2011 Mock Paper Bythemathscafe

Name : Index No :

Calculator Model :

ADDITIONAL MATHEMATICS THE MATHS CAFE Paper 2

Additional Materials: Answer Paper

4038/02 October/November 2011

1hour 30 minutes

READ THESE INSTRUCTIONS FIRST Write your index number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. 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 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 document consists of 6 printed pages including the cover page

2

© The Maths Cafe 4013/01/O/N/11 www.TheMathsCafe.com

Mathematical Formulae

1. ALGEBRA

Quadratic Equation For the equation 02 cbxax ,

aacbbx

242

.

Binomial Expansion

.......21

221 nrrnnnnn bbarn

ban

ban

aba

where n is a positive integer and

!)1)...(1(

)!(!!

rrnnn

rnrn

rn

2. TRIGONOMETRY Identities

1cossin 22 AA

AA 22 tan1sec

AAec 22 cot1cos BABABA sincoscossin)sin(

BABABA sinsincoscos)cos(

BABABA

tantan1tantan)tan(

AAA cossin22sin

AAAAA 2222 sin211cos2sincos2cos

AAA 2tan1

tan22tan

)(21cos)(

21sin2sinsin BABABA

)(21sin)(

21cos2sinsin BABABA

)(21cos)(

21cos2coscos BABABA

)(21sin)(

21sin2coscos BABABA

Formulae for ABC

Cc

Bb

Aa

sinsinsin

Abccba cos2222

Cabsin21

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3


1 (i) Write down and simplify the first three terms in the expansion, in ascending

powers of x, of nx

21 . [3]

(ii) The expansion of nxx

2132 , in ascending powers of x as far as the term in

2x , is 252 axx . Find the value of n and of a. [3]

2 If 542cos x and 2702180 x , calculate, without using calculators, the values of

(i) sec 2x, [1]

(ii) sin 4x, [2]

(iii) .tan x [2]

3 (i) Express 2)1)(2(74

xxx

in partial fractions. [4]

(ii) Hence evaluate dx

xx

127x4

1

0 2

. [3]

4 (i) Find the coordinates of all the points at which the graph of 321 xy

meets the coordinate axes. [4]

(ii) Sketch the graph of 321 xy for 30 x . [2]

(iii) Solve the equation 3213 xx . [2]


4


5 The equation of a curve is 2

273y xx

.

(i) Find dd

yx

. [1]

(ii) Show that 2

273y xx

is an increasing function for x > 0. [1]

(iii)Find the equation of the tangent to the curve at x = 3. [3] 6 The points A ( 2, 4 ) and B ( 6, 2) are at the opposite ends of a diameter of a

circle, 1C . Find the

(i) equation of the perpendicular bisector of AB, [2] (ii) equation of the circle. [3]

(iii) The circle, 1C , is reflected in the x-axis. Find the equation of the reflected circle, 2C . [2]

7 (i) Find xedxd 2 . [2]

(ii) Hence evaluate dxx

e x

4

0

2

[2]

(iii) (a) Sketch the curve 32 xey , indicating the intercepts on the axes. [2]

(b) In order to solve the equation 8ln xx , a graph of a suitable

straight line is drawn on the same axes as the graph of 32 xey .

Find the equation of this straight line. [3]


5


x°

D

C

BA

4

3

9

8 A piece of cardboard ABCD is made up of two triangles ABC and ACD as shown. It is

given that AB = 3 cm, AD = 4 cm, AC = 9 cm, BAD = 90 and CAD = x, where

0 90x .

(i) Show that the area of the cardboard, A cm2, is given by 2718sin cos2

A x x . [2]

(ii) Express A in the form sin αR x , where R > 0 and 0 < < 90. [3]

(iii)Find the value of x which gives a maximum value of A, and state this maximum

value of A. [3]

9 (i) Given that 9 5

3 dxf(x) ,

(a) evaluate 3

5

5

3 )( )( dxxfdxxf [1]

(b) find the value of k for which 25 )( 5

3 dxkxxf . [2]


6


9 (ii) The diagram shows part of the curve 12sin3 xy , meeting the x-axis at A,B and C.

(a) Find the x-coordinates of A, B and C. [3] (b) Find the area of the shaded region. [4] 10 A particle starts from a fixed point A and travels in a straight line. The velocity, v m/s, of the particle, t s after leaving A, is given by

4102 ttv (i) Find the acceleration of the particle when it is at instantaneous rest. [5]

(ii) Find the total distance travelled by the particle between t = 0 s and t = 7 s. [5]

11 In the diagram, O is the centre of the circle and

D, E, F and G are points on the circle.

AD is tangent to the circle at D and ABCD

is a parallelogram. Prove that

(i) ABD is similar to EDF [2]

(ii) DFDCDEDB [3]

End of Paper 2

A

O B C

D

E

F

G

y

x radians

12sin3 xy

A C B O


2011 Mock Paper Bythemathscafe

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