General Instructions

• Reading time – 10 minutes

• Working time – 3 hours

• NESA–approved calculators are permitted

for use

• For questions in Section II, all

mathematical reasoning and/or

calculations are to be present.

Marks will be deducted for carelessly

arranged work, or for the presentation of

minimal reasoning.

Student Number ……………………………

2020

Higher School Certificate

Sample Examination

Mathematics Extension 2

Total marks – 100

This paper has 2 sections:

Section I (10 marks)

• Attempt multiple choice Questions 1–10

• Allow about 15 minutes for this section

Section II (90 marks)

• Attempt short answer Questions 11–16

• Allow about 2 hours and 45 minutes for

this section

Please write your student number at the

bottom of this page.

Disclaimer: This is not an official examination intended to be used for examination

purposes. This is a simple practice unofficial resource provided to students to rehearse for

the upcoming Mathematics Extension 2 Higher School Certificate Examinations.

Section I

10 marks

Attempt Questions 1–10

Allow about 15 minutes for this section

Answers to these questions should be submitted on the multiple-choice answer grid provided

below. This sheet is to remain within the examination booklet – please do not discard or tear

the paper.

Multiple Choice Answer Grid

A B C D

1

2

3

4

5

6

7

8

9

10

±14

4

±14

2

±7

2

1

0

x2

x2 + 1dx

π

4

π

4

4 π

4

4 π

4

1 Let v = 2i + 3j – ak, and u = i – 3j – 2ak.

For what values of a is v perpendicular to u?

(A) a =

(B) a =

(C) a =

(D) There is no real value of a

2 Which of the following is equivalent to ?

(A)

(B) –

(C)

(D)

C

O

3 An origin O defines vectors OA, OB and OC on the Argand diagram.

The complex number w is also shown.

w

B

A

Which of the three vectors is a model of iw + w?

(A) OC

(B) OB

(C) OA

(D) This vector is not represented

a

b

b

a

b

a

a

b

4 Consider a pair of line segments:

u = 23

+ λ 23

v = 62

+ λ 14

Which coordinate is the intersection of both segments?

(A) P (–8, 6)

(B) P (1, 3)

(C) P (8, 6)

(D) P (8, –6)

5 If k ( f (t – x) )= f (kx – kt), which is necessarily equivalent to f (x) dx?

(A) f (a – x + b) dx

(B) f (x – a – b) dx

(C) f (x – a – b) dx

(D) None of the above

α

γ

6 Define four vectors; a, b, c, d, where a, b are three dimensional, and c, d are two

dimensional.

Let α, β, γ, δ be the respective angles between the vectors, as shown below in the

diagram.

k-direction

a b

c d

j-direction

i-direction

Under the assumption that cos2 α + cos2 β + cos2 γ + cos2 δ = 1, consider the

following two statements:

I. One must be a unit vector

II. They must be perpendicular

From the statements above, which are true?

(A) I is true, II is not

(B) II is true, I is not

(C) I and II are true

(D) I and II are false

β

δ

π

2

0

1

0

1

0

π

2

0

7 Let Sn be a recursion, such that

Sn = 1 + x +x2

2!+

x3

3!+ … +

xn

n!

for some finite value of n.

Let y = ln x, then:

(A) Sn < ee y

(B) Sn > ee y

(C) Sn = ee y

(D) limn → ∞

Sn < ee y

8 The base and height of a solid are formed by the graphs of y = sin x, y = –sin x and a

height of y = cos x.

z

y

x

The solid has cross sections of isosceles triangles, with side lengths of 1 unit, as shown.

Which integral correctly represents the volume of the solid, after some substitution u?

(A) u du

(B) u du

(C) 2u du

(D) 2u du

p

0

36

x4 3x2 + 9

9 Given that

36

x4 3x2 + 9=

2x + a

x2 + 3x + 3

2x b

x2 3x + 3

where a and b are real numbers, what is the limiting value of dx

as p approaches infinity?

(A) π

(B) 2 3

(C) π 3

(D) 2π 3

10 Let P (z) = z3 – 2kz2 + z – k2, where k > 0 is some real number, have one real root, being

w = 1. Two other complex roots exist to this polynomial.

What is the value of |a|, given that a is a complex root, assuming arg(a) = π

3?

(A) There is no defined modulus

(B) 2 Im(a)

(C) 2 Im(a) Re(a)

(D) 2 Re(a)

End of Section I

1

0

dx

x + 1 x2 + 1

v2

1000

Section II

90 marks

Attempt Questions 11–16

Allow about 2 hours and 45 minutes for this section

Answer each question in a separate writing booklet. Full mathematical reasoning and/or

calculations is to be shown, or marks will be deducted.

Question 11 (15 marks) Use the Question 11 Writing Booklet.

(a) Let O (0, 0, 0), P (1, 3, 2) and Q (–1, 1, 5) be fixed points.

(i) Find vectors a = OP, b = OQ and c = PQ. 1

(ii) Using a b, find the area of ∆OPQ. 2

(b) Find real numbers a, b and c such that 4

2

x + 1 x2 + 1=

a

x + 1+

bx + c

x2 + 1

and hence evaluate .

(c)

(i) In exponential form, find the solutions to z2n – i = 0, for which arg(z) is 2

a principle argument.

(ii) Hence, sketch the solutions to z3 – i = 0, using your result from (i). 2

(d) A particle of mass 0.5 kg travels in a medium encountering resistance of 4

magnitude , where v is the velocity of the particle.

Find the magnitude of the resisted force after 3 seconds.

a

–a

a

0

π

2

π

2

ex sin2 x

1 + exdx =

π

4

un<5

3

n

Question 12 (15 marks) Use the Question 12 Writing Booklet.

(a) A parameter t is used to define the vector path v = t2i – 2tj, on the restriction

0 < t < 10.

(i) By finding the cartesian equation, state the domain and range of the 2

function.

(ii) Hence, sketch a graph of this path. 1

(b)

(i) Prove that f (x) dx = f (x) + f (–x) dx. 1

(ii) Show that using the result from (i). 2

(c) The Fibonacci sequence is defined as u1 = 1, u2 = 1 and un + 2 = un + 1 + un. 3

Use mathematical induction to show that , ∀ n ∈ Z+.

Question 12 (continued)

(d) Shown below is a vector space

Given cos2 α + cos2 β + cos2 γ + cos2 δ = 1, show that:

(i) sin2 γ + sin2 δ = cos2 α + cos2 β + 1. 1

(ii) 2

cos2 α + β + γ + δ + sin2 α + β + γ + δ = cos2 α + cos2 β + cos2 γ + cos2 δ .

(you may not assume the Pythagorean identity)

Question 12 (continued)

(e) On the Argand diagram is the region z 1 i = 1. 2

Im

1

Re

1

Copy or trace this diagram into your writing booklet.

Mark the position within this locus with the greatest modulus, and hence find

the greatest modulus of z that occurs within the locus.

End of Question 12

In = 1 x2

n 12 dx

1

0

x2 1 x2n 1

2 = 1 x2n 3

2 1 x2n 1

2

In =n 1

nIn 2

P1

x

Question 13 (15 marks) Use the Question 13 Writing Booklet.

(a) Let , for n = 1, 2, 3 …

(i) Show that . 1

(ii) Hence, show that for n > 2. 3

(iii) Evaluate I4. 2

(b) Let P (x) be a polynomial with roots x = α, β, γ. 2

Show that the polynomials with roots and α2, β2, γ2 are and

x respectively.

(c) Consider your result from (b). 2

Let P (x) = x3 – 5x2 + k = 0 have roots α, β, γ.

If the polynomial with roots α+1, β+1, γ+1 is x3 – 8x2 + 13x, find k.

1

α,1

β,1

γ

π

4 4

Question 13 (continued)

(d) Consider the fixed complex number z = a + ib.

z = a + ib

θ

Denote arg(zn) as α.

(i) Show that 2

cos α = a2 + b2n2 [

n0

ann2

an 2b2 + n4

an 4b4 + … + 1k2

nk

an kbk

(ii) Find a similar expression for sin α. 1

(iii) Hence, by writing π as , find an expression for 2

cosπ

4, sin

π

4.

End of Question 13

x1 = x2f x2

f ' x2

Question 14 (15 marks) Use the Question 14 Writing Booklet.

(a) Let Pn be the set of prime numbers; P1 = 2, P2 = 3 and so on. 3

A prime number is formally defined as ‘a number which is divisible by itself and

by 1.’

Show that there are an infinite unique set of numbers which satisfy this definition.

(b) A function y = f (x) has a root at x = a. 3

Near a exist points x = x1 and x = x2, as shown below on the diagram.

y = f (x)

a x1 x2

Copy or trace this diagram into your writing booklet.

Show that .

Question 14 continues on the next page.

x4

0

A1

A1 A2

Question 14 (continued)

(c) The graph of y = f (x) is drawn below on the interval 0 ≤ x ≤ x4. 3

The points x0, x1, x2, x3, x4 are in arithmetic sequence, and the values

xn xn 1 f xn are within geometric sequence.

Rectangles are formed about these points.

y

y = f (x)

x x0 x1 x2 x3 x4

Let An denote the area of the rectangle with respective base coordinate xn.

Show that f (x) dx > .

You may not use the fact that A1 A2 < 0.

(d) Prove the equality of (1 + 2 + 3 + … + n)2 = 13 + 23 + 33 + … + n3. 4

(e) Prove that limx → c

ln f x = ln limx→c

f x . 2

x2

a2+

y2

b2= 1

Question 15 (15 marks) Use the Question 15 Writing Booklet.

(a) Let be an ellipse with 0 < b < a. 4

Point A is chosen so that its coordinates are parametrically defined as

A (a cos β, b sin β).

Similarly, the ellipses auxiliary circle, x2 + y2 = a2, is defined.

The point P (a cos α, a sin α) is a fixed point defined parametrically along the circle.

Choose points Q and Z equidistant from the origin O and enclosed by the major axis

of these conics; making QP = PO and P, A, Z collinear.

P

Q O Z

Prove that α + β = π, giving reasons.

Question 15 continues on the next page

A

x2 1

x2 + 1

1

1 + x4dx

Question 15 (continued)

(b) Find . 4

(c) Let b be a complex number and z be purely imaginary. 4

Let exp x = e x, so that b z is redefined as exp (z log b).

Taking z = i, show that b z can be written as

e- arg b cos ln b + i e- arg b sin ln b .

Find a similar expression for (1 + i) i and i i.

(d) For n > 1, use mathematical induction to show that 3

1 +1

2+

1

3+ … +

1

n< 2 n 1

End of Question 15

Question 16 (15 marks) Use the Question 16 Writing Booklet.

(a) Define a function y = f (x) over the interval a ≤ x ≤ b to be continuous and

differentiable.

Let f be a non–linear function, and choose a point X so that X is within the interval

a ≤ x ≤ b.

At X, the function is not stationary.

a X b

(i) Without assuming differentiation from first principles, show that the 2

gradient at the point X is given by

limh → 0

f X + h f X

h

(ii) Hence, for differentiable functions f (x) and g(x), show that 2

limx → c

f x

g x= lim

x → c

f ' x

g ' x

Question 16 continues on the next page.

Question 16 (continued)

(b) Let f (x) define an infinite sequence of n terms, n > –1, with an nth term modelled as

xn

n!

In your writing booklet, write down this function in an expanded form.

(i) Sketch a graph of y = f (x) up to n = 3. 2

(ii) Describe the shape of the curve as n approaches infinity. 1

(iii) Using two methods, prove that f (x) = e x. 3

(c) The graph of y = ecos1 x is rotated about the x–axis to form a solid of revolution. 5

Evaluate the volume of this solid.

End of Paper