BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS ... - Prelim … CAPS 2020 Prelim... · BEAULIEU COLLEGE Sept 2020 Mathematics Prelim Paper 2 Page 10 of 32 b) The graph of f x a bx sin

BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS 2020

MATHEMATICS GRADE 12 PAPER 2

Examiner: Mr J Ruiz–Mesa Moderator: Ms A Smith Total marks: 150 Time: 3hrs Date: 14 September 2020

Section A Section B

Question Topic Mark Allocation

Question Topic Mark Allocation

1 Analytical Geometry 15 6 Trigonometry 12

2 Analytical Geometry 21 7 Trigonometry 18

3 Trigonometry 8 8 Euclidean Geometry 22

4 Statistics 21 9 Euclidean Geometry 10

5 Euclidean Geometry 12 10 Euclidean

Geometry 5

Total 77 11 Trigonometry 6

Total 73

Instructions:

This question paper consists of 32 pages and an Information Sheet of 2 pages (i–ii). Please check that your paper is complete.

Answer all the questions on the question paper. Clearly show ALL the calculations, diagrams, graphs, et cetera you have used in determining

the answers. An approved calculator (non-programmable and non-graphical) may be used, unless stated

otherwise. If necessary, answers should be rounded off to ONE decimal place, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. Unless otherwise stated, you are required to provide reason for your statements in the

Euclidean Geometry section. It is in your own interest to write legibly and to present the work neatly. Please ensure your calculator is in DEGREE mode.

Candidate Number:

2 0 1 0 1 2 0 2 0

BEAULIEU COLLEGE Sept 2020 Mathematics Prelim Paper 2

Page 2 of 32

Question Analytical

Geometry Trigonometry

& Measurement

Statistics Euclidean Geometry

1 /15

2 /21

3 /8

4 /21

5 /12

6 /12

7 /18

8 /22

9 /10

10 /5

11 /6

Total /36 /44 /21 /49

TOTAL /150


Page 3 of 32

SECTION A

Question 1

In the diagram below, A(1 ; 4), B(-3 ; 1) and D(5 ; -2) are the coordinates of the vertices of ABD.

BD and AD intersect the x-axis at E and F respectively. The angle of inclination of BD is . The angle of inclination of AD is .

a) Calculate the gradient of AD. (2)


Page 4 of 32

b) Determine the length of the line segment AD (Leave answer in surd form if necessary). (2)

c) Determine the coordinates of M, the midpoint of AD. (2)

d) C is a point such that BC || AD. Determine the equation of the line BC in the formax by c . (3)


Page 5 of 32

e) Determine the size of . (2)

f) Hence, or otherwise, determine the size of ˆADB . (4)

[15]


Page 6 of 32

Question 2

a) In the diagram below, ABC is shown with vertices A(3 ; 5), B(7 ; 1) and C(1 ; -1). The

perpendicular bisectors of AB and BC intersect at 7 3;2 2

X

. R is the midpoint of AC.

1) Determine the equation of the perpendicular bisector of AC. (4)

2) Show that the perpendicular bisector of AC passes through X. (2)


Page 7 of 32

3) Hence, or otherwise, write down the equation of the circle through A, B and C. (4)

b) In the diagram below, circle centre A(7 ; 5) is shown. The line with equation 2 1y x , is a

tangent to the circle at P.

1) Show that the equation of line PA is 2 17y x . (3)


Page 8 of 32

2) Determine the equation of the circle centre A. (4)

3) The line with equation 2 , 1y x k k is also a tangent to the circle. Determine the

value of k. (4)

[21]


Page 9 of 32

Question 3

a) The graph of 4cos 3y x intersects the x-axis at P and Q.

Determine the coordinates of P and Q. (3)


Page 10 of 32

b) The graph of sin 4f x a bx is drawn for the interval 180 180o ox .

1) Determine the value of a and b. (2)

2) Using the graph of f, write down the x-coordinate of a point where the equation

0f x has a solution. (1)

3) The equation sin 4a bx k has four solutions in the interval 180 180o ox .

Determine the range of possible values for k. (2)

[8]


Page 11 of 32

Question 4

a) The diagram below is a cumulative frequency graph (ogive), showing the examination marks for a group of 200 students. The pass mark for the examination was 30.

1) What percentage of the students passed? (3)

2) A box and whisker plot for the examination is shown below.

From the graph, estimate the values of p, m and q. (3)


Page 12 of 32

3) A group of 10 pupils, all of whom scored marks above 60% is to be added to the dataset. Explain how the mean and median will change with the new marks added. (2)

b) In the grid below a, b, c, d, e, f and g represent values in a data set in an increasing order.

No value in the data set is repeated.

a b c d e f g The following information about the dataset is provided:

The maximum is 42. The range is 35. The median is 23 The difference between the median and the upper quartile is 14. The interquartile range is 22. 2e c . The mean is 25.

Determine the value of a, b, c, d, e, f and g. (7)


Page 13 of 32

c) The table below shows the data relating to the percentage of lessons attended by students during Term 2, 2020 and the corresponding mark for each student in the exam for that subject.

Lessons Attended (%) 70 59 85 93 78 85 84 69 70 82

Exam Mark (%) 80 62 89 98 84 91 83 72 75 85

1) Calculate the equation of the least squares regression line in the form y A Bx .

Round off your values to two decimal places. (4)

2) What will the estimated mark be for a student who attended all lessons during the terms? Comment on the reliability of your answer. (2)

[21]


Page 14 of 32

Question 5

a) Use the diagram below to prove the theorem that states: “A line drawn from the centre of the circle perpendicular to a chord, bisects the chord”

(6)


Page 15 of 32

b) In the diagram below, O is the centre of the circle. CF is perpendicular to EG. DF = 4DC and CD = 3 cm. Determine the length of EG. (6)

[12]

Total Section A: 77 Marks


Page 16 of 32

SECTION B

Question 6

a) In the diagram below, ABC is an isosceles triangle with AB = BC, AC = b and BC = a.

Prove that 2

2cos 12bBa

. (4)

b) A compost vendor, Antonio, was given the following dimensions over the telephone: “The triangular garden plot ABC has ˆCAB 44o , side AC is 8 m long and side BC is 6 m long. The garden requires a compost depth of 10 cm.” 1) Explain, showing your working, why Antonio needs more information from his client. (3)


Page 17 of 32

2) What is the maximum volume of compost needed, in m3, if the client is unable to supply the necessary information? (5)

[12]

Question 7

Please do not use a calculator to answer this question. All necessary working needs to be shown.

a) Determine the value of the expression below: sin34 cos10 cos34 sin10

sin12 cos12

o o o o

o o

(4)


Page 18 of 32

b) Given 1 sin2 cos sincos2 cos sin

x x xx x x

1) Prove the identity: (5)

2) Hence, or otherwise, calculate the following without the use of a calculator, giving your answer in its simplest surd form: cos15 sin15cos15 sin15

o o

o o

(3)


Page 19 of 32

c) Determine the general solution of: 21 4sin 5sin cos2 0x x x (6)

[18]


Page 20 of 32

Question 8

a) In the diagram below, AM is the diameter of the bigger circle AMP. RS is a common tangent to both circles at P. APB and MPN are straight lines.

1) State, no reasons required, the size of 1P (angle marked as P). (1)

2) Hence, show that BN is the diameter of the smaller circle. (2)


Page 21 of 32

3) If M = 70o, calculate the size of each of the following angles:

i. A (1)

ii. 6P (1)

iii. B (2)


Page 22 of 32

b) In the diagram below, O is the centre of the circle. AB is a diameter of the circle. CF and EB are chords which meet at D when extended. Chord EC is parallel to chord BF. CO and AC are joined.

1) Determine, in terms of x, the size of 1F . (2)

2) Prove that DF = BD. (4)


Page 23 of 32

3) Show that 1 3ˆ ˆC C (5)

4) If DF = 5 cm and OA = 6 cm. Determine the ratio, Area BFD : Area AOC. (4)

[22]


Page 24 of 32

Question 9

In the diagram, PSW and WT are tangents to circle RST at S and T respectively. PT is drawn and intersects the circle at R. RS and ST are joined. RT = TS. 1S x and 2T y .

a) Name, with reasons, THREE angles equal to y. (4)

b) Prove that PRS ||| PST (3)

x

y


Page 25 of 32

c) Hence, or otherwise, prove that PS RT = RS PT. (3)

[10]


Page 26 of 32

Question 10

In the diagram below, ABC has D on AB, F on BC and E on AC such that DE || BC and EF || AB.

CF : FE = 3 : 2 and CF : EC = 6 : 5. AD = 12 cm. BF = k.

Determine the value of k. (5)

[5]


Page 27 of 32

Question 11

In the diagram below, A and B are two hot air balloons. The angle of elevation of balloon A from C is 20o and the angle of elevation of balloon B from D is 25o. Points C and D are 2km apart.

Determine how many metres higher balloon A is than balloon B with respect to the ground. (6)

[6]

Total Section B: 73 Marks


Page 28 of 32

BLANK PAGE This page has been left blank intentionally for extra work


Page 29 of 32


Page 30 of 32


Page 31 of 32


Page 32 of 32

Documents

BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS ... - Prelim … CAPS 2020 Prelim... · BEAULIEU COLLEGE Sept 2020 Mathematics Prelim Paper 2 Page 10 of 32 b) The graph of f x a bx sin