Dunearn Prelim 2009 Am P2

Class

Number

Name : Name of your A.Maths Teacher: Mr. __________________________ DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL

DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL

DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL

DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL

DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL DUNEARN SECONDARY SCHOOL

DUNEARN SECONDARY SCHOOL Prelim Examination 2009 Add. Mathematics (4038/02) Paper 2

Secondary 4 Express & 5 Ruby Wednesday, 16 Sep 2009

0800 - 1030hrs

2 hours 30 minutes

INSTRUCTIONS TO CANDIDATES Write your name, class and register number in the spaces at the top of this page. Answer all questions. Write your answers on the writing paper provided. All working must be shown. Omission of essential working will result in loss of marks. Do not use any highlighters, correction fluid or correction tape for the paper.

INFORMATION FOR CANDIDATES Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. 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 100. The use of an electronic calculator is expected where appropriate. You are reminded of the need for clear presentation in your answers. PARENT’S SIGNATURE

FOR EXAMINER'S USE

100 Setter: Mr80 Venkat This question paper consists of 6 printed pages, including this cover page

A.Maths Paper 2 – prelim 2009

Page 1 of 6

Mathematical Formulae 1.

ALGEBRA

Quadratic Equation 2

For the equation ax  bx  c  0

 b  b 2  4ac x 2a Binomial expansion  n n n (a  b) n  a n    a n1b   a n 2 b 2  ........   a n r b r  ...  b n , 1  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 2 sin A  cos 2 A  1 sec 2 A  1  tan 2 A cos ec 2 A  1  cot 2 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 1  tan A tan B sin 2 A  2 sin A cos A

tan( A  B ) 

cos 2 A  cos 2 A  sin 2 A  2 cos 2 A  1  1  2 sin 2 A 2 tan A 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 a b c   sin A sin B sin C 2 a  b 2  c 2  2bc cos A tan 2 A 

Formulae for ABC



A.Maths Paper 2 – prelim 2009

1 ab sin C 2

Page 2 of 6

Answer all the questions 1.

2.

Find the coordinates of the turning points on the curve y = sin x cos3x for 0 < x < 

4.

[5]

Find the equation of the circle in the form x2 + y2 + 2gx + 2fy +c =0 whose center is (3,-4) and which passes through the origin O. The circle x2 + y2 – 6x – 2y – 15 = 0 contains a chord with midpoint M(2, 3). Find the equation of the chord.

[3]

(a)

Solve: 2 x  3  4 x  0

[3]

(b)

1 Without using calculator, solve 4 3 x  log 2    5 8

[3]

(a) (b)

3.

Max.Mark: 100

(a)

[3]

A pyramid has a rectangular base of area (32  5 3 ) cm2. The length of the rectangle is (10  27 ) cm. Show that the width of the rectangle is

[3]

(5  2 3 ) cm.

The height of the same pyramid is (8  2 3 ) cm. Show that the volume of the pyramid is of the form a  b 3 , where a and b are rational numbers.

5.

(b)

Prove the identity: sin2 x + tan2x sin2 x = tan2x

(a)

Solve the simultaneous equations x2 y   3, x + y = 8 4 3 Find the range values of m for which the line y = mx – 1 does not intersect the curve y = x2 - 2x + 3. State also the values of m for which this line is a tangent to the curve.

(b)

[2]

[3]

[4]

[6]

Turn over]

A.Maths Paper 2 – prelim 2009

Page 3 of 6

6.

The diagram, which is not drawn to scale, shows a right-angled triangle ABC in which angle BAC = 90º. The coordinates of A and B are (3,5) and (-1, -3) respectively. Given that the gradient of BC is ½ and that D is the foot of the perpendicular from A to BC, find

(a) (b) (c) (d)

7.

(a)

(b)

the equation of BC and of AC, the coordinates of C, the coordinates of D, the length of the perpendicular AD (in surds form)

[5] [2] [3] [1]

A particle moves in a straight line and passes a fixed point O on the line so that, t seconds after passing O, its velocity, v ms-1, is given by v = 3t2 + 2(1 – t)2 . Calculate (i) the initial velocity of the particle,

[1]

(ii)

the acceleration of the particle when t = 2,

[2]

(iii)

the displacement of the particle from O when t = 3.

[4]

Solve the cubic equation x3 – 7x2 + 4x + 12 = 0

A.Maths Paper 2 – prelim 2009

[4]

Page 4 of 6

8.

(a)

The diagram shows a rectangle block that has a volume of 72 cm3. Show 216 that the surface area, A cm2, of the block is given by A = 4 x 2  . x Hence, find the minimum value of A.

x cm

2x cm [8] y cm (b)

ln x , find 3x  6 dy the value of when x = 1, dx

Given that y  (i) (ii)

[3]

the rate of change of x when x = 1, given that y is decreasing at [2]

a constant rate of 0.1 units per second. 9.

The diagram belows shows a circle with centre P. AD is the diameter. DC is a tangent to the circle and BCD is a right angle. AB produced and DC produced meet at T. A

B P

D

A.Maths Paper 2 – prelim 2009

C

T

Page 5 of 6

(a)

[3] [3]

Show that ABD and DCB are similar with reasons Name three other triangles which are similar to ABD.

(b)

Giving the reasons clearly, show that [2]

(i)

DB 2  DA  CB ,

(ii)

DC 2  CB  ( DA  CB ) .

[2]

(b)

Find all the angles between 0º and 360 º which satisfy the equation 3 cos x – 4 sin x = 1 [5] 2 10. (a) The diagram shows part of the curve y  5 x  x . The curve meets the xaxis at the origin O and at point A. The tangent to the curve at the point B where x = 3, intersects the x-axis at C. y B

x

O A

(b)

C

(a)

Find the equation of the tangent to the curve at B.

[3]

(b)

Find the area of the shaded region.

[4]

Given that



3 0

f ( x)dx  6 ,



3 0

g ( x )dx  8 ,

5 3

f ( x )dx  4 , evaluate [2]

3

(i)

 2 f ( x)  g ( x)dx ,

(ii)



(iii)



0

0 5

f ( x)dx ,

[2]

the value of the constant m (in terms of e )for which

  f ( x )  me 3

0

2x

dx

 5.

[4]

END OF PAPER

A.Maths Paper 2 – prelim 2009

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