Topic 1: Algebra 1. Let Sn

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Topic 1: Algebra

1. Let S

_n

be the sum of the first n terms of an arithmetic sequence, whose first three terms are u

₁

, u

₂

and u

₃

. It is known that S

₁

= 7, and S

₂

= 18.

(a) Write down u

₁

.

(b) Calculate the common difference of the sequence.

(c) Calculate u

₄

. (Total 6 marks)

2. Consider the expansion of (x

²

– 2)

⁵

.

(a) Write down the number of terms in this expansion.

(b) The first four terms of the expansion in descending powers of x are x

¹⁰

– 10x

⁸

+ 40x

⁶

+ Ax

⁴

+ ...

Find the value of A.

(Total 6 marks)

3. Find the exact solution of the equation 9

^2x

= 27

^(1–x)

. (Total 6 marks)

4. (a) Given that log

₃

x – log

₃

(x – 5) = log

₃

A, express A in terms of x.

(b) Hence or otherwise, solve the equation log

₃

x – log

₃

(x – 5) = 1.

(Total 6 marks)

5. A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive row of seats has two more seats in it than the previous row.

(a) Calculate the number of seats in the 20

^th

row.

(b) Calculate the total number of seats.

(Total 6 marks)

6. A sum of $5 000 is invested at a compound interest rate of 6.3% per annum.

(a) Write down an expression for the value of the investment after n full years.

(b) What will be the value of the investment at the end of five years?

(c) The value of the investment will exceed $10 000 after n full years, (i) Write down an inequality to represent this information.

(ii) Calculate the minimum value of n.

(Total 6 marks)

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Topic 2: Functions and Equations 7. The functions f and g are defined by 𝑓𝑓: 𝑥𝑥 ⟼ 3𝑥𝑥, 𝑔𝑔: 𝑥𝑥 ⟼ 𝑥𝑥 + 2 . (a) Find an expression for (f ° g) (x).

(b) Show that f

^–l

(18) + g

^–l

(18) = 22.

(Total 6 marks)

8. The function f is defined by for –3 < x < 3.

(a) Sketch the graph of f.

(b) Write down the equation of each vertical asymptote.

(c) Write down the range of the function f.

(Total 6 marks)

9. The quadratic function f is defined by f(x) = 3x

²

– 12x + 11.

(a) Write f in the form f(x) = 3(x – h)

²

– k.

(b) The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g(x) = 3(x – p)

²

+ q.

(Total 6 marks)

10. The diagram shows the graphs of f(x) = 1 + e

^2x

, g(x) = 10x + 2, 0 ≤ x ≤ 1.5.

(a) (i) Write down an expression for the vertical distance p between the graphs of f and g.

(ii) Given that p has a maximum value for 0 ≤ x ≤ 1.5, find the value of x at which this occurs.

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The graph of y = f(x) only is shown in the diagram below. When x = a, y = 5.

(b) (i) Find f

^–1

(x).

(ii) Hence show that a = ln 2.

(5)

(c) The region shaded in the diagram is rotated through 360° about the x-axis. Write down an expression for the volume obtained.

(3)

(Total 14 marks)

11. Consider the line L with equation y + 2x = 3. The line L

₁

is parallel to L and passes through the point (6, –4).

(a) Find the gradient of L

₁

.

(b) Find the equation of L

₁

in the form y = + mx + b.

, – 9 ) 3 (

x

2

x

f =

x y

16

12

8

4 0.5 1 1.5

5 a

f g

p

x y

16

12

8

4 0.5 1 1.5

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12. The function f is given by f(x) = e

^(x–11)

–8.

(a) Find f

^–1

(x).

(b) Write down the domain of f

^–l

(x).

(Total 6 marks)

13. The graph of y = f(x) is shown in the diagram.

(a) Given f(x) above, draw a graph of the following equation:

(i) y = 2f(x).

(ii) y = f(x – 3).

(b) The point A (3, –1) is on the graph of f. The point A′ is the corresponding point on the graph of y = –f(x) + 1. Find the coordinates of A′.

(Total 6 marks)

14. The equation of a curve may be written in the form y = a(x – p)(x – q). The curve intersects the x-axis at A(–2, 0) and B(4, 0). The curve of y = f(x) is shown in the diagram below.

(a) (i) Write down the value of p and of q.

(ii) Given that the point (6, 8) is on the curve, find the value of a.

(iii) Write the equation of the curve in the form y = ax

²

+ bx + c.

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(b) (i) Find .

(ii) A tangent is drawn to the curve at a point P. The gradient of this tangent is 7.

Find the coordinates of P.

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(c) The line L passes through B(4, 0), and is perpendicular to the tangent to the curve at point B.

(i) Find the equation of L.

(ii) Find the x-coordinate of the point where L intersects the curve again.

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(Total 15 marks) 2

1 –1

–2

–2 –1 1 2 3 4 5 6 7 8 x

y

0

4

2 –2

–4

–6

–4 –2 0 2 4 6 x

y

A B

x

y

d

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Topic 3: Trigonometry 15. The graph of a function of the form y = p cos qx is given in the diagram.

(a) Write down the value of p.

(b) Calculate the value of q.

(Total 6 marks)

16. A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a second side, [AB], is 65 m and the angle between these two sides is 60°.

(a) Use the cosine rule to calculate the length of the third side of the field.

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(b) Given that sin 60° = find the area of the field in the form where p is an integer.

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Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts A

₁

and A

₂

by constructing a straight fence [AD] of length x metres, as shown on the diagram below.

(c) (i) Show that the area of A

_l

is given by . (ii) Find a similar expression for the area of A

₂

.

(iii) Hence, find the value of x in the form , where q is an integer.

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(d) (i) Explain why sin .

(ii) Use the result of part (i) and the sine rule to show that .

(5)

(Total 18 marks)

17. The following diagram shows a circle of centre O, and radius r. The shaded sector OACB has an area of 27 cm

²

. Angle AOB = θ = 1.5 radians.

(a) Find the radius.

(b) Calculate the length of the minor arc ACB.

(Total 6 marks)

2 ,

3 p 3

4 65x

3 q B D ˆ A sin C D ˆ

A =

8 5 DC BD =

O

r A

C

B

40 30 20 10

–10 –20 –30 –40

π/2 π x

y

104 m

A

B

C

65 m 30°

30°

2

1

x D

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18. Consider y = sin .

(a) The graph of y intersects the x-axis at point A. Find the x-coordinate of A, where 0 ≤ x ≤ π.

(b) Solve the equation sin = – , for 0 ≤ x ≤ 2π.

(Total 6 marks)

19. The diagram shows a triangular region formed by a hedge [AB], a part of a river bank [AC] and a fence [BC]. The hedge is 17 m long and is 29°. The end of the fence, point C, can be positioned anywhere along the river bank.

(a) Given that point C is 15 m from A, find the length of the fence [BC].

(3)

(b) The farmer has another, longer fence. It is possible for him to enclose two different triangular regions with this fence. He places the fence so that is 85°.

(i) Find the distance from A to C.

(ii) Find the area of the region ABC with the fence in this position.

(5)

(c) To form the second region, he moves the fencing so that point C is closer to point A.

Find the new distance from A to C.

(4)

(d) Find the minimum length of fence [BC] needed to enclose a triangular region ABC.

(2)

(Total 14 marks)

20. Let f(x) = sin2x + cos x for 0 ≤ x ≤ 2π.

(a) (i) Find f′(x).

One way of writing f′(x) is –2 sin

²

x – sinx + 1.

(ii) Factorize 2sin

²

x + sinx – 1.

(iii) Hence or otherwise, solve f′(x) = 0.

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The graph of y = f(x) is shown below.

There is a maximum point at A and a minimum point at B.

(b) Write down the x-coordinate of point A.

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(c) The region bounded by the graph, the x-axis and the lines x = a and x = b is shaded in the diagram above.

(i) Write down an expression that represents the area of this shaded region.

(ii) Calculate the area of this shaded region.

(5)

(Total 12 marks)

 

 

  + 9 x π

 

 

  + 9 x π

2 1

C A ˆ B

29°

A

B 15 m C

17 m

river bank

C B ˆ A

2 1

A

B

a b

x y

0 2π

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Topic 4: Vectors 21. The diagram shows a cube, OABCDEFG where the length of each edge is 5cm. Express the following vectors in terms of i, j and k.

(a) ;

(b) ;

(c) .

(Total 6 marks)

22. A triangle has its vertices at A(–1, 3), B(3, 6) and C(–4, 4).

(a) Show that

(b) Show that, to three significant figures, cos (Total 6 marks)

23. In this question, distance is in kilometers, time is in hours.

A balloon is moving at a constant height with a speed of l8 km h

^–1

, in the

direction of the vector

At time t = 0, the balloon is at point B with coordinates (0, 0, 5).

(a) Show that the position vector b of the balloon at time t is given by

b =

(6)

At time t = 0, a helicopter goes to deliver a message to the balloon. The position vector h of the helicopter at time t is given by

h =

(b) (i) Write down the coordinates of the starting position of the helicopter.

(ii) Find the speed of the helicopter.

(4)

(c) The helicopter reaches the balloon at point R.

(i) Find the time the helicopter takes to reach the balloon.

(ii) Find the coordinates of R.

(5)

(Total 15 marks) OG BD EB

9 – AC

AB • =

. 0.569 – C A ˆ

B =

 







 







0 4 3

. 0

4 . 14

8 . 10

5 0 0

 







 







 +

 





 







 =

 





 







t z

y x

 







 







 +

 





 







 =

 





 







6 24 –

48 –

0 32 49

t z

y x

A

B C

D

E

F

G

O k z

y

x i

j

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24. Find the cosine of the angle between the two vectors and . (Total 6 marks)

25. In this question the vector represents a displacement of 1 km east,

and the vector represents a displacement of 1 km north.

The diagram below shows the positions of towns A, B and C in relation to an airport O, which is at the point (0, 0).

An aircraft flies over the three towns at a constant speed of 250 km h

^–1

.

Town A is 600 km west and 200 km south of the airport.

Town B is 200 km east and 400 km north of the airport.

Town C is 1200 km east and 350 km south of the airport.

(a) (i) Find .

(ii) Show that the vector of length one unit in the direction of is . (4)

An aircraft flies over town A at 12:00, heading towards town B at 250 km h

^–1

.

Let be the velocity vector of the aircraft. Let t be the number of hours in flight after 12:00. The position of the aircraft can be given by the vector equation

.

(b) (i) Show that the velocity vector is . (ii) Find the position of the aircraft at 13:00.

(iii) At what time is the aircraft flying over town B?

(6)

Over town B the aircraft changes direction so it now flies towards town C. It takes five hours to travel the 1250 km between B and C. Over town A the pilot noted that she had 17 000 litres of fuel left. The aircraft uses 1800 litres of fuel per hour when travelling at 250 km h