Queen’s College
Mock Examination 2009 – 2010
/
Class :
Class no.:
Teacher’s Use Only
Section A Question No. / Maximum marks / marks
1 / 3
2 / 4
3 / 4
4 / 4
5 / 4
6 / 5
7 / 5
8 / 6
9 / 6
10 / 7
11 / 7
12 / 7
Section A
Total / 62
ADDITIONAL MATHEMATICS
Question Answer Book
Secondary 5 Date: 26thFebruary 2010
Time:11:00 am 1:30 pm
This paper must be answered in English
1. / Write your class and class number in the space provided on page 1.
2. / This paper consists of TWO sections, Section A and Section B. Section A carries 62 marks and Section B carries 48 marks.
3. / Answer ALL questions in Section A. Write your answers in the spaces provided in this Question-Answer Book. Do not write in the margins. Graph paper and supplementary answer sheets will be supplied on request. Write your class and class number on each sheet, and fasten them with a string INSIDE this book.
4. / Answer any FOUR questions in Section B. Write your answers in the answer book provided.
5. / The Question-Answer Book and the answer book must be handed in separately at the end of the examination.
6. / All working must be clearly shown.
7. / Unless otherwise specified, numerical answers must be exact.
8. / In this paper, vectors may be represented by bold-type letters such as u , but candidates are expected to use appropriate symbols such as in their working.
9. / The diagrams in the paper are not necessarily drawn to scale.

FORMULA FOR REFERENCE

SECTION A ( 62 marks )

Answer ALL questions in this section and write your answers in the space provided in this Question-Answer Book.

1.Given  and  are the roots of the quadratic equation ,

show that  and  are real and distinct.(3 marks )

2.If the acute angle between the lines and

is , find the value(s) of k.(4 marks)

3.Find the coefficient of in the expansion of (4 marks)

4.Given . Find f’(x) from first principle.(4 marks )

5.Solve the equation .( 4 marks )

6.If , find the equation of the tangent to the curve at .( 5 marks )

7(a). Show that .

(b) By using (a), evaluate ( 5 marks )

8(a).Show that .

(b)Hence, find the general solution of the equation (6 marks )

9.Given 3 points A(1, 4), B(3, 6), C(5, -1).

(a)Find the area of .

(b)Find the locus of a variable point P(x, y) such that area of is the same as

(6 marks )

10.In the given figure OABC is a square of length 2 units. D and E are two points on CB and AB respectively satisfying CD:DB = AE:EB = 1:3.

(a)Find ,andin terms of i and j.

(b)Hence, find , correct to the nearest degree.

( 7 marks )

11(a) Given that , by using Mathematical Induction, prove that for any positive integer n.
(b)Using (a), evaluate . (7 marks)

12.The figure below shows the curves , and the straight line .

(a)Find the coordinates of the intersection points of the curvesA, B and C as shown in the figure.

(b)Find the shaded area bounded by the curves, and the straight line .

(7 marks)

SECTION B ( 48 marks )

Answer any FOUR questions in this section. Each question carries12 marks.

Write your answers in the answer book.

13.Given the equation of the family of circlesand the straight lines, where m and k are constants.

(a)(i)Find, in terms ofm, the centre of C. Hence findthe equation of the locus of the

centre of C as m varies.

(ii)Find the radius of C.

(iii)Describe the characteristics of the family of circles C.

(iv)Find the range of values of k such that C and L intersects at two distinct points.

Hence deduce the equation of the two common tangents to the circles in C.

(if necessary, leave your answer in surd form)(8 mark)

(b)Given another circle. Find the circle(s) in C which touches H and completely encircled by it. (4 marks)

14The figure below shows a building with a horizontal base ABC such that AB = BC = CA = 10 m. P, Q and R are points vertically above A, B and C are respectively such that PA = 30 m, QB = 20 m and RC = 10 m.

(a)Show that PQR is an isosceles triangle.

(3 marks)

(b)Find tan and tan.(2 marks)

(c)M and N are points on PR and PA respectively

such that QM and NM are both perpendicular to PR.

(i)Find QM, MN and QN.

(Leave your answers in surd form.)

(ii)Hence, show that the planes PQR and

APRC are perpendicular to

each other. (7 marks)

(7 marks)

15.Fig (a) shows a piece of paper in the shape of a sector of radius cm and rad.

It is folded to form a conical vessel of radius r cm where OA coincides with OB, as shown in fig (b).

(a)Let the capacity of the vessel be V, show that .(2 marks)

(b)Find , hence deduce the value of for which the capacity of the vessel is a

maximum.(5 marks)

(c)Suppose takes the value in (b) and water is flowing into the vessel at a rate of . How fast is the water level rising when the water is 7 cm deep?

(5 marks)

  1. Given a function .

(a) (i) Find the domain of the function and

the x, y intercepts of the curve y = f(x).(2 marks)

(ii)Find the turning points of the curve. For each turning point, test whether it is a maximum or minimum point. (3 marks)

(iii)Sketch the graph of y = f(x).(2 marks)

(iv)Sketch on a separate diagram, the graph of .(2 marks)

(b)The area bounded between and the x-axis is revolved about the x-axis for 1 complete revolution. Find the volume of the solid formed. (3 marks)

  1. In the figure , OA = 15, OB = 10 and . E is a point on AB such that . Let and .

(a)Express in terms of and , hence find the length of AB.(3 marks)

(b)Find in terms of and .(1 mark)

(c)Let P and Q be the feet of perpendiculars from E to OA and OB respectively.

If and , find the values of and .(5 marks)

(d)Is PABQ a cyclic quadrilateral? Explain your answer.(3 marks)

…End of Paper…

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Additional Mathematics 2009 -2010