grade 12 core mathematics: paper ii september 2019 time: 3 ...maths.stithian.com/new caps 2019...
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St. Anne’s Diocesan College Grade 12 Core Mathematics: Paper II September 2019 Time: 3 hours Marks: 150
Please read the following instructions carefully: 1. This question paper consists of 23 printed pages and an Information sheet. Please check that
your question paper is complete.
2. Write your examination number in the space provided on this question paper.
3. Answer all the questions on the question paper and hand this in at the end of the examination.
4. All necessary working details must be clearly shown. Give reasons unless otherwise stated.
5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.
6. Round off answers to 1 decimal digit where necessary, unless otherwise stated.
7. Ensure that your calculator is in DEGREE mode.
8. Diagrams are not drawn to scale.
9. It is in your own interest to present your work neatly.
10. The last page can be used for additional working, if necessary. If this space is used, make sure that you indicate clearly which question is being answered.
EXAMINATION NUMBER:
SECTION A SECTION B
Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mark
Total 9 13 19 7 14 15 9 11 19 9 5 10 6 4
Total
Percentage
150 100
O
𝑥
𝑦
θ
QUESTION 1 Page 2 of 23
In the sketch below, the vertices of ∆ PQR are P(−8; 0), Q (−4 ; 6) and R(5; 0). S is a point on the 𝑥-axis. (a) Determine the gradient of PQ. (1) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ (b) Prove that PQR = 90°. (2) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ (c) Calculate the size of θ , rounded to one decimal place. (2) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ (d) Calculate the gradient of QS if it is given that QS bisects PQR. (4) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ [9]
P(−8; 0)
Q(−4 ; 6)
R(5; 0) S
𝑦
𝑥 O C
A
B
QUESTION 2 Page 3 of 23
In the diagram below, a circle with the equation 𝑥2 − 6𝑥 + 𝑦2 − 4𝑦 = 12, cuts the 𝑦-axis at B. Line BC is a tangent to the circle. C lies on the 𝑥-axis.
(a) Determine the co-ordinates of A, the centre of the circle. (3)
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
_______________________________________________________________________________________________________ ________________________________________________________________________________________________________ (b) Show that B is the point (0; 6). (2) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________ (c) Determine the equation of tangent BC. (3) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________
A
B
C
O 𝑥
𝑦
A
B D
E
𝑥
𝑦
C
D
A
B
C
𝑥
𝑦
Page 4 of 23
(d) the coordinates of D if ABCD is a parallelogram. (5)
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ [13]
QUESTION 3
(a) In each of the following diagrams, O is the circle centre and 𝑥 = 80°. Write down the value of 𝑦. (3) (i)
_______________________________________________ (ii) _______________________________________________
(iii)
________________________________________________
Page 5 of 23
(b) Use the diagram below to prove the theorem that states: “ The opposite angles of a cyclic quadrilateral are supplementary” (4)
Required to prove: P + R = 180° Construction: _______________________________________________________________________________________ Proof: ________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________
R
S
P
Q
R
P Q
T S
U
O •
d
b c
e
a
Page 6 of 23
(c) Refer to the diagram: RQ is a tangent to circle QTSUP with centre O. SOQ and PT are straight lines. PTS = 70° and SQT = 20°. Find, with reasons, the size of the letters marked (a) to (e). Fill your answers in the table below (solve alphabetically).
Angle Answer Reason
a
b
c
d
e
(5)
70°
20°
Page 7 of 23
(d) In the diagram below, O is the centre of the circle. Tangent RQ is produced to P such that RPS = 90°. SO is produced to meet the tangent at R. OR = 9 units and OQ = 3 units.
(i) Prove that QO || PS. (2)
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________
(ii) Calculate, giving reasons, the lengths of the following leaving answers in simplest surd form where necessary.
1. RQ (2)
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
2. QP (3) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ [19]
O Q
P S
3 units
9 units
1
2 2
1
R
QUESTION 4 Page 8 of 23
In the figure below, M is the centre of the circle and SM is perpendicular to PQ. PR and SM intersect at T and ST = RS.
(a) Prove that QMTR is a cyclic quadrilateral. (3) ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________ (b) Prove that RS is a tangent to the circle at R. (4) ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ [7]
P
M
Q
S
R
1 2
T 1
2 3
QUESTION 5 Page 9 of 23
Answer the following question without the use of a calculator. (a) If sin α = 6 cos 𝛼, determine the value of:
(i) tan α (1) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(ii) sin( α−45°)
cos α (4)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ ______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(b) If tan θ = −√3 and 0° ≤ θ ≤ 180°, determine the value of: cos( θ + 30°). (4) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
Page 10 of 23
(c) If 𝑝 = sin 25°, express each of the following in terms of p:
(i) cos 50° (3)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(ii) sin 205° (2)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
[14]
Page 11 of 23
QUESTION 6
The following frequency table shows the distribution of marks of 200 students writing the NBT exam out of 60.
NBT Mark Frequency Cumulative Frequency
0 ≤ 𝑥 ≤ 10 20
10 < 𝑥 ≤ 20 40
20 < 𝑥 ≤ 30 60
30 < 𝑥 ≤ 40 50
40 < 𝑥 ≤ 50 20
50 < 𝑥 ≤ 60 10
(a) Complete the cumulative frequency table in the space provided. (1) (b) Draw the cumulative frequency ogive on the grid below. (3)
(c) Use your graph to estimate the interquartile range. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(d) The top 40% of the students won’t need to rewrite the test. Page 12 of 23
Use the graph to determine the cut-off mark. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (e) Calculate the estimated mean of the data. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (f) It is given that the standard deviation is 12,9. The marks for the NBT test are too low and are all raised by 5 marks. Write down the new estimated mean and new standard deviation for the new set of marks. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (g) Below are two box and whisker diagrams representing the marks of 200 students each from two different Universities, for the same NBT test out of 60.
(i) What percentage of University B’s results were above 55 out of 60? (1)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (ii) University A claim that their overall results are better that those of University B.
Is this true? Refer to the diagrams to justify your answer. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ [15]
Total marks for Section A : 77
University A
University B
Marks
P
M N
O
80cm
50cm
1
1 1
2 2
SECTION B Page 13 of 23
QUESTION 7 O is the centre of the circle and P, M and N lie on the circle. MN = 80cm. The radius of the circle is 50 cm. PM = PN. Determine: (a) The size of O1, rounded to the nearest degree. ________________________________________________________________________________________________________ (3) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ (b) The length of PM, rounded to one decimal digit. (4) ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ (c) The area of ∆PNM. (2)
________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ [9]
QUESTION 8 Page 14 of 23
The diagram below is a sketch of 𝑓(𝑥) = cos(𝑥 + 45°) for 𝑥 ∈ [0°; 180°].
(a) On the axes above, sketch the graph 𝑔(𝑥) = sin 3𝑥 for 𝑥 ∈ [0°; 180°].
Label all turning points. (3)
(b) Determine the value(s) of 𝑥 for which 𝑔(𝑥) = 𝑓(𝑥) for 𝑥 ∈ [0°; 180°]. (6)
________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________
________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________
(c) Determine the values of 𝑥 for which 𝑔(𝑥) > 𝑓(𝑥), for 𝑥 ∈ [0°; 180°]. (2)
________________________________________________________________________________________________________ [11]
𝑓
𝑦
𝑥
QUESTION 9 Page 15 of 23
(a) Consider the identity: cos θ−cos 2θ+2
3 sin θ−sin 2θ =
cos θ+1
sin θ
(i) Prove the identity. (5) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(ii) For which value(s) of θ will the identity be undefined? (3) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
T
C
D
M
A
B
N
4
4
4
8 metres
𝜃
Page 16 of 23
(b) In the diagram below, TABCD is a right pyramid with a square base side of 8 metres. M is the midpoint of BC and TN is the height of the pyramid. Each of the triangular faces makes an angle of θ with the square base.
(i) Express TM in terms of θ. (2) _______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
(ii) If the total surface area of the pyramid is equal to 256 square metres, determine the
value of θ, correct to one decimal digit. (4)
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
Page 17 of 23
(c) (i) Show that 2 cos(𝑥 − 60°) = cos 𝑥 + √3 sin 𝑥 (2)
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
(ii) Determine the minimum value of cos 𝑥 + √3 sin 𝑥, and the corresponding value(s) of 𝑥 if 0° ≤ 𝑥 ≤ 360°. (3)
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
[19]
QUESTION 10 Page 18 of 23
Green garden services mow lawn for customers. To calculate the charges for their work, they measure the approximate area, 𝑥, ( in m2) of a random sample of 12 of their customers’ lawns and the time, 𝑦, in minutes that it takes them to mow the lawns. Their results are shown in the table:
Area (𝑥) (m2)
360 120 845 602 1190 530 245 486 350 1005 320 250
Time (𝑦) (minutes)
50 28 130 75 120 95 55 70 48 110 55 60
(a) Use your calculator to determine the equation of the least squares regression line. Give your answers correct to 4 decimal digits. (3) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(b) Calculate the value of 𝑟, the correlation coefficient for the data, correct to 4 decimal places. (1) _______________________________________________________________________________________________________________ (c) Given that the garden service charge a flat call-out fee of R 150, as well as R 50 per half hour (or part thereof), estimate the charge for mowing a customer’s lawn that has an area of 560m2. (For example: 100 minutes would be taken as 2 hours) (3) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (d) The local high school wants Green garden services to mow their rugby field, which is rectangular, 100m long and 70m wide. Should they use the regression equation found in (a) to calculate the time it would take to mow this area? Give a reason for your answer. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ [9]
A
B
C
D
E
M N
QUESTION 11 Page 19 of 23
In the diagram (not drawn to scale), BC is parallel to EA. MA =3
8MN and 2MB = BE.
(a) Find the value of BD:DN. (3) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(b) Determine the Area ∆BMC
Area ∆BMN (2)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
[5]
QUESTION 12 Page 20 of 23 In the diagram, AD is a diameter of the larger circle with centre C and AC is the diameter of the smaller circle with centre B. FD is a tangent to the smaller circle at E while FA is a tangent to both circles at A.
(a) Prove that ∆DEB ||| ∆DAF. (3) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (b) Hence, show that DF = 3 AF. (3) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
E
B C D
A
F
Page 21 of 23 (c) If it is further given that the diameter of the larger circle is 8 cm, determine the area of ∆DEB.
(4) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
[10]
QUESTION 13 Page 22 of 23
Two circles with centre O(0;0) and M(a;b) touch externally at B. The equation of the smaller circle
with centre O is 𝑥2 + 𝑦2 = 16. Circle centre M touches the 𝑦-axis at C(0; −8).
Determine the co-ordinates of M.
(Hint: Let CM = 𝑥)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
[6]
Page 23 of 23
QUESTION 14
In the diagram below, AC, BD and CD are tangents to the circle O with diameter AB = 4cm. CD touches the circle at point E. If AC = 𝑎 and BD = 𝑏, prove that 𝑎𝑏 = 4.
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ [4]
Total for section B: 73
O
A B
C
D
E
𝑎
𝑏
•
O Q
P S
3 units
9 units
1 2
2
1
R
O Q
P S
3 units
9 units
1
2 2
1
R
(b) The diagram below is a sketch of ℎ(𝑥) and 𝑝(𝑥). ℎ(𝑥) = cos(𝑥° + 𝑎) + 1 and 𝑝(𝑥) = 𝑏 sin(𝑐𝑥°). (Hint: ℎ(𝑥) intersects the 𝑥-axis at −210°). Determine the values of 𝑎, 𝑏 and 𝑐. (3)
________________________________________________________________________________________________________
________________________________________________________________________________________________________
________________________________________________________________________________________________________
________________________________________________________________________________________________________
𝑝(𝑥)
−210° 𝑥
𝑦
P
M N
O
80cm
50cm
1
1 1 2 2
ℎ(𝑥)
90° −270°
The following frequency table shows the distribution of marks of 200 students writing the NBT exam out of 60.
NBT Mark Frequency Cumulative Frequency
0 ≤ 𝑥 ≤ 10 20
10 < 𝑥 ≤ 20 40
20 < 𝑥 ≤ 30 60
30 < 𝑥 ≤ 40 50
40 < 𝑥 ≤ 50 20
50 < 𝑥 ≤ 60 10
(a) Complete the cumulative frequency table in the space provided. (1) (b) Draw the cumulative frequency ogive on the grid below. (3)
(c) Use your graph to estimate the interquartile range. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(d) The top 30% of the students won’t need to rewrite the test.
Use the graph to determine the cut-off mark. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (e) Calculate the estimated mean of the data. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (f) It is given that the standard deviation is 12,9. The marks for the NBT test are too low and are all raised by 5 marks. Write down the new mean and new standard deviation for the new set of marks. (2) _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ (g) Below are two box and whisker diagrams representing the marks of 200 students each from two different Universities, for the same NBT test out of 60.
(ii) What percentage of University B’s results were above 55 out of 60? (1)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________
(iii) University A claim that their overall results are better that those of University B. Is this true? Refer to the diagrams to justify your answer. (2)
_______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ [15]
A university entrance exam was written by 400 students. Scores ranged from 5 out of 50 to 50 out of 50. The data is represented below:
(a) How many students scored more than 35 out of 50 for the entrance examination? (1)
________________________________________________________________________________________________________
(b) How many students obtained marks in the interval 20 < 𝑥 ≤ 25 ? (1)
________________________________________________________________________________________________________
(c) Use the axis below to draw a box-and-whisker plot for the data. (Use the data provided to estimate values for the lower quartile, median and upper quartile). (3)
(d) Comment on the skewness of the distribution of the data. (1)
________________________________________________________________________________________________________
(e) One interpretation of the skewness of the data is that university entrance standards are dropping. Provide an alternative explanation that would account for the skewness of the data. (2) ________________________________________________________________________________________________________
________________________________________________________________________________________________________
(a) If O is the centre of the circle and 𝑥 = 80°, calculate the value of 𝑦 in each case
(no reasons required):
T
C
D
M
A
B
N
4
4
4
8 metres
𝜃
𝒚
𝒙
𝐂(𝟎; −𝟖) 𝐌(𝐚; 𝐛)
𝐎
B