Mathematics Department

29 November 2017

Paper 2

Grade: 11 Time: 3 hours

Examiner: Mrs. F. Van der Merwe Marks: 150

Moderators: Mrs. S. Hickling, Mrs. M. Van Niekerk, Mr. K. Viljoen and Mrs. A. Heuer.

NAME: …………………………………………. TEACHER: …………………………………………….

Instructions:

1. The question paper consists of 35 pages, including the cover page, and formula sheet on the last page. Please check that your paper is complete.

2. A non-programmable calculator may be used. 3. Please round off answers to two decimal places, where necessary. 4. Answer all the questions. 5. Please answer all questions in pen. 6. All calculations must be clearly shown.

Q1 (15)

Q2 (25)

Q3 (14)

Q4 (9)

Q5 (8)

Q6 (4)

Q7 (11)

Q8 (15)

Q9 (6)

Q10 (12)

Q11 (9)

Q12 (11)

Q13 (6)

Q14 (5)

Tot: /150

Grade 11 Mathematics Paper 2 Page 2 of 35

SECTION A [75]

QUESTION 1 [15]

Learners were asked the time (to the nearest minute) that they usually take to get from home to school each morning. The results are shown in the table below:

a) Complete the cumulative frequency column in the table below. (2)

b) Draw the ogive of this data on the grid provided below. (4)

Time (in minutes)

Traveling time for learners from home to school

Grade 11 Mathematics Paper 2 Page 3 of 35

c) Calculate the estimated mean time that these learners take to get to school in

the morning. (2)

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d) Calculate the standard deviation of this data. (2)

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e) Estimate the Interquartile range for this data. (2)

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f) Use the ogive to determine approximately how many learners would take more

than one standard deviation from the mean time to get to school. (3)

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Grade 11 Mathematics Paper 2 Page 4 of 35

QUESTION 2 [25]

Simplify the following expressions and show ALL the calculations without using a calculator:

a) If 5 tan A − 4 = 0 with 180° < A < 360°, calculate the value of

41 cos2 A − 5 tan A without the use of a calculator. (4)

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Grade 11 Mathematics Paper 2 Page 5 of 35

b) If sin 32∘ = k, determine, with the aid of a diagram, the value of the following in

terms of k.

1) cos 58∘ (2)

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2) sin(180∘ + 32∘) (2)

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3) sin 418∘ (2)

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Grade 11 Mathematics Paper 2 Page 6 of 35

c) Simplify the following expressions WITHOUT the use of a calculator. Show

ALL calculations.

1) sin 63∘

cos 27∘ (2)

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2) cos2 135∘

sin 240° .tan 150° (6)

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Grade 11 Mathematics Paper 2 Page 7 of 35

d) tan(180°+ 𝑥) .cos(360°− 𝑥)

sin(180°−𝑥).cos(90° + 𝑥)−cos2 𝑥 (7)

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Grade 11 Mathematics Paper 2 Page 8 of 35

QUESTION 3 [14]

In the figure, PQR is a triangle. The coordinates of P are (1; −1). The equations

of QR and PR are 𝑦 =1

3𝑥 + 2 and 𝑥 − 𝑦 − 2 = 0 respectively.

a) Show that the coordinates of Q are (0; 2). (2)

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b) Prove that PQ̂R = 90∘. (3)

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𝑦

𝑥

Q

R

P(1; −1)

𝜃

Grade 11 Mathematics Paper 2 Page 9 of 35

c) Find the coordinates of R. (4)

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d) Determine the coordinates of T, if PQRT is a parallelogram. (2)

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e) Determine the size of 𝜃. (3)

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Grade 11 Mathematics Paper 2 Page 10 of 35

QUESTION 4 [9]

In the diagram below, the graphs of 𝑓(𝑥) = cos(𝑥 + 𝑝) and 𝑔(𝑥) = 𝑞 sin 𝑥 are

shown for the interval −180° ≤ 𝑥 ≤ 180°.

a) Determine the values of 𝑝 and 𝑞. (2)

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b) The graphs intersect at A (−22,5° ; 0,38) and B. Determine the coordinates

of B. (2)

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Grade 11 Mathematics Paper 2 Page 11 of 35

c) Determine the value(s) of 𝑥 in the interval −180° ≤ 𝑥 ≤ 0° for which

𝑓(𝑥) < 𝑔(𝑥). (3)

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d) The graph of 𝑓 is shifted 30° to the left to obtain a new graph ℎ.

1) Write down the equation of ℎ in its simplest form. (1)

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2) Write down the value of 𝑥 for which ℎ has a minimum in the interval

−180° ≤ 𝑥 ≤ 180°. (1)

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Grade 11 Mathematics Paper 2 Page 12 of 35

QUESTION 5 [8]

a) Complete the following statements:

1) The angle subtended at the circle by a diameter is ________________ (1)

2) The angle between the tangent to a circle and a chord drawn from the

point of contact is equal to ___________________________________

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b) In the diagram below, O is the centre of the circle. BD is the diameter of the circle. GEH is a tangent to the circle at E. F and C are two points on the circle and FB, FE, BC, CD and BE are drawn.

Ê1 = 32° and Ê3 = 56

°

Calculate, with reasons, the values of:

1) B̂2 (1)

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Grade 11 Mathematics Paper 2 Page 13 of 35

2) Ê2 (2)

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3) D̂ (1)

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4) Ĉ (1)

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5) F̂ (1)

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Grade 11 Mathematics Paper 2 Page 14 of 35

QUESTION 6 [4]

In the diagram, O is the centre of the circle. A, B, C and D are points on the

circumference of the circle. Diameter BD bisects chord AC at E. Chords AB, CD

and AD are drawn. Ĉ = 43°.

The length of the diameter of the circle is 28 units. Calculate the length of AB.

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Grade 11 Mathematics Paper 2 Page 15 of 35

Section A

Extra lines

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Grade 11 Mathematics Paper 2 Page 16 of 35

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Grade 11 Mathematics Paper 2 Page 17 of 35

SECTION B [75]

QUESTION 7 [11]

a) Consider the following data written in ascending order and consisting of seven integer values with the middle three values missing. The mean is 12 and the median is 11.

3 ; 4 ; _____ ; _____ ; _____ ; 17 ; 26 1) If the smallest two scores are both increased by one and the largest

two scores are both reduced by one, state the effect on:

i) The standard deviation. (1)

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ii) The mean. (1)

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2) Determine an eighth data point which, when included with the original data would produce the boxplot as shown. (2)

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3 10,5 26

Grade 11 Mathematics Paper 2 Page 18 of 35

b) There are 100 students in a grade. A teacher wanted to have an understanding of the weights of the students in the grade. She took a random sample of 20 students and recorded their weights. The cumulative frequency curve below summarises the data.

1) How many students in the grade would you expect to weigh less than

70kg?

(1)

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2) If a student was selected at random from the group, what is the probability that he would weigh more than 80kg? (2)

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Weight (kg)

50 60 70 80 90 100

20

15

10

5

0

Grade 11 Mathematics Paper 2 Page 19 of 35

3) Calculate an estimate for the mean weight of the group of students.

All calculations must be shown. Use class intervals of 50 ≤ 𝑥 < 60 ;

60 ≤ 𝑥 < 70 ; 70 ≤ 𝑥 < 80 ; . . . . etc. (4)

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Grade 11 Mathematics Paper 2 Page 20 of 35

QUESTION 8 [15]

a) Given:

8

sin2 𝐴 −

4

1 + cos 𝐴 =

4

1 − cos 𝐴

1) Prove the following identity. (5)

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Grade 11 Mathematics Paper 2 Page 21 of 35

2) Determine the value(s) of A for which the identity is undefined for

0° ≤ 𝐴 ≤ 360°. (5)

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Grade 11 Mathematics Paper 2 Page 22 of 35

b) Find the general solution of 8 cos2 𝑥 − 2 cos 𝑥 = 1. (5)

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Grade 11 Mathematics Paper 2 Page 23 of 35

QUESTION 9 [6]

E is the apex of a pyramid having a square base ABCD. O is the centre of the base. EB̂A = 𝜃, AB = 3𝑚 and EO, the perpendicular height of the pyramid, is 𝑥.

If it is given that cos 𝜃 =3

2√𝑥2 + 92

and that the volume of the pyramid is

15m3, calculate the value of 𝜃. ________________________________________________________________

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𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 =1

3(𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒) × (⊥ ℎ𝑒𝑖𝑔ℎ𝑡)

Grade 11 Mathematics Paper 2 Page 24 of 35

QUESTION 10 [12] Refer to the sketch below:

D is the midpoint of AC. BC = √20 and A(k; 7), C(4; −1), and D(1; p) are given. DQ̂F = 38,66°.

a) Determine the values of p and k. (2)

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𝑦

𝑥

A(k ; 7)

D(1; p)

C(4; −1)

B

Q 38,66° F

Grade 11 Mathematics Paper 2 Page 25 of 35

b) Determine the coordinates of B. (4)

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c) Determine the coordinates of Q. (6)

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Grade 11 Mathematics Paper 2 Page 26 of 35

QUESTION 11 [9]

In ∆FGH, I is a point on FH. GÎH = 𝑎 , FĜI = 𝑏 , GH = 𝑓 and FG = ℎ.

Show that:

a) sin H = ℎ sin(𝑎−𝑏)

𝑓 (3)

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1 2

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b) GI = ℎ 𝑠𝑖𝑛(𝑎−𝑏)

𝑠𝑖𝑛 𝑎 (3)

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c) Area ∆FGI = ℎ2 𝑠𝑖𝑛(𝑎−𝑏) 𝑠𝑖𝑛 𝑏

2 𝑠𝑖𝑛 𝑎 (3)

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Grade 11 Mathematics Paper 2 Page 28 of 35

QUESTION 12 [11]

In the diagram below, ED is a diameter of the circle with centre O. ED is produced to C and CA is a tangent to the circle at B. AO intersects BE at F. BD//AO. Let Ê = 𝑥.

a) Write down, with reasons, THREE other angles each equal to 𝑥. (3)

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Grade 11 Mathematics Paper 2 Page 29 of 35

b) Determine, with reasons, CB̂E in terms of 𝑥. (2)

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c) Prove that F is the midpoint of BE. (2)

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d) Calculate the length of the diameter if it is further given that EB = 8cm and OF = 3cm. (4)

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Grade 11 Mathematics Paper 2 Page 30 of 35

QUESTION 13 [6]

In the diagram, O is the centre of the circle with diameter ST produced to U. XU is a tangent to the circle at point W and chord SW is produced to V. VU ⊥ US.

a) Prove that WVUT is a cyclic quadrilateral. (3)

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𝟐 𝟑

𝟐

𝟑 𝟒

Grade 11 Mathematics Paper 2 Page 31 of 35

b) Prove that VU is a tangent to the circle passing through V, T and S. (3)

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Grade 11 Mathematics Paper 2 Page 32 of 35

QUESTION 14 [5]

Refer to the diagram below. O is the centre of the circle ABDC. If BÂC = 60° and CB̂D = 20°, determine OD̂B.

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Grade 11 Mathematics Paper 2 Page 33 of 35

Section B

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Grade 11 Mathematics Paper 2 Page 34 of 35

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Grade 11 Mathematics Paper 2 Page 35 of 35