฀) ฀฀(฀฀)

฀฀

฀฀฀฀

In the Argand diagram above, ฀฀฀฀฀฀ is an equilateral triangle. The points ฀฀, ฀฀, ฀฀ represent the complex numbers ฀฀, ฀฀, ฀฀ respectively.

(i) (ii) (iii)

฀฀฀฀

Explain why ฀฀ − ฀฀ = (฀฀ − ฀฀ ฀)฀ 3 .

1

By squaring and adding the results in parts (i) and (ii),

2

Write down similar expressions for ฀฀ − ฀฀ and ฀฀ − ฀฀. deduce that ฀฀ 2 + ฀฀ 2 + ฀฀ 2 = ฀฀฀฀ + ฀฀฀฀ + ฀฀฀฀.

End of Question 15

Page 10 of 11

1

Practice HSC Examination

Extension 2

Question 16 (15 marks) (a)

฀฀ ฀฀

฀฀

฀฀

฀฀

฀฀

In Δ฀฀฀฀฀฀ above, ฀฀ is the point on ฀฀฀฀ such that ฀฀฀฀: ฀฀฀฀ = 2: 1 and ฀฀ is the point

on ฀฀฀฀ such that ฀฀฀฀: ฀฀฀฀ = 1: 2.

When ฀฀฀฀ is extended, it meets the extension of ฀฀฀฀ at ฀฀.

��� = ฀฀ and �฀฀฀฀ ��� = ฀฀. Let �� ฀฀฀฀ (i) (ii) (iii)

(b)

����� in terms of ฀฀ and ฀฀. Express ฀฀฀฀ ���� = 2 ฀฀ − 1 ฀฀. Show that �฀฀฀฀ 3 3

Show that ฀฀฀฀: ฀฀฀฀ = 4: 1.

1 4

Suppose that ฀฀ is a prime number and ฀฀ is a positive integer. (i)

Use the expansion of (1 + ฀฀)฀฀ to prove that (1 + ฀฀)฀ ฀ − 1 − ฀฀ ฀฀ is divisible by ฀฀.

�You may use the fact that �

(ii)

(c)

1

฀฀ � is divisible by ฀฀ for ฀ ฀ = 1, 2, … , ฀฀ − 1. � ฀฀

Hence prove by mathematical induction that ฀฀ ฀ ฀ − ฀฀ is divisible by ฀฀ for all positive integer values of ฀฀.

Suppose that ฀฀, ฀฀, ฀฀, ฀฀, ฀฀, ฀ ฀ > 0 and ฀ ฀ + ฀ ฀ = ฀ ฀ + ฀ ฀ = ฀ ฀ + ฀ ฀ = ฀฀. Prove that ฀฀฀฀ + ฀฀฀฀ + ฀฀฀฀ < ฀฀ 2 .

End of examination Page 11 of 11

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