CORE MATHEMATICS EXAM: PAPER 2 - maths.stithian.com

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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: ____________________________________________________________________________

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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.

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(b) If M is the midpoint of EC, determine the values of 𝑥 and 𝑦.

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(c) Calculate the lengh of BC. Leave your answer in surd form.

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(d) If it is further given that AE = √80, what type of quadrilateral is ABCE?

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(e) Determine the equation of AD in the form 𝑦 = 𝑚𝑥 + 𝑐

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(f) Give the angle of inclination of the line AD shown in the diagram by the symbol θ.

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(g) Show how the coordinates of Dሺ0,8 ; −3,6ሻ are derrived.

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(h) Determine the area of ABCD. (No reasons required)

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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.

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(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

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(ii) standard deviation.

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2. On how many days do the cyclists travel a distance that is within 1 standard deviation of the mean? Show all working.

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3. The median distance travelled is 125km. Determine the skewness of the data and show working.

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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

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(ii) standard deviation.

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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?

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(b) How many learners scored at least 80% for the test?

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(c) Use the graph to calculate the inter-quartile range.

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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

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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

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(b) Prove the identity: 1

tan2 𝑥− cos2 𝑥 =

cos4 𝑥

sin2 𝑥

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Question 5

Determine the general solution for 𝑥 if:

(a) 2 cos 𝑥 = −1,3

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(b) √2 sin 𝑥 . cos 𝑥 = − cos 𝑥

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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?

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(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:

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(c) For what values of 𝑥 is 𝑓ሺ𝑥ሻ ≥ 𝑔ሺ𝑥ሻ if 𝑥 ∈ [−90°; 270°]

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𝑥

𝑦

𝑓

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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.

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(b) Calculate the area of ∆ABC.

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(c) Determine the size of PQC if the ratio of BP: PC is 2: 3 and PQC is acute.

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A Q C

68°

18cm

P

B

14cm

50°

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Question 8

Use the given diagram to show that sin θ =𝑏2+𝑐2−𝑎2

2𝑏𝑐

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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.

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(b) Determine the additional volume of water required to fill up the cone.

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B

A

C

E F

D G

B

A

C

𝑏

θ

𝑐

𝑎

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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

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(b) A2

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(c) D

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(d) ACO

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A

B

C

D

E

O

51°

1

1

1

1

2

2

2

2

29°

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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.

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D

C

B

A

O

E

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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.

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(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.

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P

Q

𝑥

O

S T

24cm

7cm

A

B

1

O

C

D 30°

2

1 2

2 1

E

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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.

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CORE MATHEMATICS EXAM: PAPER 2 - maths.stithian.com

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