core mathematics exam: paper 2 - maths.stithian.com
TRANSCRIPT
Page 1 of 14
November 2020 Marks: 110 Grade 11 Time: 2 hours
St Anne’s Diocesan College
CORE MATHEMATICS EXAM: PAPER 2
Instructions:
Answer all the questions on the question paper.
Unless otherwise stated, give all answers correct to 1 decimal place.
Show all working details.
Approved calculators may be used.
This paper consists of 14 pages and a formula sheet.
Diagrams are not necessarily drawn to scale.
Write your name and the teachers name in the space provided
For Official use only:
Question 1 2 3 4 5 6 7 8 9 10 11 12 13 Total
Total 20 11 6 10 10 8 9 4 7 7 5 9 4 110
Name: ____________________________________________________________________________
Teacher: ____________________________________________________________________________
Page 2 of 14
𝑥
𝑦
Eሺ6; 𝑦ሻ
D
Mሺ𝑥; 2ሻ
Cሺ4; −2ሻ
Bሺ−4; −6ሻ
Aሺ−2; 2ሻ
O θ
Question 1
In the diagram, Aሺ−2; 2ሻ, Bሺ−4; −6ሻ, Cሺ4; −2ሻ and Eሺ6; 𝑦ሻ are the vertices of a quadrilateral
with AE ∥ BC. D lies on BC such that AD ⊥ BC and AC is drawn. Mሺ𝑥; 2ሻ is a point on EC.
(a) Calculate the gradient of BC.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (2)
(b) If M is the midpoint of EC, determine the values of 𝑥 and 𝑦.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (3)
(c) Calculate the lengh of BC. Leave your answer in surd form.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (2)
(d) If it is further given that AE = √80, what type of quadrilateral is ABCE?
___________________________________________________________________________________________________________________ (1)
(e) Determine the equation of AD in the form 𝑦 = 𝑚𝑥 + 𝑐
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (3)
Page 3 of 14
(f) Give the angle of inclination of the line AD shown in the diagram by the symbol θ.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (2)
(g) Show how the coordinates of Dሺ0,8 ; −3,6ሻ are derrived.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (4)
(h) Determine the area of ABCD. (No reasons required)
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (3) [20]
Page 4 of 14
Question 2
(a) Unordered data is shown below:
𝑥 21 38 𝑥 20 41 25 𝑥 46 13
The value of 𝑥 occurs three times and is the same unknown value. Determine the value of 𝑥 if the mean of this data is 30.
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________
_______________________________________________________________________________________________________________ (3)
(b) The data listed below shows the distance (in kms) travelled by cyclists each day of a fifteen day tour.
85 125 140 135 115
132 105 126 108 135 112 119 128 93 127
1. Use your calculator to determine the: (i) mean distance travelled
_________________________________________________________________________________________________________ (1)
(ii) standard deviation.
_________________________________________________________________________________________________________ (1)
2. On how many days do the cyclists travel a distance that is within 1 standard deviation of the mean? Show all working.
______________________________________________________________________________________________________________
______________________________________________________________________________________________________________
_________________________________________________________________________________________________________ (2)
3. The median distance travelled is 125km. Determine the skewness of the data and show working.
______________________________________________________________________________________________________________
_________________________________________________________________________________________________________ (2)
4. Before the start of each leg of the 15 day tour, the cyclists had an additional warm-up ride of 𝑥 km. Write down, in terms of 𝑥 (where applicable), the actual:
(i) mean
_____________________________________________________________________________________________________ (1)
(ii) standard deviation.
_____________________________________________________________________________________________________ (1) [11]
Page 5 of 14
Question 3
A group of learners wrote a test and the percentages they scored are shown in the cumulative frequency graph below.
(a) How many learners wrote the test?
___________________________________________________________________________________________________________________ (1)
(b) How many learners scored at least 80% for the test?
___________________________________________________________________________________________________________________ (2)
(c) Use the graph to calculate the inter-quartile range.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (3)
[6]
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 10 20 30 40 50 60 70 80 90 100 110
CU
MU
LA
TIV
E
FR
EQ
UE
NC
Y
TEST RESULTS (%)
Cumulative frequency graph of test results
Page 6 of 14
Question 4
Complete the following questions without the use of a calculator.
(a) Write the given expression in its simplest form:
cos 𝑥.tanሺ180°−𝑥ሻ.sin 430°
cosሺ−20°ሻ.sin 𝑥
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (6)
(b) Prove the identity: 1
tan2 𝑥− cos2 𝑥 =
cos4 𝑥
sin2 𝑥
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (4)
[10]
Page 7 of 14
Question 5
Determine the general solution for 𝑥 if:
(a) 2 cos 𝑥 = −1,3
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (3)
(b) √2 sin 𝑥 . cos 𝑥 = − cos 𝑥
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (7)
[10]
Page 8 of 14
Question 6
The graph of 𝑓ሺ𝑥ሻ = 2 sin 𝑥 + 1 is sketched below with 𝑥 ∈ [−90°; 270°] (a) For what values of 𝑘 will 𝑓ሺ𝑥ሻ = 𝑘 have no real solutions?
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (2)
(b) On the same set of axes shown above, sketch the graph of 𝑔ሺ𝑥ሻ = 2 cosሺ𝑥 − 30°ሻ with 𝑥 ∈ [−90°; 270°]
Show clearly the coordinates of the endpoints and the 𝑦-intercept.
Working space:
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (4)
(c) For what values of 𝑥 is 𝑓ሺ𝑥ሻ ≥ 𝑔ሺ𝑥ሻ if 𝑥 ∈ [−90°; 270°]
___________________________________________________________________________________________________________________ (2)
[8]
𝑥
𝑦
𝑓
Page 9 of 14
Question 7
In the diagram below, an acute-angled triangle ABC is drawn. A line PQ is drawn where P lies on the line BC and Q lies on the line AC. The lengths PQ = 14cm and AB = 18cm. The angles A = 68° and C = 50° (a) Show that BC = 21,79 units correct to one decimal places.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (2)
(b) Calculate the area of ∆ABC.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (3)
(c) Determine the size of PQC if the ratio of BP: PC is 2: 3 and PQC is acute.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (4)
[9]
A Q C
68°
18cm
P
B
14cm
50°
Page 10 of 14
Question 8
Use the given diagram to show that sin θ =𝑏2+𝑐2−𝑎2
2𝑏𝑐
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________ [4]
Volume formulae: 𝑉 = 𝜋𝑟2. 𝐻 𝑉 =1
3𝜋𝑟2. 𝐻 𝑉 =
4
3𝜋𝑟3
Question 9
In the diagram, the inverted cone has a height of 21cm ሺFC = 21cmሻ
and a base with a radius of 9cm ሺFE = 9cmሻ.
Water is filled to a depth of 14cm. ሺGC = 14cmሻ. (a) Determine the radius of the water surface, GD.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________ (3)
(b) Determine the additional volume of water required to fill up the cone.
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (4)
[7]
B
A
C
E F
D G
B
A
C
𝑏
θ
𝑐
𝑎
Page 11 of 14
Give full reasoning in the following questions:
Question 10
In the diagram, O is the centre of the circle.
Points A, B, C and D lie on the circumference of the circle. BOD is a diameter.
AC and BD intersect at E
A1 = 51° and B1 = 29°
Determine the size of:
(a) O1
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (1)
(b) A2
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (1)
(c) D
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (1)
(d) ACO
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (4)
[7]
A
B
C
D
E
O
51°
1
1
1
1
2
2
2
2
29°
Page 12 of 14
Question 11
Prove that the angle between a tangent to a circle and a chord drawn from the point of contact is equal to an angle in
the alternate segment.
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
[5]
D
C
B
A
O
E
Page 13 of 14
Question 12
(a) O is the centre of the circle PSQ. Chord QP such that OS ⊥ QP at T.
OT = 7cm, QT = 24cm and TS = 𝑥cm.
Calculate the length of 𝑥.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (4)
(b) O is the centre of the circle ABCD. AB is diameter
and AE is a tangent. AD = DC and A1 = 30°.
Show that DC ∥ AB.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
______________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________ (5)
[9]
P
Q
𝑥
O
S T
24cm
7cm
A
B
1
O
C
D 30°
2
1 2
2 1
E
Page 14 of 14
D
C
B
A O
E
Question 13
In the given diagram, AE is a diameter and B, C and D lie on the circumference.
Determine the value of B + D. Make one construction to help determine the value.
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
[4]