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Mathematics TRIALSPaper 2FORM 5

September 2014

TIME: 3 hours TOTAL: 150 marks

Examiner: Mrs A Gunning Moderated: Mrs B Philpot

NAME:PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY BEFORE

ANSWERING THE QUESTIONS.

This question paper consists of 24 pages. This includes 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 2 decimal places unless otherwise stated.

Ensure that your calculator is in DEGREE mode.

Diagrams have not necessarily been drawn to scale.

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SECTION A

QUESTION 1

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QUESTION 2

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QUESTION 3

(a) Two points have coordinates A (12; 4) and B (14, 8). If B is the midpoint of AC then find the

coordinates of C. (2)

(b) Points P(2 ;−3), Q(−3 ;−3) and R(5 ; y ) are given.

a. Find PQ and RQ. (2)

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b. If it is given that the distance from R to Q is twice the distance from P to Q. Find the value(s) of y. (4)

[8]

QUESTION 4In the diagram below, ABDC is a parallelogram. The equation of the line passing through A and B is

given by y= 14

x+6. The line passing through A and C also passes through the origin.

B A C=59 °, B D O= β and C D O=θ.

(a) Determine, correct to the nearest degree, the size of:i. θ (4)

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ii. β (2)

(b) Determine the equation of AC. (2)

(c) Determine the equation of BD. (2)

(d) Determine the co-ordinates of D. (2)

[12]

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QUESTION 5

(a) Simplify, without the use of a calculator.

i.2−2cos2(360 °−θ)

cos2(90 °+θ)

(4)

ii.tan 156 ° cos 114°

cos744 °− 1

sin2(−66 ° )(5)

(b) Solve for θ in each of the following equations, where θ∈[0° ;90 ° ] NO CALCULATOR may be

used in this question. All steps of working must be shown. Answer only will get only 1 mark.

i. 3 cosθ+1=0 (2)

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ii. 1−2sin2 θ=12 (2)

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SECTION BQUESTION 6In the diagram below, two concentric circles are drawn with centre A on the y axis. The smaller

circle cuts the y axis at the origin O and the point B. The line through B having equation y=32

x+6

cuts the x axis at R and the y axis at B. The larger circle passes through R.

Determine the equations of both circles. (6)

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QUESTION 7Tangents to the circle with centre the origin, touch the circle at E (4; 7) and F (8; 1). The tangents intersect at G.

Determine:(a) The equations of each of the tangents (5)

(b) The co-ordinates of G. (3)

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[8]QUESTION 8

(a) If sin 29 = √a , determine the value of the following in terms of a:

cos73cos15 + sin73sin15 (3)

(b) Determine the general solution of: sin2 x+cos2 x−cos x=0 (5)

(c) Show that cos (180°−2θ)

1−tan2θ=−cos2θ (4)

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QUESTION 9(a) On the same set of axes, sketch the graphs of f and g given that f ( x )=2cos2 x and g ( x )=1−cos x for

x∈[−180 °;180 ° ]. Show clearly all the significant points. (7)

-150 -100 -50 50 100 150 200

2,5

2

1,5

1

0,5

-0,5

-1

-1,5

-2

Using your graphs, give the value(s) of x for which:

f ( x ) . g ( x ) ≤0 (2)

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QUESTION 10(a) ∆ PQR is drawn. P=40 °. Q=74 ° and PR=12 cm. Determine, correct to the nearest cm2, the

area of ∆ PQR. (4)

(b) Shown alongside is a slice of cheesecake.E C K =60° , E K C=α, C E I=θ, MI = p units and

MICK is a rectangle.

Show that the height of the cheesecake CI ,is CI= 2 p sin∝ tan θ√3 cos∝+sin∝ (7)

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K

M

Y

Ip

C

θ

60◦

∝

E

15

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QUESTION 11

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QUESTION 12

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QUESTION 13From a point outside the circle, centre O, two tangents AD and AV are drawn. AO and VD meet in

M. BOD is a diameter of the circle. BV and VO are drawn.

Let A1+ A2=40 °

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QUESTION 14

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QUESTION 15

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QUESTION 16

In the diagram above QP is a tangent to the circle at P.(a) Prove that ∆ GTS ||| ∆ RTP (3)

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(b) Prove that ∆ QPR ||| ∆ QSP (3)

(c) Hence prove that QP . SP = QS . PR (2)

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