ST STITHIANS GIRLS’ COLLEGE

GRADE 12

MATHEMATICS: PAPER 2

24 July 2015

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ST STITHIANS COLLEGE

GRADE 12
MATHEMATICS PRELIM EXAM – PAPER 2
DATE: 24 July 2015 / TIME: 180 minutes
TOPICS: Paper 2 / TOTAL MARKS: 150
EXAMINER: Cluster N143 / MODERATOR: Cluster N143

INSTRUCTIONS:

  1. This paper consists of 10 questions. Answer ALL of the questions.
  1. This question paper consists of 22 pages.
  1. Clearly show ALL calculations you have used to determine the answers.
  1. An approved scientific calculator (non-programmable and non-graphical) may be used, unless otherwise specified.
  1. If necessary, answers should be rounded off to TWO decimal digits, unless stated otherwise.
  1. Diagrams are not necessarily drawn to scale.
  1. It is in your own interest to write legibly and to present your work neatly.

QUESTION 1[22]

1.1) is the midpoint of the line segment joining and

Find the values of and (3)

1.2) The points and are collinear.

Find the value(s) of (4)

1.3)Given points and on the Cartesian plane as sketched:

1.3.1)Calculate the size of rounded off to one decimal digit.(6)

1.3.2)Show that the midpoint, of is .(2)

1.3.3)Determine the equation of the circle which has AC as a diameter.

Give your answer in the form .(3)

1.3.4)Determine by calculation, whether point lies inside or outside this circle.

Give a reason for your answer.(2)

1.3.5) Write down the value of the shortest distance from to the circle.
(Leave your answer in surd form)(2)

QUESTION 2[18]

Refer to the diagram below. Given circle with centre O and equation .

is the centre of the larger circle. A common tangent touches the circles at

B and D respectively.

2.1)B lies on the circumference of the small circle. Determine the value of t.(3)

2.2)C is the midpoint of BD. Determine the coordinates of D.(2)

2.3)Determine the gradient of DG.(3)

2.4)Show that (3)

2.5)Determine the equation of the circle with centre G.(3)

2.6)Determine the size of angle.(4)

QUESTION 3[22]

3.1)Trigonometric functions and are given below, with :

3.1.1)Write down the equations of and g.(2)

3.1.2)Write down the period of .(1)

3.1.3)Write down the amplitude of .(1)

3.1.4)Determine the values of x where for (4)

3.2)If , and are the angles of a triangle, evaluate

without the use of a calculator:(4)

3.3)Without the use of a calculator, solve for ,

where (5)

3.4)You are riding the Colossus at Ratanga Junction and notice that consecutive peaks,

T and L , of theride are in proportion to each other.

You also notice as you are riding, that.

3.4.1)Determine the value of .

(Leaveyour answer in surd form if necessary)(3)

3.4.2)Determine the value of .(2)

QUESTION 4[9]

Mr Mears is curious to see the distribution of heights of all his History students.

The table below summarises the individual heights (in cm) of 61 History students.

4.1)Complete the table by filling in the unknown values for (a) and (b):(2)

Height Intervals in cm / Frequency / Cumulative Frequency
/ 0 / 0
/ 5 / 5
/ 11 / 16
/ (a) / 38
/ 13 / 51
/ 7 / (b)
/ 3 / 61

4.2)Below is an Ogive for the heights of the History students:

Use the Ogive to estimate the values of , and , and show on the Ogivehow you

read off your answers.

4.2.1)(1)

4.2.2)(1)

4.2.3)(1)

4.3)Which height interval(s) contain(s) heightsfrom the 90th percentile.(2)

4.4)Use the table of information to calculate an estimate for the mean oftheHistory

students’ heights.(2)

QUESTION 5[20]

5.1)In the figure is the centre of the circle and .

and are straight lines, and.

Find, with reasons, the magnitude of the following angles:

5.1.1)(3)

5.1.2)(3)

5.1.2)(2)

5.1.3)(2)

5.2)In the diagram below, parallelogram KLMN is given. is not the centre

of the circle. and . Determine the size of .(5)

5.3)An arch of a bridge is such that it is an arc of a circle and its height is 36m and

its span is 96m. (i.e. and ).

Calculate with reasons the radius OD of the arch, i.e. calculate the length of OD.

(Hint Let )(5)

QUESTION 6[16]

6.1) is a right pyramid with a square base with sides of length 4cm as shown

in the diagram below. and .

6.1.1)Determine the length of OA.(2)

6.1.2) Determine the length OE, the slant height of triangle OAB, where E is the

midpoint of AB.(3)

6.1.3)Show that the perpendicular height is …(2)

6.1.4)Hence, or otherwise, calculate the volume of the pyramid.(2)

6.2)Given AB ll CD, , , , , and

Find the Area of the shaded ΔBCD.(7)

QUESTION 7[12]

7.1.1)Prove the identity: (5)

7.1.2) Hence, determine the maximum value of , and the value

of xto give this maximum, where (2)

7.2) Determine the general solution of: (5)

QUESTION 9[9]

In the figure below, ΔABC has D and E on BC, and .

and AD ll TE.

9.1)Write down the numerical value of (1)

9.2)Show that D is the midpoint of BE.(2)

9.3)If , calculate the length of TE.(2)

9.4)Calculate the numerical value of:

9.4.1)(1)

9.4.2)(3)

QUESTION 10[10]

Refer to the figure below. , and is a tangent to the circle at .

and produced meets at . is a straight line.

10.1)Prove that .(4)

10.2)Prove that .(4)

10.3)Prove (5)

1