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Mathematics TRIALSPaper 2FORM 5

15 September 2016TIME: 3 hours TOTAL: 150 marks

Examiner: Mrs A Gunning Moderated: Ms C Mundy

NAME:

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY BEFORE ANSWERING

THE QUESTIONS.

This question paper consists of __ pages, plus an information sheet. Please check that your

question paper is complete.

Answer all questions on this question paper. Read and answer all questions carefully.

It is in your own interest to write legibly and to present your work neatly.

All necessary working which you have used in determining your answers must be clearly

shown.

Approved non-programmable calculators may be used except where otherwise stated.

Where necessary give answers correct to 1 decimal place unless otherwise stated.

Ensure that your calculator is in DEGREE mode.

Diagrams have not necessarily been drawn to scale .

Give reasons for all Euclidean Geometry questions unless otherwise stated.

The last pages can be used for additional working, if necessary. If this space is used, make

sure that you indicate clearly which question is being answered.

Grade 12 Mathematics Paper 2 September 2016 of 18

SECTION A SECTION B

Question

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Mark

Out of

Total (150) Percentage

SECTION A

QUESTION 1(a) Given A(−2;7) and B(4 ; p), the endpoints of a line segment with midpoint M (q;3),

find the values of p and q. (3)(b) The vertices of a triangle are the points P (3 ;c ) ,Q(9 ;2) and R(3 c ;11) where c is a

constant. Given that P Q R=90 °, (i) Find the value of c. (5)(ii) Show that the length of PQ is k √10 , where k is an integer which you must

calculate. (3)(iii) Find the area of ∆ PQR.

(4)(c) In the diagram, A is the point (0 ; 4) and B is the point (4 ;12 ) . The straight line CAT has

a gradient of 13 , KAB is a straight line. Determine:

(i) C T X (2)(ii) B A C (5)

[22]

QUESTION 2(c)The circle C has centre (3 ;−2) and radius 5.


(iii) Write down the equation of the circle C.(2)

(iv)The line y=2 x−3 intersects the circle at the points A and B. Find the coordinates of A and B, and hence show that AB=4√5. (6)

(d) A circle has the equation x2+ y2−6 x+2 y−15=0.(i) Find the coordinates of the centre of the circle and its radius. (3)(ii) Find the equation of the tangent to the circle at the point (7 ;2). (4)

[15]QUESTION 3

(a) Without the use of a calculator, solve sin 32 °cosx+cos32° sinx=sin 75 ° for x,

where 0 ° ≤ x ≤360 ° : (3)

(b) Each of the following curves is shown for the interval −180 °<x<180 °.

Write down the coordinates of the turning points of each curve for this interval.

(i)

(1)(ii)

(3)(iii)

(2)

(c) Given that cosA=√3−1 , and A∈(180 °;360), using a relevant sketch, and showing the working detail

(i) Find the value of sin2 A . (3)

(ii) Show that tan2 A=√32

(2)


(d) If , write in terms of p: (4)


QUESTION 4

(a) The following set of numbers {5 ;8 ;9 ; x ;16} has a mean of 9.

a. Find the value of x (2)

b. and hence the standard deviation. (3)

(b) Mr Browne is trying to compare his 2009 matric classes (27 students) results from the

school trial and the final (IEB) examinations. The box plot is the summary of the trial

examination whilst the cumulative frequency graph summarizes the results from the IEB

examination.

(i) Use the ogive and read off the value of the 60th percentile (2)

(ii) Draw the box and whisker plot for the IEB examination and indicate on this

diagram the values of the 5 number summary. (3)

(iii) How do the two sets of results compare? (3)


IEB Exams

School Trial Exams

Question 5The energy saving lightbulb, also known as a compact fluorescent light bulb (CFL), is a type

of fluorescent lamp that fits into a standard light bulb socket. An energy saving

lightbulb uses far less wattage to produce the same amount of light as a regular

incandescent light bulb (IL). The table below compares the wattage of an IL bulb to a

CFL bulb.

CFL (watts)(x)

IL bulb (watts)(y)

5 25

10 40

13 60

19 75

26 90

32 125

(a) Plot the information given in the table on the axes given below and draw in a line of best fit. (3)

(b) Use your calculator to find the equation of the line of best fit through this data and

the correlation coefficient. (3)

(c) Determine the equivalent CFL wattage for a 55W IL bulb. (1)

(d) Comment on the validity of your answer in part (c). (1)


QUESTION 6

State the skewness of the data in each case: (4)

A: 20, 35, 40, 55, 68, 75, 92

__________________________________________________________________________

B:

________________________________________________________

C:

________________________________________________________

D:


x

y

QUESTION7

In the figure O is the centre of the circle and DB=DF .

AF , BEand BF are straight lines, and F=20 °.

Find, with reasons, the magnitude of the following angles:D2 (3) A (3) O2 (2) C1 (2)


F

C2

1

34 3 B2

1

D 21

23 1

4O

E A

QUESTION 8

In the diagram below, parallelogram KLMN is given. T is not the centre of the circle.

L=66o and N1=24o. Determine the size of M 1. (5)


1

4

3

3

3

2

2

2

2

2

1

1

1

1

LP

K

T

M

N

QUESTION 9

(a) In ∆ PQR, X and Y are drawn on PQ and PR respectively such that XY // QR. Prove

the theorem which states that a line drawn parallel to one side of a triangle cuts the

other two sides in proportion.

Complete the diagram and proof given below.


(b) In ∆XYZ, R is the midpoint of YZ with P and Q on XY such that PZ // QR. XR cuts

PZ at M. PXXY

=13 and PZ = 36 cm. Let XP = k .

Calculate the length of MZ. (5)

SECTION B

QUESTION 10

(a) If x and y are complementary angles, calculate the numerical value

sin xcos y+cos x sin y (2)

(b) If θ, 2θ and 3 θ are the angles of a triangle, evaluate cos2θ+cos2 2θ+cos2 3θ

without the use of a calculator: (4)

(c) Show that the equation cos2 x=sin (x−30 °) is an alternate version of the equation

4 cos2 x−2−√3 sin x+cos x=0 (4)

(d) The diagram given below shows the curves y=2sin x and y=3cos x for the interval 0 ° ≤ x ≤360 °.

Find correct to 1 decimal place the coordinates of the points where the curves intersect in

this interval. (5)

(e) Determine the general solution of sin x=cos2 x−1 (6)


(f) The expression K =m sinθ+n has a maximum value of 3 and a minimum value of -5. Determine all the possible values of m and n if m>0 (4)

QUESTION 11

Let a , b , c and d be integers such that a<b<c and c=d

The mode of these four numbers is11

The range of these four numbers is 8

The mean of these four numbers is 8

Calculate the values of a , b , c and d (5)

QUESTION 12

EITHER

The diagram shows circles C1 and C2 which both pass through the point P and the common tangent to the circles at P is the line l

.

C1 has the equation x2+ y2−4 y−16=0. and

C2 has the equation x2+ y2−2x−8 y−60=0

(i) Find the equation of the straight line through the centres of C1and C2.(ii) Find the equation of the tangent l to the circles at P..

OR

The diagram shows the straight line l with equation x+2 y−20=0 and the straight line m which is perpendicular to l and which passes through the origin.

(i) Find the coordinates of A and B where A and B meet the x-axis and y-axis respectively. (2)

(ii) Given that lines l and m intersect at C ,find the ratio of the area of ∆ OAC to the area of ∆ OBC . (5)


QUESTION 13

QUESTION 14

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 15

Given AB // CD, , , , , and


C

• O

BA

D

Find the Area of the shaded ΔBCD. (7)


42°

35°

6

9

15

ED

C

B

A

12

QUESTION 16In the diagram, MT is a vertical netball post and P and Q are two points on the ground (in the same horizontal plane as point T).The angle of inclination of the top of the post M from Q is θ.

PQ = k metres and PM = 2PQ.

(a) It is also given that M P Q = 2 θ. Show that the area of MPQ = 2 k2 . sin θ .cosθ (3)

(b) Show also that MQ = k √1+8 sin2 θ . (4)


Question 17

In the figure, O is the centre of the circle. MCT is a tangent. OM || DC.

(a) Prove:

(i) BK = KC (4)

(ii) BOCM is a cyclic quad. (3)

(iii) . (3)

(b) Calculate BC (in surd form) if OK = 1 and MK = 7. (4)

[14]


Question 18

The diagram below illustrates the tangent-secant theorem (N.B. this is not one of the standard theorems you learn in class):

AB is a tangent to the circle at A. The line BCD cuts the circle at C and D. By the tangent-secant theorem:

(Note that a secant is a line which cuts a curve in two places)(a) By making suitable construction lines and using similar triangles,

prove the tangent-secant theorem. (7) (b) Refer to the diagram below. AB is a tangent to the circle at B. AEF is a secant which

cuts the circle at E and F. ADC is a secant which cuts the circle at C and D.

(i) Using the tangent-secant theorem, or otherwise, prove:

(3)

(ii) You are now given that AE = 4 and AD = FE = 5.


A

B

C

D

C

D

A

B

F E

Find:

(1) the length of the tangent AB (2)

(2) the length of CD. (2)

[14]


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