Class Index Number

Candidate Name

ANDERSON SECONDARY SCHOOL

2010 Preliminary Examination Secondary Four Express / Five Normal Academic

ADDITIONAL MATHEMATICS Paper 1 4038/01

2 September 2010 2 hours Candidates answer on writing papers. Additional materials: Writing paper (10 sheets)

READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . At the end of the examination, fasten all your work securely together. Hand in the question paper and your answer scripts separately. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80.

This document consists of 4 printed pages. ANDSS 4E5N Prelim 2010 Add Math (4038/01) [Turn over

Mathematical Formulae

1. ALGEBRA Quadratic Equation

For the equation , 02 =++ cbxax

aacbbx

242 =

Binomial expansion

nrrnnnnn bbarn

ban

ban

aba ++

++

+

+=+ ......

21)( 221 ,

where n is a positive integer and !

)1(...)1()!(!

!r

rnnnrnr

nrn +==

2. TRIGONOMETRY

Identities 1cossin 22 =+ AA

AA 22 tan1sec +=

AAcosec 22 cot1+=

BABABA sincoscossin)sin( =

BABABA sinsincoscos)cos( m=

BABABA

tantan1tantan)tan( m

=

AAA cossin22sin =

AAAAA 2222 sin211cos2sincos2cos ===

AAA 2tan1

tan22tan =

)(cos)(sin2sinsin 2121 BABABA +=+

)(sin)(cos2sinsin 2121 BABABA +=

)(cos)(cos2coscos 2121 BABABA +=+

)(sin)(sin2coscos 2121 BABABA += Formulae for ABC

Cc

Bb

Aa

sinsinsin==

Abccba cos2222 +=

Cab sin21=

2 ANDSS 4E5N Prelim 2010 Add Math (4038/01) [Turn over

1. (a) Given that A = 3 41 2

and B =

7 100 3

,

(i) If A2 B = kC , where k and all the elements of C are integers, find the greatest value of k and the matrix C. [2]

(ii) Write down the inverse matrix of A. [1] (b) If a pair of simultaneous equations is represented by a matrix equation

2 34 h

xy

=

1k

,

state the value of h and of k such that the equations have infinite number of solutions. [2]

2. (a) Find the range of values of k for which the expression kx 2 + 2kx + 3k 4x 2

is never positive for all real values of x. [3]

(b) The line is a tangent to the curve y = mx +1 y = x + 1x

.

Show that the value of m = 34

. [3]

3. It is given that sin = 25

and is obtuse. Find the exact value of (a) tan A , given that tan(A + ) = 4 . [2] (b) tan B , given that 3cos(B ) sin(B + ) = 0 [3]

4. Solve the equation 22x +2 2x+2 = 9(2x 1) 1. [6] 5. A curve has the equation y = 1+ 4 x 3x 2 + x 3.

(a) Express dydx

in the form a(x + b)2 + c where a, b and c are constants. [2] (b) Explain why the curve is an increasing function for all real values of x. [1] (c) With the values of a, b and c obtained, sketch the graph of dy

dx= a(x + b)2 + c ,

labelling its turning point and intercepts. [3]

6. (a) Find the term independent of x in the expansion of x 2 2x

9

. [4]

(b) Find the coefficient of x 9 in the expansion of x 2 2x

9

(4x 9 + 3) . [3] 7. If 4 x 2 +11x 3 is a factor of )314 , find the remaining factor. ()74(2 + xxx

Find the roots of the equation . 0)314()74(2 =+ xxxHence, solve the equation e2x (4ex + 7) =14ex 3. [7]

3 ANDSS 4E5N Prelim 2010 Add Math (4038/01) [Turn over

8. The length of each side of a regular hexagon is x cm. Find the area of the hexagon, A, in terms of x. If x is increasing at the rate of 0.4 cm/s, find the rate of change of A when x = 5. [6]

9. (a) Solve the equation 2x 1 3 = 2x . [3]

(b) On the same graph, sketch the graphs 12 = xy and 32 += xy for 1 x 3, labelling the vertex and all intercepts clearly. [3]

(c) Hence solve the inequality 2x 1 2x 3. [2] 10. Solve the following equations, giving all angles between 0 and 360.

(a) 1)50 [3] 2sin(2 =x(b) 5 [4] tan5sec3 2 = xx

11. A circle C passes through points P(0, 2), Q(7, 3) and R(8, 4) where PQRS is a

square. Find (a) the centre and the radius of the circle C, [3] (b) the equation the circle that is the reflection of the circle C in the line y = x. [3]

12. The figure shows a trapezium ABCD in which the coordinates of A, C and D are

(0, 4), (5, 3) and (6, 5) respectively. AD is parallel to BC and is perpendicular to AB. (a) Show that the coordinates of B are (3, 6). [4]

(b) The point E lies on the line AD such that 32=

ABDofAreaABEofArea .

Find the coordinates of E. [2] (c) The point F with coordinates (1, 12) lies on the line CB produced.

Show that AFBE is a parallelogram. [2] (d) Find the ratio of area of AFBE : area of EBD. [2] y

B

A (0, 4)

C (5, 3)

x 0

D (6, 5)

4 ANDSS 4E5N Prelim 2010 Add Math (4038/01) [End of paper

kokhoongText Boxx

kokhoongText Box0

2010 Preliminary ExaminationSecondary Four Express / Five Normal Academic

READ THESE INSTRUCTIONS FIRST