Prelim Website CAPS 2018 Prelim Papers... · Web viewThe circle with centre C(2;-3) passes through point K(6;-1) and S , which lies on the y- axis. M(4;e) is a point such that MK

Mathematics Paper 2Grade 12

Preliminary Examination 2018

Duration: 180 min Examiner: R. ObermeyerMarks: 150 Moderator: A. JanischDate: 3 September 2018 External Moderator: I. Atteridge

INSTRUCTIONS:

See overleaf for Instructions. This paper consists of 23 pages (including cover) and an information sheet.

NAME: ____________________________________

ASSESSMENT

Question Level Tested Topic Time

AllocationPossible

markActualmark

SECTION A1 1 – 4 Statistics 11 mins 92 1 – 4 Analytical Geometry 11 mins 93 1 – 4 Trigonometry 14 mins 124 1 – 4 Euclidean Geometry 19 mins 165 1 – 4 Euclidean Geometry 10 mins 96 1 – 4 Euclidean Geometry 5 mins 4

SECTION B7 1 – 4 Statistics 16 mins 138 1 – 4 Analytical Geometry 12 mins 109 1 – 4 Analytical Geometry 16 mins 13

10 1 – 4 Trigonometry 19 mins 1611 1 – 4 Trigonometry 13 mins 1112 1 – 4 Euclidean Geometry 18 mins 1513 1 – 4 Euclidean Geometry 10 mins 814 1 – 4 Euclidean Geometry 6 mins 5

TOTAL: 150

PERCENTAGE:

Teacher’s Signature: ________________________

Controller’s Signature: _______________________

Moderator’s Signature: _______________________

Mathematics Paper 2 Grade 12Preliminary Exam 2018 Examiner: Ms. R. Obermeyer

Instructions

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 23 pages (including the cover page) and an Information Sheet of 2 pages. Please check that your question paper is complete.

2. Read the questions carefully.3. Answer ALL the questions on the question paper and hand this in at the end of the

examination. 4. You will NOT receive extra paper for working out. Use the provided space wisely.5. Diagrams are not necessarily drawn to scale.6. You may use an approved non-programmable and non-graphical calculator, unless

otherwise stated.7. All necessary working details must be clearly shown.8. Round off your answers to one decimal digit where necessary, unless otherwise stated.9. Ensure that your calculator is in DEGREE mode.10. It is in your own interest to write legibly and to present your work neatly.

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SECTION AQuestion 1

a) A parachutist jumps out of a helicopter and his height was recorded at various times after his parachute was released.The following table gives the results where y represents his height above the ground (in metres) and t represents the time (in seconds) after he opened his parachute.

1) Determine the equation of the least squares regression line for this data. (2)______________________________________________________________________________

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2) Write down the correlation coefficient and comment on the strength of the relationship. (3)

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b) The ogive (cumulative frequency curve) below shows the performance of students who wrote a basic programming skills course. The test had a maximum of 100 marks.

1) How many students wrote the test? (1)______________________________________________________________________________

2) How many learners achieved a mark above 70% for the test? (1)____________________________________________________________________________________________________________________________________________________________

3) Only the top 25% of the students are allowed to proceed to the advanced programming course. What is the minimum mark achievable to still be part of this group? (2)

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Question 2

The circle with centre C (2;−3) passes through point K (6 ;−1) and S, which lies on the y−¿axis.M (4 ;e) is a point such that MK is a tangent to the circle at K .

a) Determine the equation of the circle in the form: ( x−a )2+ ( y−b )2=r2 (4)______________________________________________________________________________

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b) Determine the equation of tangent MK . (3)______________________________________________________________________________

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c) Determine the value of e. (2)______________________________________________________________________________

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[9]Question 3

a) Given: sin 25°=u

Determine, without the use of a calculator:

1) tan25 ° (3)______________________________________________________________________________

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2) sin 130 ° (3)______________________________________________________________________________

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b) Evaluate without the use of a calculator. Show all working out.

sin2(−x)+cos(360 °−x )∙ cos (x−180 °)cos2 x+cos (90°+x) ∙ sin(180 °−x)

(6)

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All steps and reasons need to be shown for Questions 4 – 6

Question 4

a) In the figure, S is the centre of circle PQR. Prove the theorem that states:

Q S R=2×Q P R

Given: Circle PQR with centre S

Required to prove: _________________________________________________ (1)

Construction: _________________________________________________ (1)

Proof: (4)_________________________________________________________________________

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b) In the diagram, O is the centre of the circle and WXYZ is a cyclic quadrilateral such that OW bisects X W Z .XZ is joined and W 2=p.

1) Give two other angles also equal to p. (3)______________________________________________________________________________

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2) Give Y 1 in terms of p. (2)______________________________________________________________________________

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3) Find X2 in terms of p. (3)______________________________________________________________________________

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4) Can WX be a tangent to the circle through XYZ? Give a reason. (2)______________________________________________________________________________

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[16]Question 5

In the figure, AB is a diameter of the circle ABED and AC is a tangent to the circle at A.BDF and BEC are straight lines.A1=x

a) Give a reason why B1=x. (1)______________________________________________________________________________

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b) Express E1 in terms of x. (4)______________________________________________________________________________

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c) Prove that CFDE is a cyclic quadrilateral. (4)______________________________________________________________________________

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[9]Question 6

Refer to the diagram below where MN = 16 cm and NB = 20 cm.BS = (x+1) cm and ST = x cm.MN // ST

Determine the value of x.______________________________________________________________________________

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SECTION B

Question 7

a) Five numbers in ascending order are given.

2 ; x ;7 ; y ;18

The numbers have a mean of 9 and an interquartile range of 12.Determine the values of x and y. (4)

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b) At the 2017 IAAF World Athletics Championships in London, 44 runners completed the men’s 100m sprint heats. The time taken by the slower runner was 10,75 seconds and the time of the fastest runner was 9,99 seconds. The speeds of all 44 runners are represented in the histogram below.

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1) Describe the skewness of the distribution. (1)______________________________________________________________________________

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2) Using the frequency table below to assist you, sketch an ogive (cumulative frequency curve) of the data, on the grid provided below. (4)

3) Eighteen runners ran times of less than x seconds. Estimate x. (2)______________________________________________________________________________

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4) After the race, it is discovered that the timing equipment was faulty. It added 0,1 seconds to each runner’s time. How does this error influence:i) The range of the data set? (1)

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ii) The mean of the data set? (1)______________________________________________________________________________

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[13]Question 8

The equation of a circle is given as x2+ y2−2 px+4 py+4=0.

a) Determine the co-ordinates of the centre of the circle and its radius in terms of p. (4)______________________________________________________________________________

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b) If (sinα+p; cosα−2 p ) is a point on the circle for all values of α , determine the value(s) of p. (3)

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c) For which values of p does the equation NOT represent a circle? (3)______________________________________________________________________________

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Mathematics Paper 2 Grade 12Preliminary Exam 2018 Examiner: Ms. R. ObermeyerQuestion 9

A circle with centre A is inscribed in the isosceles triangle OPR, with OP = PR.AM is a vertical line.OP, OR and PR are all tangents to the circle.

a) Determine the equation of line OP. (2)______________________________________________________________________________

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b) Determine the equation of AN. (4)______________________________________________________________________________

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c) Determine the co-ordinates of A. (2)______________________________________________________________________________

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d) Given that the equation of the inscribed circle with centre A is ( x−5 )2+( y−52 )2

=374 ,

determine whether the circle x2−10 x+ y2+4 y−20=0 is a tangent to the inscribed circle with centre A. (5)

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[13]Question 10

a) Determine the general solution of the following equation:

sin 2θ−4sin2θ=0 (6)______________________________________________________________________________

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b) Simplify fully:sin 3 psin p

− cos3 pcos p (4)

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c) cos (A+B)=13 and cos (A – B)=1

2

Find without the use of a calculator, the value of ( tan A ∙ tan B). (6)______________________________________________________________________________

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Mathematics Paper 2 Grade 12Preliminary Exam 2018 Examiner: Ms. R. ObermeyerQuestion 11

a) In quadrilateral ABCD, AB = BC = x and CD = DA = 2 x.

1) Prove that : AC=x √2(1−cosB). (3)______________________________________________________________________________

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2) Hence, or otherwise, prove that: 4 (1−cosD)=1−cosB. (4)______________________________________________________________________________

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b) A yield sign consists of two equilateral triangles. The length of each side of the inner triangle is 50 cm and the length of each side of the outer triangle is 80 cm.

Calculate the area of the shaded part of the yield sign. (4)______________________________________________________________________________

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All steps and reasons need to be shown for Questions 12 – 14

Question 12

a) In the diagram below: PA, PB and QR are tangents to the circle at A, B and C respectively.

Show that the perimeter of ∆ PQR = 2 PA. (7)______________________________________________________________________________

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b) XY is a common tangent to circles CBD and ACE at C.FD // ACAFE is a straight line.

1) Prove that AE // BD. (4)______________________________________________________________________________

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2) If it is given that CB : CA = 3 : 5, find the ratio of EF : EA. (4)______________________________________________________________________________

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Question 13

In the diagram below, AB is a chord of the circle with centre O.AO is produced to C such that AC ⊥ BC.Also, OD ⊥ AB.

a) Prove that ∆ ABC /// ∆ AOD. (3)______________________________________________________________________________

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b) Hence, or otherwise, prove that 2 AD2=OA 2+OA ∙OC. (5)______________________________________________________________________________

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Question 14

The diagram below shows three semicircles which are mutually tangent and have their diameters on the line AC.The line DB is perpendicular to AC, with DA and DC intersecting the two smaller circles at P and Q respectively.If DB = 10 units, determine the length of PQ.

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Documents

Prelim Website CAPS 2018 Prelim Papers... · Web viewThe circle with centre C(2;-3) passes through point K(6;-1) and S , which lies on the y- axis. M(4;e) is a point such that MK