Gr 11 P2 Nov 2014 Page 1

ST BENEDICT’S COLLEGE

SUBJECT Mathematics DATE 12 November 2014 GRADE 11 MARKS 150

EXAMINER Mrs Sillman MODERATORS Gr 11 Teachers NAME

DURATION 3 hrs

CLASS

TEACHER

QUESTION NO. ASS STANDARD

DESCRIPTION

TOTAL ACTUAL

1 - 3 Data Handling 22

4 - 5 Analytical Geometry 23

6 Analytical Geometry 10

7 Trigonometry 31

8 - 9 Trigonometry 18

10 Euclidean Geometry 4

11 -14 Euclidean Geometry 30

15 Measurement 12

150

Gr 11 P2 Nov 2014 Page 2

INSTRUCTIONS

1. This question paper consists of 21 pages (15 questions). Please check that your paper is complete.

2. Read the questions carefully.

3. Answer all the questions.

4. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.

5. Round off your answers to one decimal digit unless otherwise stated.

6. All the necessary working details must be clearly shown. Answers only, without relevant calculations, may incur penalties.

7. It is in your own interest to write legibly and to present your work neatly.

Formulae

𝑑 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

𝑀 (𝑥1 + 𝑥2

2;𝑥1 + 𝑥2

2)

𝑦 = 𝑚𝑥 + 𝑐

𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

𝑚 =𝑦2 − 𝑦1

𝑥2 − 𝑥1

𝑚 = 𝑡𝑎𝑛𝜃

In ΔABC: 𝑎

𝑠𝑖𝑛𝐴=

𝑏

𝑠𝑖𝑛𝐵=

𝑐

𝑠𝑖𝑛𝐶

𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐. 𝑐𝑜𝑠𝐴

𝐴𝑟𝑒𝑎 ∆𝐴𝐵𝐶 =1

2𝑎𝑏. 𝑠𝑖𝑛𝐶

Gr 11 P2 Nov 2014 Page 3

QUESTION 1

The following data represents the number of goals scored by 12 premier league soccer

players:

11 12 14 8 9 12 18 15 16 4 10 14

a) Determine the standard deviation of the data (correct to 2 dec places) (2) b) What percentage of players scored within one std deviation of the mean?

(rounded to the nearest whole) (4)

c) The table below represents how many games were played in the season by each of the

goal scorers.

No of goals (x) 11 12 14 8 9 12 18 15 16 4 10 14

No of games (y) 40 38 38 36 38 40 40 39 36 28 32 36

Find the: (i) line of best fit (regression line) (ii) the correlation coefficient of the data (3)

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Gr 11 P2 Nov 2014 Page 4

QUESTION 2

Consider the box-and-whisker plot below:

a) Write down the five-number summary and say what each number represents. (5) b) Comment on the skewness of the data. (2)

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Gr 11 P2 Nov 2014 Page 5

QUESTION 3

The histogram below shows the distribution of Mathematics results for 200 Gr 11’s.

a) Calculate the estimated mean. (3) b) Identify the modal class (1) c) In which class does the median lie? (2)

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Gr 11 P2 Nov 2014 Page 6

QUESTION 4

ABCD is a quadrilateral with vertices A ( -2 ; 3 ) B ( p ; q ) C ( 11 ; 4 ) and D ( 4 ; 0 )

a) Calculate the gradient of AC. (2) b) Determine the angle of inclination of AC, correct to one decimal place. (2) c) Calculate the length of AC correct to 2 decimal places. (3) d) Determine the equation of AC. (3) e) If AC and BD bisect each other, find the values of p and q. (5)

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Gr 11 P2 Nov 2014 Page 7

QUESTION 5

K ( -3 ; 5 ) L ( t ; 2 ) and M ( 1 ; -5 ) are points in the Cartesian plane. Calculate t if:

a) K, L and M are collinear. (4) b) KM KL (4)

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Gr 11 P2 Nov 2014 Page 8

QUESTION 6

P ( 7 ; 6 ) R ( 10 ; 3 ) and N ( -2 ; 3 ) are the vertices of ΔPRN with EDPN. D is the

midpoint of PN.

a) Determine the co-ordinates of point D (2) b) Calculate the co-ordinates of K if N is the midpoint of PK. (4) c) For which values of ‘r’ will the straight line with equation 𝑦𝑟 − 2𝑟 = 7𝑥 be:

(i) parallel to PN (ii) perpendicular to PN

(4)

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Gr 11 P2 Nov 2014 Page 9

QUESTION 7

a) Given 5 cos 𝑥 + 4 = 0, where 𝑥 is an obtuse angle, use a diagram and without a

calculator, determine the value of 5 sin 𝑥 + 3 tan 𝑥

(5)

b) Simplify without a calculator:

cos(900 + 𝑥). tan(2250)

cos(1800 − 𝑥) . tan(𝑥 − 3600). 𝑐𝑜𝑠2400

(5)

Gr 11 P2 Nov 2014 Page 10

c) If 5𝑠𝑖𝑛𝜃 = 1,835 determine the value of cos (3𝜃 − 750) correct to 2 decimal

places. ( < 900)

(4)

d) Determine the general solution for:

𝑡𝑎𝑛(𝑥 − 30°) = 1,812

(4)

Gr 11 P2 Nov 2014 Page 11

e) Prove that 𝑠𝑖𝑛𝜃

1 − 𝑐𝑜𝑠𝜃−

𝑐𝑜𝑠𝜃

𝑠𝑖𝑛𝜃=

1

𝑠𝑖𝑛𝜃

(4)

f) Determine the value(s) of 𝑥 for 𝑥 ∈ [1800 ; 3600 ] if 6 cos 𝑥 − 5 = 4

cos 𝑥

(9)

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Gr 11 P2 Nov 2014 Page 12

QUESTION 8

A construction company is commissioned to build a bridge over a river. The width of the river

is BR. A surveyor (standing at point P) takes measurements and draws the following

diagrammatic representation:

PQ = 65m BQ = 55m 𝐵�̂�𝑅 = 12,10 𝐵�̂�𝑃 = 500

a) Calculate the length of BP (3)

b) Hence, calculate the length of the bridge so that it extends 55m beyond the bank on both sides (starting from point Q)

(7)

NOTE: Space for this answer continues on the next page

Q

P

B

R

Gr 11 P2 Nov 2014 Page 13

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Gr 11 P2 Nov 2014 Page 14

QUESTION 9

The sketch below contains the graphs of 𝑔(𝑥) = cos(𝑥

2) 𝑎𝑛𝑑 ℎ(𝑥) = sin 𝑥 for

𝑥 ∈ [−1800 ; 1800]. The co-ordinates of A, a point of intersection of the two graphs is

(600; √3

2)

a) Use your graph to solve the following equation:

cos(𝑥

2) = sin 𝑥 for 𝑥 ∈ [−1800 ; 1800].

(3)

b) What is the period and amplitude of g(𝑥)? (2) c) If the equation of ‘h’ is changed to ℎ(𝑥) = 2 𝑠𝑖𝑛𝑥 + 1, explain clearly how the graph

of ‘h’ will change. You must refer to range, amplitude and period. (3)

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Gr 11 P2 Nov 2014 Page 15

QUESTION 10

In quadrilateral BLOG if 𝑥 + 𝑦 = 1800. Then BLOG would be a cyclic quadrilateral.

Give two more conditions for a quadrilateral to be cyclic. Illustrate each of your answers with

a suitable sketch for explanation.

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Gr 11 P2 Nov 2014 Page 16

QUESTION 11

Using the diagram below, prove the theorem that states, “The angle between the tangent to

a circle and the chord drawn from the point of contact is equal to the angle in the alternate

segment”

Given:

RTP:

Proof:

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Gr 11 P2 Nov 2014 Page 17

QUESTION 12

In the diagram below, AE is a tangent to the circle. B and Q are points on the circle and

BQ is produced to P. PE//QA

a) Prove that 𝐴1̂ = �̂� (3)

b) What can you conclude about ABPE in light of what was proved in (a). Give a reason for your answer

(2)

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Gr 11 P2 Nov 2014 Page 18

QUESTION 13

a) In the diagram below, AOC is the diameter of circle with centre O.

𝐴1̂ = 500

Determine, with reasons, the sizes of �̂�2 and 𝐴�̂�𝐵

(3)

Gr 11 P2 Nov 2014 Page 19

b) In the diagram below, two circles intersect at A and B. Chord AC of the smaller circle is a tangent to the larger circle at A. DB and BC are both cards. Chord AD of the larger circle intersects the smaller circle at E. EB is produced to F. AB and CE are drawn. AC = EC.

Let 𝐴1̂ = 𝑦 𝑎𝑛𝑑 𝐴2̂ = 𝑥

(i) Name one other angle equal to 𝑥, with a reason.

(ii) Express 𝐸1̂ in terms of 𝑥 𝑎𝑛𝑑 𝑦. (iii) Show that BC bisects 𝐴�̂�𝐹

(2) (3) (4)

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Gr 11 P2 Nov 2014 Page 20

QUESTION 14

In the diagram O is the centre of the circle. X, Y, L and M are points on the circumference.

OY//ML and 𝑌�̂�𝐿 = 400. Calculate, with reasons, the size of 𝑌�̂�𝑀.

(Hint: a construction will be necessary)

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Gr 11 P2 Nov 2014 Page 21

QUESTION 15

A solid cylinder with a height of 10cm is cut in half as shown in the diagram. A triangular

prism is attached to it. The end of the prism is an isosceles triangle with a height of 3cm.

a) Calculate the radius of the cylinder. (2) b) Calculate the volume of the shape. (6) c) If the dimensions of the shape are doubled, by what factor will:

(i) The surface area increase? (ii) The volume increase?

(4)

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