QUEEN’S COLLEGE



QUEEN’S COLLEGE

Half-yearly Examination, 2009 – 2010

MATHEMATICS PAPER 1

Question-Answer Book

Secondary 4 Date : 12-1-2010

Time: 8:30 – 10:00

[pic]

1. Write your class, class number in the spaces provided on this cover.

2. This paper consists of TWO sections, A and B. Section A and B carry 80 and 40 marks respectively.

3. Attempt ALL questions in this paper. Write your answer in the spaces provided in this Question-Answer Book. Supplementary answer sheets will be supplied on request. Write your class and class number on each sheet and put them inside this book.

4. Unless otherwise specified, all working must be clearly shown.

5. Unless otherwise specified, numerical answers should either be exact or correct to 3 significant figures.

6. The diagrams in this paper are not necessarily drawn to scale.

|Class | | |

|Class Number | | |

| |Teacher’s Use Only |

|Section A |Max Marks |Marks |

|Question No. | | |

|1 |5 | |

|2 |5 | |

|3 |4 | |

|4 |6 | |

|5 |6 | |

|6 |8 | |

|7 |10 | |

|8 |10 | |

|9 |13 | |

|10 |13 | |

|Section A |80 | |

|Total | | |

|Section B |Max Marks |Marks |

|Question No. | | |

|11 |20 | |

|12 |20 | |

|Section B |40 | |

|Total | | |

|Teacher’s |Paper I Total | |

|Use Only | | |

SECTION A (80 marks)

Answer ALL questions in this section and write your answers in the spaces provided.

1. Convert the recurring decimal[pic] into a rational number. (5 marks)

2. Express [pic] in terms of i, where [pic] (5 marks)

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3. Give the domain of the following functions,

(i) [pic] (2 marks)

(ii) [pic] (2 marks)

4. If[pic], find the values of a and b. (6 marks)

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5. When a polynomial f(x) is divided by [pic], the quotient is [pic] and the remainder is 7. Find f(x). (6 marks)

6. If the equation [pic] has equal roots, find the value of k and solve the equation for x. (8 marks)

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7. Given [pic]

(a) Find [pic] and f(2x)

(b) Solve the equation [pic]

(leave your answer in surd form if necessary) (10 marks)

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8. (a) Express [pic] in the form of [pic].

(b) Hence evaluate [pic] (10 marks)

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9. The roots of the equation [pic] are [pic] and [pic].

a) Without solving the equation, find the values of

(i) [pic] and [pic]

(ii) [pic]

(b) Form a quadratic equation with roots [pic] and [pic]. (13 marks)

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10. When the polynomial [pic] is divided by (x-1), the remainder is -12.

f(x) is divisible by (x + 3).

a) Find the values of a and b.

b) Solve the equation f(x) = 0. (13 marks)

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SECTION B (40 marks)

Answer both questions in this section and write your answers in the spaces provided.

11 Given a line [pic].

(a) If L cuts the x-axis and y-axis at P and Q respectively, find the coordinates of

P and Q. (3 marks)

(b) Find the coordinates of R which divides PQ in the ratio of 2:1. (2 marks)

(c) Find the equation of the line [pic] which passes through R and is

perpendicular to L. (3 marks)

(d) If S(3, k) is a point on [pic], find k. (2 marks)

(e) Find the equation of the line [pic] which passes through S and with

x-intercept =[pic]. (2 marks)

(f) Find the intersection point T of [pic]and [pic]. (3 marks)

(g) Find the area of [pic]. (5 marks)

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12(a). Fig(a) below shows the graph of [pic]. It passes through the points (1, 25) and (4, 19) with y-intercept equal to 35.

[pic]

i) Find the values of a, b and c.

ii) Find the vertex of the graph.

iii) Insert the x-axis in a possible position in fig(a) (12 marks)

(b) Fig (b) shows a rectangle ABCD of dimension 5cm x 7cm. Points P, Q, R and S are points on AB, BC, CD and DC respectively such that AP = BQ = CR = DS = x cm.

i) Find the area of [pic] and [pic] in terms of x.

ii) Deduce the area of the parallelogram PQRS in terms of x.

iii) By using the result of (a), find the value of x such that the area of the parallelogram is a minimum. What is the minimum area. (8 marks)

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END OF PAPER

Solution

Section A:

1. Let [pic] … (1) 1

[pic] … (2) 1

(2) – (1), 1

[pic] 1

[pic] 1

2. [pic] (1+1+1)

= -7+11i (1+1)

3(i). Domain = [pic] or all real numbers except 1. 2A

Domain: [pic] 2A – deduct 1 for x>3.

4. [pic][pic]

[pic] 1M – for axpanding

[pic] 1M –(for equating coeff)

1A + 1A

From (1), a = -5 1A

Sub into (2), b = -10 1A

5. [pic][pic] 1M+1A

[pic] (1+1+1+1)

6. For [pic] to have equal roots,

[pic] 1M + 1A

[pic] 1M+1A

k = -4 1A

k = -4, [pic] 1M

[pic] 1A

[pic]

x = 1 1A

|Alternatively, |

|sum of roots = [pic] 1M+1A |

|since it has equal roots, [pic]x=1. 1A |

7(a). [pic]

[pic] 1A

[pic] 1M

[pic] 1A

[pic] 1A

[pic] 1A

(b) [pic] 1M + 1A [pic]

[pic] 1A

[pic]

[pic], 2 1A+1A

8(a). [pic] 1

[pic] 1M+[pic]

[pic] 1A

(b) [pic]

[pic] 1M

[pic] 1A+1A+1A

[pic] 1A

9(a) (i) [pic] and [pic] 1A+1A

(ii) [pic] 1M +1A

[pic] 1M +1A

[pic] 1A

(b) [pic] + [pic] = [pic] 1A

= [pic] 1M + 1A

([pic])([pic]) = 1 1A

The equation with [pic] and [pic] as roots is

[pic] 1M +1A

10. [pic] 1M

a + b + 4 = 0 ……… (1) 1A

[pic] 1M

[pic] …….. (2) 1A

(1)-(2)

[pic] 1M

[pic] 1A

sub [pic] into (1)

[pic]

[pic] 1A

(b) [pic]

(x + 3) is a factor of f(x) 1M

[pic] 2A

[pic][pic] 1A

[pic]-3, 2. 2A

Section B

11(a). Rewrite L in the intercepts form [pic] 1M

P is (-3, 0) and Q is (0, 6) 1A + 1A

(b) Let R be (a, b)

[pic] 1A

[pic] 1A

[pic]R is (-1, 4)

(c) Slope of L = 2, [pic]slope of [pic] is [pic] 1A

Equation of [pic]: [pic] 1M (for pt slope form)

x + 2y – 7 = 0 1A

(d) sub (3, k) into [pic], 3 + 2k – 7 = 0 1M

k = 2 1A

(e) equation of [pic] 1M (for 2 points form)

4x - 7y +2 = 0 1A

(f) [pic]: 2x - y + 6 = 0 ……. (1)

[pic]: 4x - 7y + 2 = 0 …... (2)

From (1), y = 2x + 6 1M

Sub into (2), 4x – 7(2x + 6) + 2 = 0

-10x – 40 = 0

x = -4 1A

y = -2

T is (-4, -2) 1A

(g) R (-1, 4), S(3, 2), T(-4, -2)

RST is a right-angled triangle with [pic] 1

Area of RST = [pic] 1M+1A

= [pic]sq units 1A + 1A

12(a) (i). y-intercept equal to 35, [pic]c = 35 1A

sub the points into[pic], 1M

[pic] … (1) 1A

[pic] … (2) 1A

(2) – (1),

3a – 6 = 0

a = 2 1A

b = -12 1A

[pic]

(ii) [pic] 1M

[pic] 1M

[pic] 2A

vertex is (3, 17) 1A

(iii) x-axis should be below the vertex. 1A

(b) (i) Area of [pic][pic] and 1M+1A

area of [pic][pic] 1A – deduct 1m without unit

(ii) Area of PQRS = 35-x(7-x)-x(5-x)[pic] 1A

=[pic] 1A

By (a), for minimum area x should be 3 1M +1A

and the minimum area is 17[pic] 1A

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[pic]

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Fig (a)

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7 cm

Fig (b)

x cm

S

R

Q

[pic]

[pic]

(4, 19)

(1, 25)

35

y

[pic]

P

Page total

D

C

B

A

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