CSEC® Additional Mathematics Past Papers · CSEC® Additional Maths Past Papers LIST OF CONTENTS Paper 02 (03 May 2012) 3 Paper 032 (12 June 2012) 11 Paper 02 (07 May 2013) 13 Paper

Additional Mathematics

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CSEC® Additional Maths Past Papers

LIST OF CONTENTS

Paper 02 (03 May 2012) 3

Paper 032 (12 June 2012) 11

Paper 02 (07 May 2013) 13

Paper 032 (12 June 2013) 21

Paper 02 (06 May 2014) 23

Paper 032 (09 June 2014) 33

Paper 02 (05 May 2015) 38

Paper 032 (08 June 2015) 48

Paper 02 May/June 2016 51

Paper 032 May/June 2016 74

TEST CODE 01254020FORM TP 2012037 MAY/JUNE 2012

C A R I B B E A N E X A M I N A T I O N S C O U N C I L

SECONDARY EDUCATION CERTIFICATEEXAMINATION

ADDITIONAL MATHEMATICS

Paper 02 – General Proficiency

2 hours 40 minutes

READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

1. DO NOT open this examination paper until instructed to do so.

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

3. Answer ONE question in Section IV.

4. Write your solutions with full working in the booklet provided.

Required 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.

Copyright © 2011 Caribbean Examinations CouncilAll rights reserved.

01254020/F 2012

03 MAY 2012 (p.m.)

- 2 -

GO ON TO THE NEXT PAGE01254020/F 2012

SECTION I

Answer BOTH questions.

1. (a) The functions f and g are defined by

f(x) = x3 + 1, 0 < x < 3

g(x) = x + 5, x ∈ R

where R is the set of real numbers.

(i) Determine the composition function g(f(x)). (1 mark)

(ii) State the range of g(f(x)). (1 mark)

(iii) Determine the inverse of g(f(x)). (2 marks)

(b) If x + 2 is a factor of f(x) = 2x3 – 3x2 – 4x + a, find the value of a. (2 marks)

(c) Solve the equation 32x – 9(3–2x) = 8. (5 marks)

(d) (i) Express x3 = 10x–3 in the form log10x = ax + b. (2 marks)

(ii) Hence, state the value of the gradient of a graph of log10x versus x. (1 mark)

Total 14 marks

2. (a) The quadratic equation x2 – 4x + 6 = 0 has roots α and β.

Calculate the value of α2 + β2. (5 marks)

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

——— > 0. (4 marks)

(c) A customer repays a loan monthly by increasing the payment each month by $x. If the customer repaid $50 in the 5th month and $70 in the 9th month, calculate the TOTAL amount of money repaid at the end of the 24th month. (5 marks)

Total 14 marks

2x – 53x + 1

- 3 -


SECTION II


3. (a) The equation of a circle is given by x2 + y2 – 4x + 6y = 87.

(i) A line has equation x + y + 1 = 0. Show that this line passes through the centre of the circle. (3 marks)

(ii) Find the equation of the tangent to the circle at the point A (– 6, 3). (4 marks)

(b) Given OA = a, OB = b, AP = — OA,

where a = and b = .

(i) Write BP in terms of a and b. (2 marks)

(ii) Find |BP|. (3 marks)

Total 12 marks

→ → → →12

21

32

→

→

- 4 -


4. (a) The diagram shows a sector of a circle centre O with an adjoining square. The radius of the circle is 4 m.

If the sector AOC subtends an angle — at O, calculate, giving your answer in terms of π

(i) the area of the shape OACMN

(ii) the perimeter of the shape OACMN. (5 marks)

(b) Given that sin — = —–, cos — = — and sin — = cos — = —–, evaluate without using

calculators, the exact value of cos —–. (3 marks)

(c) Prove the identity

——–——— ≡ ———— . (4 marks)

Total 12 marks

�3

�3

�4

12

√ 32

�4

√ 22

7�12

�3

1

1 – sin θcos θ1

cos θ + tan θ

- 5 -


SECTION III


5. (a) Differentiate the following expression with respect to x, simplifying your answer.

——— (4 marks)

(b) The point P (2, 10) lies on the curve y = 3x2 + 5x – 12. Find equations for

(i) the tangent to the curve at P

(ii) the normal to the curve at P. (5 marks)

(c) The length of the side of a square is increasing at a rate of 4 cms–1. Find the rate of increase of the area when the length of the side is 5 cm. (5 marks)

Total 14 marks

6. (a) Evaluate (16 – 7x)3 dx. (4 marks)

(b) The point Q (4, 8) lies on a curve for which —– = 3x – 5. Determine the equation of the curve. (3 marks)

(c) Calculate the area between the curve y = 2 cos x + 3 sin x and the x-axis from x = 0 to —.

(3 marks)

(d) Calculate the volume of the solid formed when the area enclosed by the curve y = x2 + 2 and the x-axis, from x = 0 to x = 3, is rotated through 360° about the x-axis.

[Leave your solution in terms of � ]. (4 marks)

Total 14 marks

3x + 4x – 2

∫2

1

dydx

�3

- 6 -


SECTION IV

Answer only ONE question.

7. (a) A survey carried out in a town revealed that 25% of the households surveyed owned a laptop computer and 70% owned a desktop computer. In addition, it was found that 12% owned both a laptop and a desktop computer.

If a sample of households from the town is selected at random, determine the proportion that own NEITHER a laptop NOR a desktop computer.

(4 marks)

(b) A bag contains 4 red marbles, 3 black marbles and 3 blue marbles. Three marbles are drawn at random without replacement from the bag.

Find the probability that the marbles

(i) drawn are ALL of the SAME colour (3 marks)

(ii) contain EXACTLY 1 red marble. (3 marks)

(c) The probability of hiring a taxi from garage A, B or C is 0.3, 0.5 and 0.2 respectively. The probability that the taxi ordered will be late from A is 0.07, from B is 0.1 and from C is 0.2.

(i) Illustrate this information on a tree diagram showing the probability on all branches. (3 marks)

(ii) A garage is chosen at random, determine the probability that

a) the taxi will arrive late (3 marks)

b) the taxi will come from garage C given that it is late. (4 marks)

Total 20 marks

8. (a) A car starting from rest at a point A, moves along a straight line reaching a velocity of 24 ms–1 by a constant acceleration of 6 ms–2. The car maintains this constant velocity of 24 ms–1 for 5 seconds and is then brought to rest again by a constant acceleration of –3 ms–2.

(i) Using the graph sheet provided, draw a velocity-time graph to illustrate the motion of the car. (3 marks)

(ii) Determine the TOTAL distance travelled by the car. (3 marks)

(iii) A second car, moving at a constant velocity of 32 ms–1 drives past point A, 3 seconds after the first car left point A. Calculate the length of time after the first car started that this second car meets it.

[Assume that the cars meet during the time when the first car is moving at a constant velocity.] (4 marks)

(b) A particle moves in a straight line with acceleration given by a = (5t – 1) ms–2 at any time t seconds. When t = 2 seconds, the particle has velocity 4 ms–1 and is 8 m from a fixed point O. Determine

(i) its velocity when t = 4 (5 marks)

(ii) its displacement from O when t = 3. (5 marks)

Total 20 marks

END OF TEST

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

- 7 -

01254020/F 2012






Answer Sheet for Question 8. (a) (i) Candidate Number ............................................

ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET

01254020/F 2012






ALTERNATIVE TO SBA

90 minutes

Answer all parts of the given question.



01254032/F 2012

12 JUNE 2012 (p.m.)

- 2 -

01254032/F 2012

(a) Two sports clubs, P and Q, wish to use 600 m of fencing to enclose a court. They wish to determine which design gives the maximum area.

Sports Club P uses the 600 m of fencing to make a rectangular court measuring 3x m by 2y m as shown in the diagram below.

3x m

2y m

Sports Club Q uses the 600 m of fencing to make six equal-sized rectangular courts that are adjacent to each other as shown in the diagram below. Each court measures x m by y m.

x m x m x m

y m

y m

For Sports Club P the mathematical problem is to maximize the area of enclosure to satisfy its perimeter and the following conditions:

Maximize A = 6xySubject to 6x + 4y = 600

(i) Formulate the mathematical problem for Sports Club Q. (2 marks)

(ii) Determine the MAXIMUM area of the court for Sports Club Q. (3 marks)

(iii) Show that Sports Club P has the maximum area when a square enclosure is used and determine the MAXIMUM possible area. (4 marks)

(iv) Suggest which sports club design should be used. (1 mark )

(b) The numbers log (a3 b7), log (a5 b12) and log (a8 b15) are the first three terms of an arithmetic series. The 12th term of the series is log bn. Calculate the value of n. (10 marks)

Total 20 marks

END OF TEST




CARIBBEAN SECONDARY EDUCATION CERTIFICATE®

EXAMINATION



2 hours 40 minutes

07 MAY 2013 (p.m.)


1. DO NOT open this examination paper until instructed to do so.

2. This paper consists of FOUR sections. Answer ALL questions in Section 1, Section 2 and Section 3.

3. Answer ONE question in Section 4.


5. A list of formulae is provided on page 2 of this booklet.


Electronic calculator (non programmable)Geometry SetMathematical Tables (provided)Graph paper (provided)



01254020/F 2013


- 2 -

LIST OF FORMULAE


- 3 -

SECTION 1


All working must be clearly shown.

1. (a) Let f(x) = x3 – x2 – 14x + 24.

(i) Use the factor theorem to show that x + 4 is a factor of f(x). (2 marks)

(ii) Determine the other linear factors of f(x). (3 marks)

(b) A function f(x) is given by f(x) = .

(i) Find an expression for the inverse function f –1(x). (3 marks)

(ii) The function g is given by g(x) = x +1. Write an expression for the composite function, fg(x). Simplify your answer. (2 marks)

(c) Given that 53x –2 = 7x + 2, show that

x = . (4 marks)

Total 14 marks

2. (a) Let f(x) = 3x2 + 6x – 1.

(i) Express f(x) in the form a(x + h)2 + k where h and k are constants. (3 marks)

(ii) State the minimum value of f(x). (1 mark)

(iii) Determine the value of x for which f(x) is a minimum. (1 mark)

(b) Find the set of values of x for which 2x2 + 3x – 5 ≥ 0. (4 marks)

(c) Find the sum to infinity of the following series:

– + – + – + – + ...

Note: This series can be rewritten as the sum of two geometric series. (5 marks)

Total 14 marks

2x – 1x + 2

2(log5 + log7)(log125 – log7)

14

242

143

244

- 4 -


SECTION 2



3. (a) (i) A circle, C, has centre with coordinates A (2,1) and passes through the point B (10,7).

Express the equation of the circle in the form x2 + y2 + hx + gy + k = 0, where h, g and k are integers to be determined. (3 marks)

(ii) The line l is a tangent to the circle C at the point B. Find an equation for l.(3 marks)

(b) The position vectors of two points, P and Q, relative to a fixed origin, O, are 10i – 8j and λi + 10j respectively, where λ is a constant.

Find the value of λ such that OP and OQ are perpendicular. (3 marks)

(c) The position vectors of A and B with respect to a fixed origin, O, are given by OA = –2i +5j and OB = 3i – 7j respectively.

Find the unit vector in the direction of AB. (3 marks)

Total 12 marks

4. (a)

The diagram shows a sector cut from a circle of centre O. The angle at O is —. If the

perimeter of the sector is — (12 + π) cm, what is its area? (4 marks)

(b) Solve the equation 2 cos2θ + 3 sin θ = 0 for 0 ≤ θ ≤ 360°. (5 marks)

(c) Given that tan (θ – α) = – and that tan θ = 3, use the appropriate compound angle formula to find the value of the acute angle α. (3 marks)

Total 12 marks

→ →

56

π6

12

→ →


- 5 -

SECTION 3



5. (a) Given that y = x3 – 3x2 + 2. Find

(i) the coordinates of the stationary points of y (5 marks)

(ii) the second derivative of y and hence determine the nature of EACH of the stationary points. (5 marks)

(b) Differentiate y = (5x + 3)3sin x with respect to x, simplifying your result as far as possible. (4 marks)

Total 14 marks

6. (a) Find (5x2 +4) dx. (2 marks)

(b) Evaluate (4 marks)

(c) A curve passes through the points P(0, 8) and Q(4, 0) and is such that — = 2 – 2x.

Find the area of the finite region bounded by the curve in the first quadrant. (8 marks)

Total 14 marks

dydx

0

2

∫ 3sinx – 5cos x( )dxπ

(3 sin x – 5 cos x) dx.

0

2

∫ 3sinx – 5cos x( )dx


- 6 -

SECTION 4

Answer ONLY ONE question.


7. (a) Of the persons buying petrol at a service station, 40 per cent are females. Of the females, 30 per cent pay for their petrol with cash, and of the males, 65 per cent pay for their petrol with cash.

(i) Copy and complete the following tree diagram, by putting in all the missing probabilities, to show this information. (2 marks)

(ii) What is the probability that a customer pays for petrol with cash?(3 marks)

(iii) Determine which is the more likely event:

Event T: Customer is female, GIVEN that the petrol is paid WITH cash. Event V: A male customer does NOT pay for petrol with cash. (4 marks)

(b) The marks obtained by 30 students on an English exam are given as

58 92 41 89 72 66 51 63 80 40

69 45 83 76 53 56 75 50 99 50

85 63 58 75 66 56 81 74 51 94

(i) State ONE advantage of using a stem and leaf diagram versus a box and whiskers plot to display the data. (1 mark)

(ii) Construct a stem and leaf diagram to show the data. (3 marks)

(iii) Determine the median mark. (2 marks)

(iv) Calculate the semi inter-quartile range of the marks. (3 marks)

(v) Two students are chosen at random from the class.

Determine the probability that both scored less than 50 on the exam.(2 marks)

Total 20 marks

01254020/F 2013

- 7 -

8. (a) A particle starts from rest and accelerates uniformly to 20 m s–1 in 5 seconds. It continues at this velocity for 10 seconds. It then accelerates again uniformly to a velocity of 60 m s–1 in 5 seconds. The particle then decelerates uniformly until it comes to rest, 15 seconds later.

(i) On the graph paper provided, draw a velocity-time graph to illustrate the motion of the particle. (3 marks)

(ii) From your graph determine

a) the total distance, in metres, travelled by the particle (4 marks)

b) the average velocity of the particle for the entire journey. (2 marks)

(b) A particle travels in a straight line in such a way that after t seconds its velocity, v, from a fixed point, O, is given by the function v = 3t2 – 18t + 15.

Calculate

(i) the values of t when the particle is instantaneously at rest (3 marks)

(ii) the distance travelled by the particle between 1 second and 3 seconds(3 marks)

(iii) the value of — when

a) t = 2 seconds (2 marks)

b) t = 3 seconds. (1 mark)

(iv) Give an interpretation for the value in

a) 8 (b) (iii) a) (1 mark)

b) 8 (b) (iii) b). (1 mark)

Total 20 marks

END OF TEST


dvdt

01254020/F 2013




EXAMINATION



Answer Sheet for Question 8 (a) Candidate Number .................................





EXAMINATION



Alternative Paper

1 hour 30 minutes

12 JUNE 2013 (p.m.)

Answer the given questions.



01254032/F 2013

Answer the given questions.

1. (a) A circle is drawn with centre at origin, O, and radius 6 cm. Find the coordinates of all intersections of the circle with an origin centred square of side length 10 cm whose sides are parallel to the coordinate axes as illustrated in Figure 1.

Figure 1(10 marks)

(b) The cuboid shown in Figure 2 has width x m and its length is twice its width. The volume of the cuboid is 720 m3.

Figure 2

(i) Find an expression for the height, h, of the cuboid in terms of x. (3 marks)

(ii) Show that an expression for the surface area, A, of the cuboid is given by

A = + 4x2. (3 marks)

(iii) Hence show that A has a stationary value when x = 3√10. (4 marks)

Total 20 marks

END OF TEST


01254032/F 2013

- 2 -

O

x

3

2160x




EXAMINATION



2 hours 40 minutes


1. This paper consists of FOUR sections. Answer ALL questions in Section 1, Section 2 and Section 3.

2. Answer ONE question in Section 4.




Electronic Calculator (non programmable)Geometry SetMathematical Tables (provided)Graph Paper (provided)



01254020/F 2014

06 MAY 2014 (p.m.)

- 2 -


LIST OF FORMULAE

Arithmetic Series Tn = a + (n – 1)d Sn = — [2a + (n – 1)d]

Geometric Series Tn = arn–1 Sn = ———— S∞ = ——, –1 < r < 1 or |r| < 1

Circle x2 + y2 + 2fx + 2gy + c = 0 (x + f)2 + (y + g)2 = r2

Vectors v = — cos θ = ——— |v| = √ (x2 + y2) where v = xi + yj

Trigonometry sin (A + B) ≡ sin A cos B + cos A sin B

cos (A + B) ≡ cos A cos B sin A sin B

tan (A + B) ≡ ——————–

Differentiation ––– (ax + b)n = an(ax + b)n–1

––– sin x = cos x

––– cos x = –sin x

Statistics x = —— = ——— , S2 = ————— = ———— – (x)2

Probability P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Kinematics v = u + at v2 = u2 + 2as s = ut + — at2

n2

a(rn – 1)r –1

v|v|

^ a•b|a| × |b|

+

tan A + tan B1 tan A tan B+

ddx

ddx

ddx

–

n

i = 1nΣ xi

n

i = 1Σ fi xi

n

i = 1Σ fi

n

n

i = 1Σ (xi – x)2–

–

12

n

i = 1Σ fi

n

i = 1Σ fi xi

2

a1 – r

- 3 -


SECTION 1


ALL working must be clearly shown.

1. (a) (i) The function f is defined by f : x → 1 – x2, x ∈ .

Show that f is NOT one-to-one. (1 mark)

(ii) The function g is defined by g : x → — x – 3, x ∈ .

a) Find fg(x), and clearly state its domain. (2 marks)

b) Determine the inverse, g–1, of g and sketch on the same pair of axes, the graphs of g and g–1. (3 marks)

(b) When the expression 2x3 + ax2 – 5x – 2 is divided by 2x – 1, the remainder is –3.5.

Determine the value of the constant a. (3 marks)

(c) The length of a regtangular kitchen is y m and the width is x m. If the length of the kitchen is half the square of its width and its perimeter is 48 m, find the values of x and y (the dimensions of the kitchen). (5 marks)

Total 14 marks

2. (a) Given that f(x) = –2x2 – 12x – 9.

(i) Express f(x) in the form k + a (x + h)2, where a, h and k are integers to be determined. (3 marks)

(ii) State the maximum value of f(x). (1 mark)

(iii) Determine the value of x for which f(x) is a maximum. (1 mark)

(b) Find the set of values of x for which 3 + 5x – 2x2 < 0. (4 marks)

(c) A series is given by 0.2 + 0.02 + 0.002 + 0.0002 + ...

(i) Show that this series is geometric. (3 marks)

(ii) Find the sum to infinity of this series, giving your answer as an exact fraction. (2 marks)

Total 14 marks

12

- 4 -


SECTION 2



3. (a) (i) Determine the value of k such that the lines x + 3y = 6 and kx + 2y = 12 are perpendicular to each other. (3 marks)

(ii) A circle of radius 5 cm has as its centre the point of intersection of the two perpendicular lines in (i). Determine the equation for this circle. (3 marks)

(b) RST is a triangle in the coordinate plane. Position vectors R, S, and T relative to an

origin, O, are , and respectively.

(i) Show that TRS = 90°. (4 marks)

(ii) Determine the length of the hypotenuse. (2 marks)

[Hint: A rough drawing of RST might help]. Total 12 marks

^

1 3 4 1 –1 4

- 5 -


4. (a) Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians.

A

B

H

CO

9 cm

0.7 rad

Figure 1.

(i) Find the area of the sector OAB. (2 marks)

(ii) Hence, find the area of the shaded region, H. (4 marks)

(b) Given that sin — = — and cos — = —–, show that

cos x + — = — ( √ 3 cos x – sin x), where x is acute. (2 marks)

(c) Prove the identity ——–——– ≡ 1 + ——– . (4 marks)

Total 12 marks

12

�6

�6

√ 32

�6

12

tan θ sin θ1 – cos θ

1cos θ

- 6 -


SECTION 3



5. (a) The equation of a curve is y = 3 + 4x – x2. The point P (3, 6) lies on the curve.

Find the equation of the tangent to the curve at P, giving your answer in the form

ax + by + c = 0, where a, b, c, ∈ . (4 marks)

(b) Given that f(x) = 2x3 – 9x2 – 24x + 7.

(i) Find ALL the stationary points of f(x). (5 marks)

(ii) Determine the nature of EACH of the stationary points of f(x). (5 marks)

Total 14 marks

6. (a) Evaluate x (x2 – 2) dx. (4 marks)

(b) Evaluate (4 cos x + 2 sin x) dx, leaving your answer in surd form. (4 marks)

(c) A curve passes through the point P (2, –5) and is such that —– = 6x2 – 1.

(i) Determine the equation of the curve. (3 marks)

(ii) Find the area of the finite region bounded by the curve, the x-axis, the line x = 3 and the line x = 4. (3 marks)

Total 14 marks

∫0

�3–

∫4

2

dydx

- 7 -


SECTION 4

Answer ONLY ONE question.


7. (a) There are 60 students in the sixth form of a certain school. Mathematics is studied by 27 of them, Biology by 20 of them and 22 students study neither Mathematics nor Biology. If a student is selected at random, what is the probability that the student is studying

(i) both Mathematics and Biology? (3 marks)

(ii) Biology only? (2 marks)

(b) Two ordinary six-sided dice are thrown together. The random variable S represents the sum of the scores of their faces landing uppermost.

(i) Copy and complete the sample space diagram below.

6 9

5 7

4 10

3 8

2 6

1 2

1 2 3 4 5 6

Sample space diagram of S (1 mark)

(ii) Find

a) P (S > 9) (2 marks)

b) P (S < 4). (1 mark)

- 8 -


16

29

19

(iii) Let D be the difference between the scores of the faces landing uppermost. The table below gives the probability of each possible value of d.

d 0 1 2 3 4 5

P (D = d) — a — b — c

Find the values of a, b and c. (3 marks)

(c) The aptitude scores obtained by 51 applicants for a supervisory job are summarized in the following stem and leaf diagram.

5|1 means 51

3 1 5 9

4 2 4 6 8 9

5 1 3 3 5 6 7 9

6 0 1 3 3 3 5 6 8 8 9

7 1 2 2 2 4 5 5 5 6 8 8 8 9 9

8 0 1 2 3 5 8 8 9

9 0 1 2 6

(i) Find the median and quartiles for the data given. (4 marks)

(ii) Construct a box-and-whisker plot to illustrate the data given and comment on the distribution of the data. (4 marks)

Total 20 marks

- 9 -


8. (a) Figure 2 below, not drawn to scale, shows the motion of a car with velocity, V, as it moves along a straight road from Point A to Point B. The time, t, taken to travel from Point A to Point B is 90 seconds and the distance from Point A to Point B is 1410 m.

Figure 2.

(i) What distance did the car travel from Point A towards Point B before starting to decelerate? (2 marks)

(ii) Calculate the deceleration of the car as it goes from 25 m s–1 to 10 m s–1. (5 marks)

(iii) For how long did the car maintain the speed of 10 m s–1? (1 mark)

(iv) From Point B, the car decelerates uniformly, coming to rest at a Point C and covering a further distance of 30 m. Determine the average velocity of the car over the journey from Point A to Point C. (2 marks)

- 10 -

01254020/F 2014

(b) A particle travels along a straight line. It starts from rest at a point, P, on the line and after 10 seconds, it comes to rest at another point, Q, on the line. The velocity v m s–1 at time t seconds after leaving P is

v = 0.72t2 – 0.096t3 for 0 < t < 5

v = 2.4t – 0.24t2 for 5 < t < 10

At maximum velocity the particle has no acceleration.

(i) Find the time when the velocity is at its maximum. (3 marks)

(ii) Determine the maximum velocity. (2 marks)

(iii) Find the distance moved by the particle from P to the point where the particle attains its maximum velocity. (5 marks)

Total 20 marks

END OF TEST





EXAMINATION



ALTERNATIVE

1 hour 30 minutes


1. This paper consists of ONE question. Answer the given question.





01254032/F 2014

09 JUNE 2014 (p.m.)

- 2 -


LIST OF FORMULAE







tan (A + B) ≡ ——————–




Statistics x = —— = ——— , S2 = ————— = ———— – (x)2



n2

a(rn – 1)r – 1

v|v|

^ a•b|a| × |b|

+


ddx

ddx

ddx

–

n

i = 1nΣ xi

n

i = 1Σ fi xi

n

i = 1Σ fi

n

n

i = 1Σ (xi – x)2–

–

12

n

i = 1Σ fi

n

i = 1Σ fi xi

2

a1 – r

- 3 -


1. (a) A student has to compute the area under the graph of a function. He reasons that he can do so by subdividing the area into an infinitely large number of rectangles. To help himself, he investigates by finding the area under the graph of the function f(x) = x over the interval [0,1], using the method of circumscribed rectangles as shown in Figure 1.

f(x) = xf(x)

x0 x1 x2 x3 xn-1 1...

Figure 1. Circumscribed Rectangles

(i) The student subdivides the interval [0, 1] into n equal subintervals. Calculate the width, ∆ x, of each subinterval. (1 mark)

(ii) Let the points of subdivision be x0 = 0, x1, x2, x3, ..., xn–1, xn = 1 as shown in Figure 1.

Find the values of x1, x2, x3, ..., xn–1 in terms of n. (1 mark)

(iii) Determine the heights h1, h2, h3, ..., hn of the circumscribed rectangles over each of the respective n subintervals. (2 marks)

(iv) Determine the area A1, A2, A3, ..., An of the respective circumscribed rectangles. (2 marks)

(v) Show that the sum, Sn, of the areas of these circumscribed rectangles is given by

Sn = ——– . (3 marks)

(Hint: You will need to evaluate the sum of a series. State any theorem used.)

(vi) a) Compute S(n) for n = 10, 20, 50 and 100, giving your answers to three decimal places. (2 marks)

b) What number does S(n) approach as n gets larger? (1 mark)

n + 12n

(b) The variables x and y are related by a law of the form y = axn, where a and n are integers. The approximate values for y, corresponding to the given values of x are shown in Table 1.

Table 1

x 2 3 4 5 6 7

y 50 250 775 1875 3900 7200

(i) Use logarithms to reduce this relation to a linear form, giving your values of lg x and lg y correct to two decimal places where appropriate. (2 marks)

(ii) Using the graph paper provided and a scale of 2 cm to represent 0.1 units on the x-axis, and 1 cm to represent 0.2 units on the y-axis, plot a suitable straight line graph of lg y against lg x. (2 marks)

(iii) Use your straight line graph to estimate the value of the constant a and the value of the constant n. (4 marks)

Total 20 marks

END OF TEST


- 4 -

01254032/F 2014




EXAMINATION



ALTERNATIVE

Graph Sheet for Question 1 (b) (ii) Candidate Number .............................................


01254032/F 2014

05 MAY 2015 (p.m.)




EXAMINATION



2 hours 40 minutes







Electronic Calculator (non-programmable)Geometry SetMathematical Tables (provided)Graph Paper (provided)



01254020/F 2015

- 2 -


LIST OF FORMULAE


Geometric Series Tn = arn–1 Sn = ———— S∞ = —— , –1 < r < 1 or |r| < 1





tan (A + B) ≡ ——————–




Statistics x = —— = ——— , S2 = ————— = ———— – (x)2



n2

a(rn – 1)r – 1

v|v|

^ a•b|a| × |b|

+


ddx

ddx

ddx

–

n

i = 1nΣ xi

n

i = 1Σ fi xi

n

i = 1Σ fi

n

n

i = 1Σ (xi – x)2–

–

12

n

i = 1Σ fi xi

2

a1 – r

n

i = 1Σ fi

- 3 -


SECTION I



1. (a) The functions f and g are defined by

f (x) = x2 + 5, x > 1

g (x) = 4x – 3, x ∈ R

where R is the set of real numbers.

Find the value of g (f (2)). (2 marks)

(b) The function h is defined by h(x) = 3x + 5x – 2 where x ∈ R, x ≠ 2.

Determine the inverse of h(x). (3 marks)

(c) Given that x – 2 is a factor of k(x) = 2x3 – 5x2 + x + 2, factorize k(x) completely. (3 marks)

(d) Solve the following equations:

(i) 16 x+2 = 14 (2 marks)

(ii) log3 (x + 2) + log3 (x – 1) = log3 (6x – 8) (4 marks)

Total 14 marks

- 4 -


2. (a) Given that f (x) = 3x2 – 9x + 4:

(i) Express f (x) in the form a (x + b)2 + c, where a, b and c are real numbers. (3 marks)

(ii) State the coordinates of the minimum point of f (x). (1 mark)

(b) The equation 3x2 – 6x – 4 = 0 has roots α and β. Find the value of 1α + 1

β . (4 marks)

(c) Determine the coordinates of the points of intersection of the curve

2x2 – y + 19 = 0 and the line y + 11x = 4. (3 marks)

(d) An employee of a company is offered an annual starting salary of $36 000 which increases by $2 400 per annum. Determine the annual salary that the employee should receive in the ninth year. (3 marks)

Total 14 marks

- 5 -


SECTION II



3. (a) The equation of a circle is given by x2 + y2 – 12x – 22y + 152 = 0.

(i) Determine the coordinates of the centre of the circle. (2 marks)

(ii) Find the length of the radius. (1 mark)

(iii) Determine the equation of the normal to the circle at the point (4, 10). (3 marks)

(b) The position vectors of two points, A and B, relative to an origin O, are such that OA = 3i – 2j and OB = 5i – 7j. Determine

(i) the unit vector AB (3 marks)

(ii) the acute angle AOB, in degrees, to one decimal place. (3 marks)

Total 12 marks

- 6 -


4. (a) The following diagram shows a circle of radius r = 4 cm, with centre O and sector AOB which subtends an angle, θ = π

6 radians at the centre.

If the area of the triangle AOB = 12 r2 sin θ, then calculate the area of the shaded region.

(4 marks)

(b) Solve the following equation, giving your answer correct to one decimal place.

8 sin2 θ = 5 – 10 cos θ, where 0o < θ < 360o (4 marks)

(c) Prove the identity

sin θ + sin 2θ

1 + cos θ + cos 2θ ≡ tan θ. (4 marks)

Total 12 marks

- 7 -


SECTION III



5. (a) Differentiate the following expression with respect to x, simplifying your answer.

(2x2 + 3) sin 5x (4 marks)

(b) (i) Find the coordinates of all the stationary points of the curve y = x3 – 5x2 + 3x + 1. (3 marks)

(ii) Determine the nature of EACH point in (i) above. (2 marks)

(c) A spherical balloon of volume V = 43 π r3 is being filled with air at the rate of 200 cm3 s–1.

Calculate, in terms of π, the rate at which the radius is increasing when the radius of the balloon is 10 cm. (5 marks)

Total 14 marks

- 8 -


6. (a) Evaluate ∫ 3 cos θ dθ. (4 marks)

(b) A curve has an equation which satisfies dydx = kx(x – 1) where k is a constant.

Given that the value of the gradient of the curve at the point (2, 3) is 14, determine

(i) the value of k (2 marks)

(ii) the equation of the curve. (4 marks)

(c) Calculate, in terms of π, the volume of the solid formed when the area enclosed by the curve y = x2 + 1 and the x-axis, from x = 0 to x = 1, is rotated through 360o about the x-axis.

(4 marks)

Total 14 marks

π

3

π

6

- 9 -


SECTION IV



7. (a) There are three traffic lights that a motorist must pass on the way to work. The probability that the motorist has to stop at the first traffic light is 0.2, and that for the second and third traffic lights are 0.5 and 0.8 respectively. Find the probability that the motorist has to stop at

(i) ONLY ONE one traffic light (4 marks)

(ii) AT LEAST TWO traffic lights. (4 marks)

(b) Use the data in the following table to estimate the mean of x.

x 5–9 10–14 15–19 20–24

f 8 4 10 3

(4 marks)

(c) Research in a town shows that if it rains on any one day then the probability that it will rain the following day is 25%. If it does not rain one day then the probability that it will rain the following day is 12%. Starting on a Monday and given that it rains on that Monday:

(i) Draw a probability tree diagram to illustrate the information, and show the probability on ALL of the branches. (4 marks)

(ii) Determine the probability that it will rain on the Wednesday of that week. (4 marks)

Total 20 marks

- 10 -

01254020/F 2015

8. (a) A particle moving in a straight line has a velocity of 3 m s at t = 0 and 4 seconds later its velocity is 9 m s .

(i) On the answer graph sheet, provided as an insert, draw a velocity–time graph to represent the motion of the particle. (3 marks)

(ii) Calculate the acceleration of the particle. (3 marks)

(iii) Determine the increase in displacement over the interval t = 0 to t = 4. (4 marks)

(b) A particle moves in a straight line so that t seconds after passing through a fixed point O, its acceleration, a, is given by a = (3t – 1)m s . The particle has a velocity, v, of 4 m s when t = 2 and its displacement, s, from O is 6 metres when t = 2. Find

(i) the velocity when t = 4 (5 marks)

(ii) the displacement of the particle from O when t = 3. (5 marks)

Total 20 marks

END OF TEST


–1

–2

–1

–1




EXAMINATION



ALTERNATIVE

90 minutes


1. This paper consists of ONE question. Answer the given question.





01254032/F 2015

08 JUNE 2015 (p.m.)

12

63

33

5038

- 2 -


LIST OF FORMULAE


Geometric Series Tn = arn–1 Sn = ———— S∞ = —— , –1 < r < 1 or |r| < 1





tan (A + B) ≡ ——————–




Statistics x = —— = ——— , S2 = ————— = ———— – (x)2



n2

a(rn – 1)r – 1

v|v|

^ a•b|a| × |b|

+


ddx

ddx

ddx

–

n

i = 1nΣ xi

n

i = 1Σ fi xi

n

i = 1Σ fi

n

n

i = 1Σ (xi – x)2–

–

12

n

i = 1Σ fi

n

i = 1Σ fi xi

2

a1 – r

- 3 -

01254032/F 2015

Answer this question.

1. (a) The daily profit, f (x), in hundreds of dollars, for a company that manufactures x radios daily is given by f (x) = –x2 + 40x – 300.

Determine the maximum daily profit of the company. (6 marks)

(b) The combined area of a square and a rectangle is 128 cm2. The width of the rectangle is 3 cm more than the length of a side of the square, and the length of the rectangle is 3 cm more than its width. Detemine the dimensions of the square and the rectangle.

(5 marks)

(c) A manufacturer of tanks makes a tank, with a rectangular base, as shown in the diagram below, where h is the height of the isosceles triangle and x the width of the rectangular base.

4

4 4

x

h

(i) Obtain an expression for the area A of the shaded section (top surface) as a function of the distance x between the parallel sides. (5 marks)

(ii) Hence, find the domain of A (that is, the range of values of x ). (4 marks)

Total 20 marks

END OF TEST





EXAMINATION



2 hours 40 minutes




3. Write your answers in the spaces provided in this booklet.

4. Do NOT write in the margins.


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

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


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



01254020/F 2016‘‘*’’Barcode Area”*”Sequential Bar Code

- 4 -


‘‘*’’Barcode Area”*”Sequential Bar Code

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LIST OF FORMULAE







tan (A + B) ≡ ——————–




Statistics x = —— = ——— , S2 = ————— = ———— – (x)2



n2

a(rn – 1)r – 1

v|v|

^ a•b|a| × |b|

+


ddx

ddx

ddx

–

n

i = 1nΣ xi

n

i = 1Σ fi xi

n

i = 1Σ fi

n

n

i = 1Σ (xi – x)2–

–

12

n

i = 1Σ fi

n

i = 1Σ fi xi

2

a1 – r

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SECTION I



1. (a) The domain for the function f (x) = 2x – 5 is {–2, –1, 0, 1}.

(i) Determine the range of the function.

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

................................................................................................................................. (2 marks)

(ii) Find f –1 (x).

.................................................................................................................................

.................................................................................................................................

................................................................................................................................. (1 mark)

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(iii) Sketch the graphs of f (x) and f –1 (x) on the same axes.

(2 marks)

(iv) Comment on the relationship between the two graphs.

................................................................................................................................. (1 mark)

(b) Solve the equation 22x + 1 + 5 (2x) – 3 = 0.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

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..............................................................................................................................................

..............................................................................................................................................

.............................................................................................................................................. (4 marks)

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(c) (i) Given that T = k p , make c the subject of the formula.

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

................................................................................................................................. (2 marks)

(ii) Solve the equation

log (x + 1) + log (x – 1) = 2 log (x + 2). .................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

.................................................................................................................................

................................................................................................................................. (2 marks)

Total 14 marks

hc—

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2. (a) (i) Determine the nature of the roots of the quadratic equation 2x2 + 3x – 9 = 0.

.................................................................................................................................

.................................................................................................................................

................................................................................................................................. (1 mark)

(ii) Given that f (x) = 2x2 + 3x – 9, sketch the graph of the quadratic function, clearly indicating the minimum value.

(5 marks)

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(b) Evaluate ∑ 3–n.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

.............................................................................................................................................. (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 2011, he earned $100, $115 and $130 respectively. Calculate the total dividends on his investment by the end of 2016.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

.............................................................................................................................................. (5 marks)

Total 14 marks

25

1

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SECTION II



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

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A – –

– –

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

respectively, where a = , and b = . P lies on AB such that PB = — AB.

Find the coordinates of OP.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

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.............................................................................................................................................. (4 marks)

Total 12 marks

1 –3 2 5

14

→ → →

→

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4. (a) The following diagram (not drawn to scale) shows two sectors, AOB and DOC. OB and OC are x cm and (x + 2) cm respectively and angle AOB = θ.

If θ = —– radians, calculate the area of the shaded region in terms of x.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

.............................................................................................................................................. (4 marks)

(b) Given that cos 30° = —– and sin 45° = —– , without the use of a calculator, evaluate

cos 105°, in surd form, giving your answer in the simplest terms.

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

..............................................................................................................................................

.............................................................................................................................................. (5 marks)

2π9

√ 22

√ 32

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(c) Prove that the identity ——–——– ≡ tan θ + tan α.

..............................................................................................................................................

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.............................................................................................................................................. (3 marks)

Total 12 marks

sin (θ + α)cos θ cos α

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SECTION III



5. (a) Find —– given that y = √ 5x2 – 4, simplifying your answer.

..............................................................................................................................................

..............................................................................................................................................

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..............................................................................................................................................

..............................................................................................................................................

.............................................................................................................................................. (4 marks)

(b) The point P (1, 8) lies on the curve with equation y = 2x (x + 1)2. Determine the equation of the normal to the curve at the point P.

..............................................................................................................................................

..............................................................................................................................................

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.............................................................................................................................................. (5 marks)

dydx

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(c) Obtain the equation for EACH of the two tangents drawn to the curve y = x2 at the points where y = 16.

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Total 14 marks

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6. (a) (i) Find ∫ (3 cos θ – 5 sin θ) dθ.

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................................................................................................................................. (3 marks)

(ii) Evaluate ∫3

1 — – 3 + 2x3 dx.

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2x2

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(b) The following figure shows the finite region R bounded by the lines x = 1, x = 2 and the arc of the curve y = (x – 2)2 + 5.

Calculate the area of the region R.

..............................................................................................................................................

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(c) The point P (1, 2) lies on the curve which has a gradient function given by —– = 3x2 – 6x. Find the equation of the curve.

..............................................................................................................................................

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.............................................................................................................................................. (3 marks)

Total 14 marks

dydx

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SECTION IV



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

15 16 18 18 20 21 22 22

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

.................................................................................................................................

................................................................................................................................. (1 mark)

(iii) Determine the mode.

................................................................................................................................. (1 mark)

(iv) Determine the median.

.................................................................................................................................

.................................................................................................................................

................................................................................................................................. (2 marks)

(v) Determine the interquartile range.

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(3 marks)

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(b) Two events, A and B, are such that P(A) = 0.5, P(B) = 0.8 and P(A ∪ B) = 0.9.

(i) Determine P(A ∩ B).

.................................................................................................................................

.................................................................................................................................

................................................................................................................................. (2 marks)

(ii) Determine P(A | B).

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................................................................................................................................. (2 marks)

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

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DO

NO

T W

RIT

E IN

TH

IS A

RE

A

DO

NO

T W

RIT

E IN

TH

IS A

RE

A

DO

NO

T W

RIT

E IN

TH

IS A

RE

A

(c) A bag contains 3 red balls, 4 black balls and 3 yellow balls. Three balls are drawn at random with replacement from the bag. Find the probability that the balls drawn are all of the same colour.

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.............................................................................................................................................. (4 marks)

Total 20 marks

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DO

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T W

RIT

E IN

TH

IS A

RE

A

DO

NO

T W

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E IN

TH

IS A

RE

A

DO

NO

T W

RIT

E IN

TH

IS A

RE

A

8. (a) A motorist starts from a point, X, and travels 100 m due North to a point, Y, at a constant speed of 5 m s–1. He stays at Y for 40 seconds and then travels at a constant speed of 10 m s–1 for 200 m due South to a point, Z.

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

Time (seconds)

Dis

plac

emen

t (m

etre

s)

(5 marks)

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DO

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TH

IS A

RE

A

DO

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E IN

TH

IS A

RE

A

DO

NO

T W

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E IN

TH

IS A

RE

A

(ii) Calculate the average speed for the whole journey.

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................................................................................................................................. (3 marks)

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

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DO

NO

T W

RIT

E IN

TH

IS A

RE

A

DO

NO

T W

RIT

E IN

TH

IS A

RE

A

DO

NO

T W

RIT

E IN

TH

IS A

RE

A

- 25 -


(b) A particle starting from rest, travels in a straight line with an acceleration, a, given by a = cos t where t is the time in seconds.

(i) Find the velocity of the particle in terms of t. .................................................................................................................................

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................................................................................................................................. (3 marks)

(ii) Calculate the displacement of the particle in the interval of time t = π to t = 2π.

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................................................................................................................................. (6 marks)

Total 20 marks

END OF TEST





EXAMINATION



ALTERNATIVE TO SCHOOL-BASED ASSESSMENT

1 hour 30 minutes


1. This paper consists of ONE compulsory question. Answer the given question.

2. Write your answers in the spaces provided in this booklet.

3. Do NOT write in the margins.


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

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




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IS A

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IS A

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DO

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E IN

TH

IS A

RE

A

LIST OF FORMULAE







tan (A + B) ≡ ——————–




Statistics x = —— = ——— , S2 = ————— = ———— – (x)2



n2

a(rn – 1)r – 1

v|v|

^ a•b|a| × |b|

+


ddx

ddx

ddx

–

n

i = 1nΣ xi

n

i = 1Σ fi xi

n

i = 1Σ fi

n

n

i = 1Σ (xi – x)2–

–

12

n

i = 1Σ fi

n

i = 1Σ fi xi

2

a1 – r

- 5 -



DO

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E IN

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IS A

RE

A

DO

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E IN

TH

IS A

RE

A

DO

NO

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E IN

TH

IS A

RE

A

A manufacturing plant wants to minimize the amount of material used to produce paint containers. The project team has the task of designing a container to hold 19 litres (5 gallons) of paint. The team suggests three designs, Design I, Design II and Design III.

(a) Express the volume of paint in m3.

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............................................................................................................................................................ (1 mark)

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E IN

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IS A

RE

A

DO

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E IN

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IS A

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DO

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E IN

TH

IS A

RE

A

Design I – A cylindrical container of height, h, and radius, r.

Surface area of a cylinder = 2 π r (h + r)

Take π = 3.142

(b) Using the required volume, formulate the appropriate equation for Design I to show that the minimum surface area is 0.394 m2.

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DO

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T W

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E IN

TH

IS A

RE

A

DO

NO

T W

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E IN

TH

IS A

RE

A

DO

NO

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E IN

TH

IS A

RE

A

Design II – A rectangular container of height, h, with a top and a base of area x2.

(c) Using the required volume, formulate the appropriate equation for Design II to determine the minimum surface area.

..........................................................................................................................................................

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............................................................................................................................................................ (7 marks)

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DO

NO

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E IN

TH

IS A

RE

A

DO

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E IN

TH

IS A

RE

A

DO

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IS A

RE

A

Design III – A hexagonal container of height, h, and radius, r.

Cross-sectional area of the hexagonal container

= — √ 3 r2

h

r

(d) Given that the equation for the surface area of Design III is

A = 5.20 r2 + ——–, determine the minimum surface area.

..........................................................................................................................................................

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............................................................................................................................................................ (4 marks)

32

0.044r

- 9 -


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IS A

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IS A

RE

A

01254032/F 2016

(e) Recommend to the project team the design which is BEST in order to minimise the amount of material used to produce the paint containers. Include the dimensions of the selected design.

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............................................................................................................................................................ (3 marks)

Total 20 marks

END OF TEST


Documents

CSEC® Additional Mathematics Past Papers · CSEC® Additional Maths Past Papers LIST OF CONTENTS Paper 02 (03 May 2012) 3 Paper 032 (12 June 2012) 11 Paper 02 (07 May 2013) 13 Paper