r % TEST CODE OI254O2O-lMAY/JUNE 20I6FORM TP 2016037CARIBBEAN EXAMINATIONS COUNCIL

CARIBBEAN SECONDARY EDUCATION CERTIFICATE@EXAMINATION

ADDITIONAL MATHEMATICS

Paper 02 - General Proficiency

2 hours 40 minutes

Reouired Examination Materials

Electronic calculator (non programmable)Geometry setMathematical tables (provided)Graph paper (provided)

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.

1

READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

This paper consists of FOUR sections. Answer ALL questions in Section I,Section II and Section III.

Answer ONE question in Section IV.

Write your answers in the spaces provided in this booklet

Do NOT write in the margins.

A list of formulae is provided on page 4 of this booklet.

If you need to rewrite any answer and there is not enough space to do so on theoriginal page, you must use the extra page(s) provided at the back of this booklet.Remember to draw a line through your original answer.

If you use the extra page(s) you MUST write the question number clearly inthe box provided at the top of the extra page(s) and, where relevant, includethe question part beside the answer.

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-lr 4LIST OF FORMULAE

Arithmetic Series

Geometric Series

Circle

Vectors

Trigonometry

Differentiation

Statistics

Probability

Kinematics

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-lr -5-SECTION I

Answer BOTH questions.

ALL working must be clearly shown.

The domain for the function/(x):2x - 5 is {-2, -1, 0, I }

(i) Determine the range of the function.

(ii) Find / '(x)

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(2 marks)

(1 mark)

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l. (a)

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JL 0125402005

r -6-(iii) Sketch the graphs of f (x) andfa (x) on the same axes.

(iv) Comment on the relationship between the two graphs

(b) Solve the equation 22t+t + 5 (2')- 3:0

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(2 marks)

(1 mark)

(4 marks)

GO ON TO THE NEXT PAGE

IL 0125402006

-7 -

rh)l;l(c) (i) Given that T: k p r, make c the subject of the formula.

(ii) Solve the equation

log (x + 1) + log (x- l) : 2 log (x + 2).

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(2 marks)

(2 marks)

Total 14 marks

GO ON TO THE NEXT PAGE012540201F 2016

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r -8-2. (a) (i) Determine the nature of the roots of the quadratic equation 2f + 3x - 9:0

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(1 mark)

(ii) Given thatf (x) :2f -r 3x - 9, sketch the graph of the quadratic function, clearlyindicating the minimum value.

(5 marks)

GO ON TO THE NEXT PAGE

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012540201F 2016

JL 0125402008

t- -9 -25

(b) Evaluate ZZ'.

(3 marks)

(c) A man invested $x in a company in January 2010, on which he earns quarterly dividends.At the end of the second, third and fourth quarter in 201 l, he earned $100, $1 l5 and $130respectively" Calculate the total dividends on his investment by the end of 2016.

(5 marks)

Total 14 marks

GO ON TO THE NEXT PAGE012s40201F 2016

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JL 0125402009

r -10-SECTION II

Answer BOTH questions.

ALL working must be clearly shown.

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3. (a)

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(i) The points M (3,2) and N(-1, 4) are the ends of a diameter of circle C. Determinethe equation of circle C.

(5 marks)

(ii) Find the equation of the tangent to the circle C at the point P (-1,6)

(3 marks)

GO ON TO THE NEXT PAGE

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t- -ll-(b) The position vector of two points A and.B, relative to a fixed origin, O, are a and b

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respecrivery, where ,:l:1. "ro

,: [;] . p ties onissuch that rt: +Ii

---+

Find the coordinates of OP.

GO ON TO THE NEXT PAGE01254020/F 2016

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(4 marks)

Total 12 marks

L 0125402011

r -12- -lThe following diagram (not drawn to scale) shows two sectors, AOB and DOC. OB andOC are x cm and (x + 2) cm respectively and angle AOB : 0.

D

4. (a)

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2tr_Trf 0: radians, calculate the area of the shaded region in terms of x,

(4 marks)

(b) Given that cos 30" and sin 45o:fr , without the use of a calculator, evaluate2cos 105o, in surd form, giving your answer in the simplest terms

(5 marks)

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JL 0125402012

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r - 13 - -lsin (d + a)(c) Prove that the identitycos d cos a = tan 0+tana

(3 marks)

Total 12 marks

GO ON TO THE NEXT PAGE0r2540201F 2016

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t- -14-SECTION III

Answer BOTH questions.

ALLworking must be clearly shown.

5. (a) nina $ given thaty : 'l-5f 4,simplifying your answer.

(4 marks)

(b) The point P (1, 8) lies on the curve with equati on y:2x (x + I )2. Determine the equationof the normal to the curve at the point P.

(5 marks)

GO ON TO THE NEXT PAGE01254020/F 2016

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IL 0125402014

-lt- - 15 -Obtain the equation for EACH of the two tangents drawn to the curve)/: f at the pointswhere y: 16.

(5 marks)

Total 14 marks

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(c)

0t254020tF 2016

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-lr - 16-J6. (a) (i) Find (3cos0-5sin0)d0

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(3 marks)

(4 marks)

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(ii) Evaluate I l+ -3 +2r)dx

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-lt- -17 -The following figure shows the finite region R bounded by the lines x : l, x: 2 and thearc of the curve / : (x - 2)2 + 5 .

y-axis

(b)

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(0,9)

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Calculate the area of the region R

(4 marks)

GO ON TO THE NEXT PAGE

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t- - 18 -The point P (1 ,2) lies on the curve which has a gradient function given byFind the equation of the curve.

GO ON TO THE NEXT PAGE

-l(c)

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9-:3x2 - 6x.dx

(3 marks)

Total 14 marks

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-lr - 19 -SECTION IV

Answer only ONE question.

ALL working must be clearly shown

7. (a) Use the data set provided below to answer the questions which follow.

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22 25 28 30 30 32 35 40

41 52 54 59 60 65 68 75

(i) Construct a stem-and-leaf diagram to represent the given data.

(3 marks)

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-lr -20-(ii) State an advantage of using the stem-and-leaf diagram to represent the given data.

(l mark)

(iii) Determine the mode

(l mark)

(iv) Determine the median.

(2 marks)

(v) Determine the interquartile range.

(3 marks)

GO ON TO THE NEXT PAGE

012540201F 2016

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r -21 -Two events, A and B, are such that P(A):0.5, P(B): 0.8 and P(A w B) : 0.9

(i) Determine P(A a B'1.

(ii) Determine P(A I B\

(iii) State, giving a reason, whether or not A and.B are independent events.

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-l(b)

01254020tF 2016

(2 marks)

(2 marks)

(2 marks)

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r -22-A bag contains 3 red balls, 4 black balls and 3 yellow balls. Three balls are drawn atrandom with replacement from the bag. Find the probability that the balls drawn are allof the same colour.

(4 marks)

Total20 marks

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-l(c)

01254020/F 2016

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r -23 -A motorist starts from a point, X, and travels 100 m due North to a point, Y, at a constantspeed of 5 m s-r. He stays atYfor 40 seconds and then travels at a constant speed ofl0 m s-' for 200 m due South to a point, Z.

(i) On the following grid, draw a displacement-time graph to display this information.

(5 marks)

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-l8. (a)

01254020/F 2016

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-lI- -24 -(ii) Calculate the average speed for the whole journey

(iii) Calculate the average velocity of the whole journey.

(3 marks)

(3 marks)

GO ON TO THE NEXT PAGE01254020/F 2016

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r -25 -(b) A particle starting from rest, travels in a straight line with an acceleration, a, given by

a : cos / where r is the time in seconds.

(i) Find the velocity of the particle in terms of r

(3 marks)

(ii) Calculate the displacement of the particle in the interval of time t : r to t : 2r.

(6 marks)

Total20 marks

END OF TEST

IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.

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-lr -26 -EXTRA SPACE

If you use this extra page, you MUST write the question number clearly in the box provided.

Question No.

01254020/F 2016

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