Pierce Prelim 2009 Am1

Class Candidate Name

Index Number

_____________________________________

PEIRCE SECONDARY SCHOOL Department of MATHEMATICS Preliminary Examination II for Secondary Four Express/ Five Normal Academic Additional Mathematics Paper 1 Friday

4038/1

4 September 2009

1030 – 1230

Additional materials: A4 writing paper (6 sheets) Graph paper (1 sheet) Electronic calculator

TIME

2 hours

INSTRUCTIONS TO CANDIDATES 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. Show your working on the same page as the the answer. Do notall use staples, paper clips, highlighters, gluerest or of correction fluid. Omission of essential working will result in loss of marks. Answer all questions. Write your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of Angles angle in degrees, unless a different level of accuracy is specified in the question. The use of a scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. 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 question paper consists of 6 printed pages, including this cover page. Final Copy by Han MH

[Turn over

2

Mathematical Formulae

1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0  b  b 2  4ac x= 2a

Binomial expansion  n (a + b)n = an +   an  1b + 1

 n n  2 2  n   a b + ......... +   an  r br + … + bn ,  2 r  n n! n( n  1)...(n  r  1) where n is a positive integer and   = = r!  r  r !(n  r )! 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin(A  B) = sin A cos B  cos A sin B cos(A  B) = cos A cos B  sin A sin B tan A  tan B tan(A  B) = 1  tan A tan B sin 2A = 2 sinA cosA cos 2A = cos2 A  sin2 A = 2 cos2 A  1 = 1  2 sin2 A 2 tan A tan 2A = 1  tan 2 A 1 1 sin A + sin B = 2 sin (A + B)cos (A  B) 2 2 1 1 sin A  sin B = 2 cos (A + B)sin (A  B) 2 2 1 1 cos A + cos B = 2 cos (A + B)cos (A  B) 2 2 1 1 cos A  cos B = 2 sin (A + B)sin (A  B) 2 2 Formulae for ABC a b c = = sin A sin B sin C a2 = b 2 + c2  2bc cos A 1  = ab sin C 2

3 2

1

 9  Given that a  b 7    , where a and b are rational numbers, find, without 5 7  using a calculator, the value of a and of b.

2

3

Given that  is acute and that cos  

1 , find, without using a calculator, 3

(i)

cosec (90   ) ,

[2]

(ii)

1 sin  . 2

[2]

The expression 9 x 4  5ax 3  7bx 2  5 x  6 has a factor x – 1 and leaves a remainder of 40 when divided by x + 1. Calculate the value of a and of b.

4

(i)

(ii)

(b)

[5]

Explain why the method fails when applied to the lines 7 x  y  13 and

14 x  2 y  5 .

(a)

[5]

 1 7  Given that A =   , find A-1 and hence use it to find the coordinates  3 11   of the point of intersection of the lines 7 x  y  13 and 11x  3 y  9 .

5

[4]

[1]

Find the range of values of x for which (i)

2 x 2  9 x  5 is negative,

[2]

(ii)

2  x  3 x 2 is negative.

[2]

Find the range of values of k for which x 2  2kx  3 is always positive, giving your answer in surd form.

[2]

4 6

20

S

P

Q





M

N

20

20

R

The diagram shows the cross-section of a right trapezoidal prism PQRS. The sides  PS = QR = MN = 20 cm and the angles SPM = RQN =  radians, where 0    . 2 (i)

(ii)

7

(i)

Show that the area of the cross-section, A cm2, of the prism is given by 1   A  400  cos   sin 2  . [3] 2   Given that  can vary, find the value of  that will give the maximum area of the cross-section. [4]

Find the first three terms in the expansion, in ascending powers of x, of

2  x  . 2 6

[2] n

(ii)

(iii)

 1  The binomial expansion of 1  x  , where n > 0, in ascending powers of x  2  1 3 is 1  nx  x 2 +… . Find the value of n. [3] 2 2

Using the expansions found in part (i) and (ii), find the coefficient of x 2 in the n

6  1  binomial expansion of 2  x 2 1  x  .  2 



8



sin A  cosec A  cot 3 A . cos A  sec A

(i)

Show that

(ii)

Find all the angles between 0 and 360 which satisfy the equation sin A  cosec A  5 cos 3 A . cos A  sec A

[2]

[3]

[4]

5 9

R P A

S

Q

B T

The diagram shows two circles intersecting at A and B. The line PQ, which is the tangent to the smaller circle at A, meets RS at A. (i)

Prove that AQ = RQ.

[3]

(ii)

Prove that triangles ABS and QRS are similar. Hence show that SA  SR  SB  SQ .

[2]

Another line is drawn from S to form the tangent to the bigger circle at T. (iii)

Show that the result SA  SR  SB  SQ may be proven using the tangentsecant theorem.

10

11

(a)

Given that p  log 2 q , express log 2

(b)

Solve the equation

[2]

32 in terms of p. q

[2]

(i)

log 2 ( x  3)  2 log 4 x  log 6 36 ,

[3]

(ii)

e x (3e x  2)  5 .

[3]

(a)

The roots of the quadratic equation 3 x 2  5 x  1  0 are  and . Find the quadratic equation whose roots are    and   . [6]

(b)

Find the value of m such that one root of the equation x 2  8 x  m  7  0 is three times the other.

(c)

[2]

Given that tan A and tan B are the roots of the equation x 2  x  (1  2 )  0 , 1 show that tan( A  B)  . [2] 2

6 12

The table shows experimental values of two variables x and y. x y

2 34

4 221

6 528

8 1096

10 1845

It is known that x and y are related by the equation y  ax n , where a and n are constants. (i)

Using the vertical axis for ln y and the horizontal axis for ln x, draw a straight line graph of ln y against ln x, for the given data.

[3]

(ii)

Use the graph to estimate the value of a and of n.

[4]

(iii)

By drawing a suitable line on your graph, solve the equation ax n  x 4 .

[2]

~ End of Paper ~