# Math Paper

**Topics:**Anno Domini, Pallavolo Modena, Series

**Pages:**17 (1928 words)

**Published:**June 16, 2013

H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N

Mathematics

General Instructions • Reading time – 5 minutes • Working time – 3 hours • Write using black or blue pen • Board-approved calculators may be used • A table of standard integrals is provided at the back of this paper • All necessary working should be shown in every question

Total marks – 120 • Attempt Questions 1–10 • All questions are of equal value

212

BLANK PAGE

– 2 –

Total marks – 120 Attempt Questions 1–10 All questions are of equal value Answer each question in the appropriate writing booklet. Extra writing booklets are available.

Marks Question 1 (12 marks) Use the Question 1 Writing Booklet.

(a)

Evaluate 2 cos

π correct to three significant figures. 5

2

(b)

Factorise 3x 2 + x – 2.

2

(c)

Simplify

2 1 . − n n +1

2

(d)

Solve 4 x − 3 = 7.

2

(e)

Expand and simplify

(

3 −1 2 3 + 5 .

)(

)

2

(f)

Find the sum of the first 21 terms of the arithmetic series 3 + 7 + 11 + ··· .

2

– 3 –

Marks Question 2 (12 marks) Use the Question 2 Writing Booklet.

(a)

Differentiate with respect to x : (i) (ii) (iii)

( x 2 + 3) 9

x 2 loge x sin x . x+4

2 2 2

(b)

Let M be the midpoint of (–1, 4) and (5, 8). 1

Find the equation of the line through M with gradient

−

. 2

2

(c)

(i)

⌠ dx Find

⎮ . x ⌡

+ 5

π ⌠ 12

1

(ii)

2 Evaluate ⎮ sec

3x dx

.

⌡0

3

– 4 –

Marks Question 3 (12 marks) Use the Question 3 Writing Booklet.

(a)

y C (1, 5) D

O A (0, –1)

2x –y –1 =

B (0, 3)

NOT

TO

SCALE

0

x

In the diagram, ABCD is a quadrilateral. The equation of the line AD is 2x – y – 1 = 0. (i) (ii) (iii) (iv) (v) Show that ABCD is a trapezium by showing that BC is parallel to AD. The line CD is parallel to the x-axis. Find the coordinates of D. Find the length of BC. Show that the perpendicular distance from B to AD is 4 5 . 2 1 1 2 2

Hence, or otherwise, find the area of the trapezium ABCD.

(b)

(i)

Differentiate loge (cos x) with respect to x.

2

(ii)

⌠4 Hence, or otherwise, evaluate

⎮ tan x dx .

⌡0

π

2

– 5 –

Marks Question 4 (12 marks) Use the Question 4 Writing Booklet.

(a)

P NOT TO SCALE X Y

2

Q In the diagram, XR bisects ∠PRQ and XY QR .

R

Copy or trace the diagram into your writing booklet. Prove that ΔXYR is an isosceles triangle.

(b)

The zoom function in a software package multiplies the dimensions of an image by 1.2 . In an image, the height of a building is 50 mm. After the zoom function is applied once, the height of the building in the image is 60 mm. After a second application, its height is 72 mm. (i) Calculate the height of the building in the image after the zoom function has been applied eight times. Give your answer to the nearest mm. The height of the building in the image is required to be more than 400 mm. Starting from the original image, what is the least number of times the zoom function must be applied? 2

(ii)

2

(c)

Consider the parabola x 2 = 8( y – 3) . (i) (ii) (iii) (iv) Write down the coordinates of the vertex. Find the coordinates of the focus. Sketch the parabola. Calculate the area bounded by the parabola and the line y = 5. – 6 – 1 1 1 3

Marks Question 5 (12 marks) Use the Question 5 Writing Booklet.

(a)

The gradient of a curve is given by the point (0, 7) . What is the equation of the curve?

dy = 1 − 6 sin 3 x . The curve passes through dx

3

(b) Consider the geometric series 5 + 10x + 20x 2 + 40x 3 + ··· . (i) For what values of x does this series have a limiting sum? (ii) The limiting sum of this series is 100. Find the value of x. 2 2

(c)

Light intensity is measured in lux. The light intensity at the surface of a lake is 6000 lux. The light intensity, I lux, a distance s metres below the surface of the lake is given by I = Ae –ks where A...

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