Candidate’s Number:

Maths Set Teacher’s Initials:

BISHOPS

GRADE12

MATHEMATICS

PAPER 2

Time:3 hoursMock 2016

Total:150 marks SMcP/GG

INSTRUCTIONS AND INFORMATION:

Read the following instructions carefully before answering the questions.

  1. This paper consists of 10 questions.
  2. Answer ALL the questions in the Answer Booklet provided, writing neatly and legibly.
  3. Clearly show ALL calculations, diagrams, graphs et cetera that you have used to

determine your answers.

  1. Diagrams are NOT necessarily drawn to scale.
  2. Answers only will NOT necessarily be awarded full marks.
  3. You may use an approved scientific calculator (non-programmable and

non-graphical), unless stated otherwise.

  1. Where appropriate, round off answers to TWO decimal places, unless stated otherwise.
  2. AnINFORMATION SHEETis printed on the last page.

QUESTION 1

The histogram above shows the ages of staff in a school.

1.1Use the histogram to complete the cumulative frequency table below.(2)

Age / Frequency / Cumulative Frequency
25 < A≤30 / 2 / 2
30 < A≤35 / 8 / 10
35 < A≤40
40 < A≤45
45 < A≤50
50 < A≤55
55 < A≤60
60 < A≤65 / 6

1.2Draw a cumulative frequency graph on the set of axes provided to represent
the data in the table.(3)

1.3Use your cumulative frequency graph to find an estimate for the median age.(2)

1.4Use your cumulative frequency graph to find an estimate for the percentage of teachers older than 50 years. (2)

1.5 Use your cumulative frequency graph to draw a box and whisker diagram
for the given data.Use the number line provided.(3)

1.6Comment on the skewness of the data.(1)

[13]

QUESTION 2

In the table below, the scores in beam and floor eventsof 13 gymnasts who participated at the Rio 2016 Games are given. A typical score under today's rules ranges from
13 to 16 points.

A scatterplot diagram is also given to show the correlation between the two events.

Beam
x / 13.666 / 13.066 / 13.2 / 13.6 / 14.366 / 13.7 / 14.866 / 13.2 / 13.866 / 13.9 / 13.8 / 13.8 / 14.666
Floor
y / 14.733 / 14.133 / 13.233 / 13.766 / 13.9 / 14.3 / 15.433 / 13.833 / 13.933 / 14.133 / 14.233 / 14.075 / 14.9

2.1Use your calculator to determine the equation of the least squares regression
line y = A + B x. Give your answers correct to 4 decimal places.(3)

2.2Calculate the value of r, the correlation coefficient for the data, correct to
4 decimal places.(2)

2.3Discuss the correlation between the two sets of data.(2)

2.4Use the least squares regression line found in 2.1 to predict what agymnast is likelyto score for the floor event if she were to obtain a score of 14,5for the
beam event. (2)

[9]

QUESTION 3

In the diagram below, A(−3; 4) , B(4; 8), C(5; 0) and D are the vertices of
parallelogram ABCD. BC is extended to E to meet DE which is parallel to the x-axis.

3.1Determine the equation of line BE.(4)

3.2Determine the coordinates of D.(2)

3.3Determine the coordinates of P, where P is the point of intersection of
the diagonals of ABCD.(2)

3.4Prove that ABCD is a rhombus. (3)

3.5Calculate the size of. (4)

3.6Calculate the length of DE.(3)

3.7Calculate the area of ∆ABC. (4)

[22]

QUESTION 4

The circle with centre P has the equation .
QS is a tangent to the circle at Q and SR is a tangent to the circle at R(6; 9)
S is the point of intersection of the two tangents.

Answers to this question should be left in surd form where necessary.

4.1Write down the coordinates of P and the length of PQ.(3)

4.2If the equation of the tangent to the circle at Qisy = 2x, determine the
coordinatesof Q.(5)

4.3Show that the tangent at R has the equation .(4)

4.4Determine the length of SQ.(3)

[15]

QUESTION 5

NO CALCULATOR MAY BE USED IN THIS QUESTION

5.1If determine:

5.1.1(2)

5.1.2(2)

5.1.3(4)

5.2Given:

5.2.1 Prove the identity.(5)

5.2.2 Determine all the values ofx for which the identity is undefined.(3)

5.3Simplify (4)

[20]

QUESTION 6

The graph ofis drawn below

6.1Write down the period of f.(1)

6.2Write down the range of .(2)

6.3Draw the graph of on the same set ofaxes as f(x).
Show all turning points and intercepts with the axes.(3)

6.4Solve for x if f(x) = g(x) and .(6)

6.5Use the graphs to determine the value(s) of x for which:

6.5.1 in the interval (1)

6.5.2in the interval (2)

[15]

QUESTION 7

7.1

O is the centre of the circle. AB = 18 cm, AC = 14 cm and BC = 8 cm.

7.1.1Calculate the size of .(3)

7.1.2Calculate the radius of the circle correct to one decimal place.(4)

7.2In the diagram,and .

7.2.1 Find in term of α and β. (1)

7.2.2 Prove that . (6)

[14]

QUESTION 8

8.1O is the centre of the circle, and ST is atangent to the circle at T.

Use the diagram to prove the theoremwhich states that .(5)

8.2CD and CE are produced to A and B respectively so that AE is a tangent to the

circle and AB = AE..

8.2.1Calculate, giving reasons, the size of

a)(2)

b)(2)

8.2.2Prove that ABED is a cyclic quadrilateral.(3)

8.2.3Prove that AB is a tangent to the circle through B, D and C. (3)

8.2.4Calculate, giving reasons, the size of .(2)

[17]

QUESTION 9

In D is the midpoint of AB, CD || EF and .

9.1Determine, with reasons, the value of .(4)

9.2Find the value of (no reasons required).(3)

[7]

QUESTION 10

10.1In the diagram below, ∆ABC and ∆PQR are given with .

Line XY is drawn so that AX = PQ and AY = PR.

Use the diagram to prove

10.1.1XY || BC (4)

10.1.2(2)

10.2

In the diagram the circle with centre O has a radius twice that of the circle with centre T.

SRQ is a tangent to both circles. POTQ is a straight line.

10.2.1Prove that ∆ PST ||| ∆ SRT.(4)

10.2.2If the radius of the smaller circle is r, find the length of ST in terms of r. (4)

10.2.3Determine the size of .(3)

10.2.4Hence determine the length of SQ in terms of r.(1)

[18]

INFORMATION SHEET

; ;

M

In ABC:

P(A or B) = P(A) + P(B) – P(A and B)