Grade 12 June Paper II 2014

ST TERESA’S SCHOOL

MID YEAR EXAMINATION

JUNE 2014

GRADE 12 MATHEMATICS: PAPER 2

(ANALYTICAL GEOMETRY, EUCLIDEAN GEOMETRY, TRIGONOMETRY AND STATISTICS)

Time: 3 hours150 Marks

Examiner: K MOFFATModerator: K MARSH

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 26 pages including a two page formula sheet. Please check that your paper is complete.

2. Answer all questions on the QUESTION PAPER.

3. You may use an approved, non-programmable, and non-graphics calculator, unless otherwise stated.

4. Answers must be rounded off to two decimal places, unless otherwise stated.

5. It is in your own interest to show all your working details, to write legibly and to present your work neatly.

6. Diagrams are not drawn to scale.

“There are no secrets to success. It is the result of preparation, hard work and learning from failure.”

General Colin Powell

SECTION A: [75 Marks]

QUESTION 1: [10]

The ogive curve (cumulative frequency) below represents the finishing times of the 590 runners who completed the RAC Old Parks 10km race on 1 June 2008.

(a) Estimate in how many minutes a runner would have had to complete the race in order to place in the 15th percentile or better.(2)

(b) If a silver medal is awarded to all runners completing the race in under 40 minutes, estimate the number of runners who would receive a silver medal.(3)

(c) Draw a box and whisker plot to summarise the data represented on the graph. (5)

QUESTION 2: [15]

(a) If , find the value of the following, without the use of a calculator:

(1)(2)

(2) (1)

(3)the value of if and :(5)

(b) Evaluate the following, without the use of a calculator:(7)

QUESTION 3: [6]

(a) Sketch the following trigonometric graphs for on the same set of axes provided below.(4) and

(b) From your graph, write down the value(s) of for which:(2)

QUESTION 4: [13]

In the diagram below, PQRS is a quadrilateral on the Cartesian Plane with and . The diagonals of PQRS bisect each other at right angles, at M. T is the point of intersection of line PR with the y-axis and P is the x-intercept of line PR.

(yxOQ(2;10)S(6;2)MTPR)

(a) Determine the gradient of PR.(3)

(b) Show that the equation of PR is given by .(3)

(c) Determine the co-ordinates of R.(4)

(d) Calculate the size of .(3)

QUESTION 5: [11]

, and are three points in the Cartesian Plane.

(a) Determine k if:

(1) P, Q, and R are collinear.(4)

(2) units.(4)

(b) If P, Q and R are collinear, determine the equation of the line perpendicular to PQR at P.(3)

QUESTION 6: [20]

(a) Use the diagram below to prove the theorem that states that:

“Opposite angles of a cyclic quadrilateral are supplementary.”(5)

Given:Circle centre O and points and D on the circumference of the circle

RTP:

Proof:

(b) (12111222)O is the centre of the circle. AE is a diameter. BA is a tangent to the circle at A. BCE is a secant to the circle.

(1) Prove that (4)

(2) Prove that AODB is a cyclic quadrilateral.(6)

(3) Prove that .(5)

SECTION B: [75]

QUESTION 7:[8]

(a) Given that , deduce the compound angle formula for .(3)

(b) Prove that:.(5)

QUESTION 8: [8]

Three numbers 2, x and y have a mean of 5 and a standard deviation of .

Determine x and y.

QUESTION 9: [10]

(2) (2) (1) (1) (MACBD)

In the diagram above, and are tangents to the base of a right cylindrical silo with and the points of contact. is a point metres above and the angle of elevation of from is . is the centre of the base that has a radius of metres. ., , and are all in the same horizontal plane.

(a) Find , giving your answer in degrees.(1)

(b) Find in terms of . (1)

(c) Find in terms of .(2)

(d) Prove that .(6)

(yxO. MAPCBD)QUESTION 10: [15]

In the diagram above is the midpoint of the radius OP and lies on the circumference of the smaller circle with centre M and diameter OA. The equation of the straight line BC is with B and C the x- and y-intercepts respectively.

(a) Find the co-ordinates of P.(2)

(b) Find the equation of the larger circle.(2)

(c) Determine the equation of the smaller circle.(5)

(d) Calculate the equation of the tangent to the circle AOD at D, the point of intersection of the circle and the x-axis.(6)

QUESTION 11: [13]

(2) (1)

(1) (2)

In the diagram above, O is the centre of the circle with radius r. CA is a tangent to the circle. BOD is a diameter that extends to the point C, such that .

.

(a) Prove that .(5)

(b) Determine AC in terms of r.(4)

(c) Express in terms of and hence determine the area of in terms of r and .(4)

QUESTION 12: [11]

(2)

(1)

(2)

(2) (1)

(1)

In the above diagram, KT bisects so that . Note that H is not the centre of the circle.

(a) Prove that TL is a tangent to circle LHK.(3)

(b) Show that (4)

(c) Show that (4)

QUESTION 13: [10]

(1) (1)

(3) (2) (2)

(3)

(2)

(3) (1)

(4)

(2) (1)

(2) (1)

In the above diagram, the diagonal DB of the rhombus BCDE is produced to A.

A is joined to E.

O is the point of intersection of the two diagonals.

(a) Prove that (5)

(b) Show that .(5)

MATHEMATICS

INFORMATION SHEET

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