# Math Paper

Topics: Series, Summation, Mobile phone / Pages: 8 (1928 words) / Published: Jun 16th, 2013
2008
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

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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

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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

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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