QUEEN’S COLLEGE

Yearly Examination, 2006– 2007

MATHEMATICS PAPER 1

Question-Answer Book

Secondary 4Date:14 –6– 2007

Time:8:30 am – 10:00 am

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

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

3.Attempts ALL questions in this paper. Write your answer in the spaces provided in this Question-Answer 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
Question No. / Max. Marks / Marks
1 / 6
2 / 5
3 / 6
4 / 8
5 / 8
6 / 10
7 / 12
8 / 12
9 / 13
10 / 20
11 / 20
Total / 120

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SECTION AShort questions. (80 marks)

Answer ALL questions in this section.

1.Simplify and express your answer with positive indices.(6 marks)

1M for fractional index
1M for multiplication rule
1A
1M, 1A
1A

2.If , where a and b are constants, is divisible by x– 2, find a : b.

(5 marks)

Let 1
1M
1A
1A
1A

3.In the figure, and. Find.(6 marks)

(at centre is twiceat ) 1M, 1A
1A
1
(at centre is twiceat ) 1M for giving correct reasoning
1A, [u]

4.(a).Prove that for all real values of a, the roots of the equation are real.

(b).(i)Find the value of a such that the equation in (a) has equal roots.

(ii)Hence find the equal roots.

(8 marks)

(a) 1M, 1A
1M for perfect square, 1A
the equation has real roots for all real a
(b)(i) 1M
1A
(ii) If , then 1M for substitution
1A

5.At 2:00 pm, a ship S is 200 km due south of a harbour H. If S moves at a speed of 40 km/h in the direction:

(a)find the shortest distance between the ship and the harbour;

(b)find the time at which the ship is the nearest to the harbour.

(Correct to the nearest minute.)(8 marks)

(a) The shortest distance = HP 1M for recognize HP
HP = 1M, 1A
= 34.7296
= 34.7 km 1A
(b) SP = 1A
Time required = 1M
= 4 hour 55 minutes 1A
The ship is the nearest to the harbour at 6:55 pm 1A

6.(a).Factorize .

(b)Hence solve the equation . (10 marks)

(a) 1M
= 1A
= 1A
(b)
1M for
1M, 1A
1M, 1A
or rejected 1A, 1A

7.The figure shows a cyclic quadrilateral ABCD with,AB = 16cm, AD = 14cmandCD = 18 cm.

(a)Find the length of BD.

(b)Find .

(c)Find the area of the quadrilateral ABCD.

(12 marks)

(a) 1M, 1A
cm 1A
(b) 1
Consider,
1M, 1A
or rejected 1A, 1A
1A
(c) Area
1M, 1A
1A [u]
  1. The monthly cost $C of running a Jewellery Shop is partly constant and partly varies as the number of employees N. The monthly costs are $520 000 and $680 000 when there are 40 and 60 employees respectively.

(a)Express C in terms of N.

(b)The original monthly cost of running the shop is $800 000. It is decided that the monthly cost should be reduced to $400 000. Find the number of employees that should be dismissed.

(12 marks)

(a) where are non-zero constants 1
C = 520 000, N = 40
……….(1) 1M, 1A
C = 680 000, N = 60
……….(2) 1A
(2) – (1) 160000 = 20 1M, 1A
= 8000 1A
Sub = 8000 into (1) 520000 =
= 200000 1A
(b) If ,
1A
If ,
1A
No. of employees should be dismissed = 75 – 25 = 50 1M, 1A

9.Let for .

(a).Rewrite in the form , where a, b and c are integers.

(b).If ,(i)find the value of ,

(ii)hence find the values of .(13 marks)

(a)
1M
1A
1A
(b)(i)
1M, 1A, 1A
rejected 1A, 1A
(ii)
1M, 1A
1A
1M
1A

SECTION B Long Questions. (40 marks)

Answer ALL questions in this section. Each question carries 20 marks.

10.In the figure, TP and TQ are tangents drawn from the point T to a circle with centre O and radius = 2. TO is produced to cut the circle at R.

(a)Given that , express the following angles in terms of x:

(i)

(ii)

(iii)

(iv)

(12marks)

(a)(i) (tangentradius) 1R [R: reasoning]
1A
In ()
1A
(ii) (tangent properties) 1R
1A
1A
(iii) (at centre is twiceat ) 1M, 1R
1A
(iv) (in alt. segment) 1M, 1R
1A

(b)Given that TPRQ is a parallelogram, find

(i)the value of x,

(ii)the length of TR.

(8 marks)

(b)(i) (tangent properties) 1A
(opp., //gram) 1M
2A
(ii) In
1M
1A
TR = TO + OR
TR= 6 2A

11.In the figure, OT is a vertical tower, O, A, B and C are points on the horizontal ground. A is due east of O and OA = 300 m. C is due south of O and due west of B. The compass bearing of B from A is and AB = 100 m. The angle of elevation of T from B is.

(a)Find(i)

(ii)OB in surd form,

(iii)the angle of elevation of T from A correct to the nearest degree.

(7marks)

(a)(i) 1A
(ii) 1M, 1A
m 1A
(iii) In
1A
1M
1A

(b) Find BC.(4marks)

1M
1A
1M
m 1A

(c) Peter and Mary, both starting from B, want to go to D and E respectively (Not shown in the figure). Their routes are as follows:

Peter walks in the direction of AB from BtoD.

Mary walks due west from B to E.

If the angles of elevation of T from D and E are both , determine whose route is shorter. (9 marks)

For Peter’s route
1M,1A
1A
x = 200 or x = 100 rejected (x>100) 1A, 1A
BD =x - AB
= 200 – 100
= 100 1A
For Mary’s route
CE = BC 1M
for recognizing CE = BC
BE = 2BC
= 500 1A
i.e., Peter’s route is shorter 1A

END OF PAPER

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