Specialist Mathematics 1 - Sample questions

VET EQUINE STUDIES (SAMPLE)

VCE Specialist Mathematics

Written examination 1 – End of year

Sample questions

These sample questions are intended to demonstrate how new aspects of Units 3 and 4 of

VCE Specialist Mathematics written examination 1 may be examined. They do not constitute

a full examination paper.

Question 1 (4 marks)

Consider the statement

1

2

1

4

1

2

1

2

1

8

 ...

nn

, where n ∈ N.

a. Show that if n = 1, the statement is true. 1 mark

b. Assume that the statement is true for n = k.

Write down the assumption in terms of k. 1 mark

c. Hence, prove by mathematical induction that

1

2

1

4

1

2

1

2

1

8

 ...

nn

, where n ∈ N. 2 marks

SM EXAM 1 (SAMPLE)

Question 2 (4 marks)

a. Consider the inequality 2

n

> n

2

for n ≥ n

0

, where n ∈ N.

Show that n

0

= 5. 1 mark

b. Prove by mathematical induction that 2

n

> n

2

for n ≥ 5, where n ∈ N. 3 marks

SM EXAM 1 (SAMPLE)

Question 3 (4 marks)

Prove by mathematical induction that the number 9

n

− 5

n

is divisible by 4 for all n ∈ N.

Question 4 (3 marks)

Use proof by contradiction to prove that if n is odd, where n ∈ N, then n

3

+ 1 is even.

SM EXAM 1 (SAMPLE)

Question 5 (3 marks)

Use proof by contradiction to prove that

35

11



.

Question 6 (4 marks)

The curve given by

yx

4

2

, where x ∈ [–1, 1], is rotated about the x-axis to form a solid of revolution.

Find the surface area of this solid of revolution.

SM EXAM 1 (SAMPLE)

Question 7 (5 marks)

The curve given by

yx

=

3

is rotated about the y-axis to form a solid of revolution.

Find the surface area of the part of this solid of revolution where x ∈ [0, 8].

SM EXAM 1 (SAMPLE)

Question 8 (4 marks)

Determine the surface area obtained by rotating the curve dened by the parametric equations

x = sin

3

(θ), y = cos

3

(θ), where



















0

2

,

, about the y-axis.

SM EXAM 1 (SAMPLE)

Question 9 (3 marks)

Find the surface area of revolution formed when the curve dened by the parametric equations

xt

4

3

1

3

()

,

yt

=

1

2

, where 0 ≤ t ≤ 1, is rotated about the x-axis.

Question 10 (7 marks)

The population of bacteria, P(t), in a Petri dish satises the logistic dierential equation

dP

dt

P















26

8000

where t is measured in hours and the initial population is 4000 bacteria.

a. Find the maximum number of bacteria predicted by this model. 1 mark

b. Find the number of bacteria when the population is growing at its fastest rate. 2 marks

SM EXAM 1 (SAMPLE)

c. Solve the dierential equation to nd P as a function of t. 4 marks

Question 11 (4 marks)

Find

xxdx

2

2cos( )

∫

.

SM EXAM 1 (SAMPLE)

Question 12 (3 marks)

The vectors



aijk

23

and



bijk

423

lie in a plane that passes through the point (3, 2, 1).

Find the Cartesian equation of this plane.

Question 13 (6 marks)

a. Find the equation of the plane that passes through the points P(3, 3, 6), Q(1, −1, 2) and

R(5, 2, 0). 4 marks

SM EXAM 1 (SAMPLE)

b. Find the point of intersection of the line given by





ri

ki

jk

25 243

t()

, where t ∈ R,

with the plane given by 2x − 2y + z = 6. 2 marks

Question 14 (3 marks)

Find the angle between the plane given by 2x + y + z = 7 and the line given by









rijk ijk



11 43 2

t()

, where t ∈ R.

SM EXAM 1 (SAMPLE)

Question 15 (5 marks)

a. Find the vector equation of the line through the points A(3, 1, −1) and B(5, 2, −6). 2 marks

b. Find the sine of the angle that this line makes with the plane given by x + 2y − z = 9. 3 marks

SM EXAM 1 (SAMPLE)

Question 16 (4 marks)

The position of a particle after t seconds is given by





ri

jk

()

tt tt t

22

51

6

, where t ≥ 0 and components

are measured in metres.

Find the time at which the minimum speed occurs and calculate the minimum speed. Give your answer in

m s

−1

.

Question 17 (3 marks)

Two planes have equations x + y − z = 3 and 2x − y − 2z = 4.

Given that the angle between the two planes is θ, nd sec(θ).

SM EXAM 1 (SAMPLE)

Question 18 (3 marks)

The position vectors



aijk

242

and



bi jk 23

form two sides of a triangle.

Find the area of the triangle in the form

cd

, where c, d ∈ N.

Question 19 (4 marks)

A parallelogram, OABC, has vertices at O(0, 0, 0), A(1, 2, −1) and C(3, m, 1), where m ∈ R.

Find the value(s) of m if the area of the parallelogram is

45

.