grade 12: mathematics prelim paper 2 - st stithians college caps 2018 prelim papers... · 2018. 10....

Grade 12: Mathematics Prelim

Paper 2

3 Hours154 Marks

July, 2018

ExaminerM Klein

ModeratorA LiebenbergJ Lottering

Name:

Teacher:

1. This paper consists of 20 pages and an information sheet.

2. Show ALL calculations, answers only will NOT be awarded full marks.

3. Approved non-programmable calculators are permissible unless stated oth-erwise.

4. Round off answers to 2 decimal places, unless stated otherwise.

5. Diagrams are NOT necessarily drawn to scale.

Question 1 2 3 4 5 6 7 8 9 10 Total

MarkAchieved

PossibleMark

Marker

Grade 12 Paper 2 July, 2018

Section A 80 Marks

Question 1 [17 Marks]

A teacher, not happy with the test results of her class, decided to do a survey todetermine the amount of time spent watching TV each day. The results of thesurvey are shown below, together with the test results for the class.

Time Watching TV (Hours) Test Results (%)3 49

1,5 783 501 72

2,5 633,5 470,5 834 482 755 36

(a) Draw a scatter plot to show the relationship between these 2 variables [5]

TestResults%

Hours Watching TV

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(b) Determine the equation of the line of best fit correct to 2 decimal places. [3]

(c) Determine the correlation coefficient for this set of data correct to 2 decimalplaces. [1]

(d) Describe the correlation between the time spent watching TV and the resultsobtained in the test. [2]

(e) If it is given that a pupil watches 112

hours of TV a day, predict the test markthat will be achieved. [2]

(f) What is the mean number of hours of TV watched per day? [2]

(g) What is the standard deviation for the number of hours watched? Give theanswer correct to 2 decimal places. [2]

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D

A(-1;4)

C(2;1)

A is the point (−1; 4) and C is the point (2; 1)

(a) Determine the equation of the line AC. [3]

(b) Determine the co-ordinates of D, the mid-point of AC. [2]

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(c) Determine the angle of inclination of the line AC. [3]

(d) Determine the equation of the circle centre C, passing through D. [4]

(e) Determine the equation of the tangent to the circle at D. [3]

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(a) Prove that cosθsin2θ

− cos2θ2sinθ

= sinθ. [5]

(b) For what values of θ is the above identity undefined? [6]

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If sin400 = a and cos400 = b, determine the value of the following in terms of aand/or b without using a calculator.

(a) sin(80) [3]

(b) sin(1600) [4]

(c) tan(2300) [4]

(d) b in terms of a. [2]

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(a) Solve for x ∈ [−1200; 1200] if cos(x− 300) = sin(3x) [6]

(b) On the axes provided below, sketch the graphs of f(x) = cos(x − 300) andg(x) = sin(3x) for x ∈ [−1200; 1200]. [8]

1

90

-1

-90

(c) Use your graph to determine the values of x ∈ [−1200; 1200] for whichcos(x− 300) > sin(3x). [3]

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Prove the theorem which states that if two triangles are equiangular then theircorresponding sides are proportional.A = D, B = E and C = F .

ProveAB

DE=AC

DF[7]

A

B C

D

E F

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Section B 74 Marks


Solve for x: [10]

5sinx+2

tanx− 5cosx = 2

.

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In the diagram, the inverted cone has a height of 21 cm and a base with a radiusof 9 cm. Water is poured in to a depth of 14 cm. (i.e. GC = 14 cm.

A

B

C

D

EF

G

(a) Determine the radius of the water surface. [2]

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

volume of a cone is given by the formula V =πr2h

3[3]

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A climber wants to determine the height MN of a mountain. From A a point onthe same horizontal plane as N , the foot of the mountain, he measures MAN = α,the angle of elevation of the summit M . He measures MAB = β and then walksd metres to point B in the same horizontal plane as A and N and measuresABM = θ.

M

NA

B

d

(a) Show that the height of the mountain MN =dsinα.sinθ

sin(β + θ). [5]

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(b) Find MN without the use of a calculator if d = 1000 metres, α = 900−β andβ = θ. [4]

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The radius of the larger circle centre P is twice that of the smaller circle, centreQ. SR is a tangent to both circles. ST , SR and QR are drawn.

D

E F

S

T

P

R

(a) Prove ∆TSQ|||∆SRQ. [5]

(b) If the radius of the smaller circle is r, find the length of SQ in terms of r. [5]

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(c) Calculate the size of T SR. [4]

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O is the centre of the circle in the diagram below.RS = 2 units, S1 = θ and P = 2S1

⍬1

1

2

12

1

P

Q

RS

O

(a) Prove that sin(3θ) =1

r, where r is the radius of the circle. [9]

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(b) Find the value of θ if the radius of the circle is 2 units. [3]

(c) Determine the area of ∆ORS [4]

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In the diagram below, you are given a circle with diameter ON and the curvey2 = px.The point L(8; 4) is a point of intersection of the circle and the curve.

x

y

O

L

N

Determine:

(a) the value of p. [2]

(b) the equation of the circle. [6]

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(c) the equation of the line LN . [3]

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In this question, leave your answers in simplified surd form where necessary

A(2; 2� )

B

MC

(a) Find the equation of the circle, centre O which passes through the pointA(2; 2

√3). [2]

(b) A second circle, centre M touches the first circle at B and the y − axis atC(0;−8). Determine the equation of this second circle. [7]

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grade 12: mathematics prelim paper 2 - st stithians college caps 2018 prelim papers... · 2018. 10....

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