Parklands College

 JUNE EXAMINATION

MATHEMATICS GRADE 12 PAPER 2

Date: June 2015

Time: 3 hours Total: 150 marks Examiners: Mr F A du Preez ; Moderator: Mr S Joubert Miss B Malan ; Mrs M Naidoo ; Mr F Bredell ; Mr S Loseby

INSTRUCTIONS: 1. This paper consists of 10 QUESTIONS and 12 PAGES, including a

DIAGRAM  SHEET and an INFORMATION (FORMULA) SHEET.

2. Each question (e.g. QUESTION 2) must be numbered in the middle of the page,

and the sub-sections (e.g. 1.2), 1.3.2)) in the left hand margin. 3. All the steps must be shown.

4. A right hand margin must be drawn on each page.

5. A line must be drawn after QUESTION 1, QUESTION 2, but not after 1.1

and 1.2, etc.

6. Unless stated otherwise, calculators (non-programmable) may be used,

in which case answers must be correctly approximated to two decimals.

7. Where technology is used, all the steps must still be shown by the candidate.

8. Assume all denominators to be non-zero, unless restrictions are required.

9. A DIAGRAM SHEET is supplied for QUESTIONS 8, 9 and 10. Detach the

DIAGRAM SHEET from the main paper and hand it in with your ANSWER

BOOK.

10. The diagrams are not drawn to scale.

- 2 -

QUESTION 1 P(− 5 ; y) , Q(− 7 ; − 4) , R(x ; 0) and S(11; 6) are the vertices of a parallelogram, with T the intersection point of the diagonals. (Note : QS does not pass through the origin of the axes.)

1.1. Calculate the following : 1.1.1. The coordinates of T (2) 1.1.2. The values of x and y (4) 1.1.3. The equation of SV , a median of Δ SQR (5) 1.1.4. The length of (in simplified surd form) (2)  1.1.5. The inclination of PR (4) 1.2. A line passing through the point A(− 2 ; 7) bisects the area of the parallelogram. 1.2.1. Calculate the gradient of this line. (2) 1.2.2. Show whether this line and QS are perpendicular. (2) 1.2.3. If A , S and E( p ; 8) are collinear, calculate the value of p . (4)

[25]

QS

A(-2;7)

P(- 5;y)

S(11;6)

Q(-7;-4)

V

TX

O

Y

R(x;0)

- 3 -

QUESTION 2

The centre of a circle is M (− 3; 2) , and A(2 ; − 3) is a point on the circle. 2.1. Determine the following : 2.1.1. The equation of the circle (3) 2.1.2. The equation of the tangent to the circle at A (5) 2.1.3. The equation of the tangent to the circle at B (2) 2.2. Show whether D(4 ; 0) is a point on, inside or outside the circle with centre M . (3) 2.3. If the Y -axis is a tangent to another circle with centreM , write down the equation of this new circle. (2)

[15]

BA(2;-3)

M(-3;2)

OX

Y

- 4 -  QUESTION 3

The equations of the circles with centres A and B are x2 + y2 −10x + 24y + 69 = 0 and x2 + y2 = 16 respectively. 3.1. Write the equation of circle A in centre-radius form. (4) 3.2. Hence, determine whether the circles intersect, touch or do not intersect at all, by showing all the calculations. (4)

[8]

QUESTION 4

4.1. Given : sin(A+ 45°) + cos(A+ 45°) = 2 cosA 4.1.1. Prove this identity for all angles A . (3)

4.1.2. Hence, prove that sin65° + cos65° = 2 cos20° , without using a calculator. (3)

4.2. Prove that sin(180°− 3A)sin(90°− A)− cos(− 3A)sinA

− 2sin(360°− A)sin(180° + A)+1= tan2A (8)

4.3. If cos22° = p , determine the following in terms of p : 4.3.1. cos44° (2) 4.3.2. sin 22° (2) 4.3.3. cos412° (4)

[22] QUESTION 5

Determine the general solution of x :

5.1. (cos x + sin x)(cos x − sin x) =32

(4)

5.2. sin2x = 3sin2 x (7)

[11]

- 5 -

QUESTION 6

In the diagram, the trigonometric functions f (x) = cos3x and g(x) = sin x , − 90° ≤ x ≤180° , are represented. 6.1. Write down the period of f . (1) 6.2. Determine the general solution of the equation f (x) = g(x) . (6) 6.3. Determine the x -coordinates of the points A and B . (2) 6.4. Hence, determine the values of x for which f (x) ≥ g(x) , if − 60° ≤ x ≤ 60° . (2) 6.5. If the graph of f is shifted horizontally by 15° , determine the new equation. (1) [12]

B

A

g

f XO

Y

- 6 - QUESTION 7

7.1. Sketches may be used for the following:

7.1.1. Show that, in any triangle Δ ABC : cosA = b2 + c2 − a2

2bc. (2)

7.1.2. Hence, show that ccosA+ acosC = b . (4) 7.2. In the diagram, Q , E and R points on the same horizontal plane.

PE is perpendicular to this plane. EQR = x and QER = y , and the angle of elevation from Q to P is θ . PQ = k units . 7.2.1. Express QR in terms of QE , x and y . (2)

7.2.2. Hence, prove that QR = k sin y.cosθsin(x + y)

. (3)

7.2.3. Hence, if RQ = RE , prove that QR = k cosθ2cos x

(2)

[13]

yxθ

k

E

R

Q

P

- 7 - QUESTION 8

8.1. O is the centre of the circle. Redraw the diagram and prove the theorem which states that the line segment which joins the centre of the circle and the midpoint of a chord, is perpendicular to the chord.

(6) 8.2. In each diagram, O is the centre of the circle. Calculate, giving reasons, the values of x , y and p . 8.2.1.

(8) 8.2.2.

(5)

[19]

yx

35° O

E

D

C

B

A

C BA

O

230°

25°

2

11

2 p

D

C

B

A

O

- 8 -

QUESTION 9

T and Q are the centres of the circles. BD is a common tangent, AQ and CT are also tangents to the circles. 9.1. Prove that TAOD is a cyclic quadrilateral. (4) 9.2. Prove that TACQ is a cyclic quadrilateral. (3) 9.3. Name, without proof, one more cyclic quadrilateral which has D as a vertex. (1) 9.4. If a circle is drawn through the points T , A and Q , what will TQ be called in this circle ? (1) 9.5. If AD and CD are drawn and BTC = x , write down, with reasons, 3 more angles each equal to x . (5)

[14]

TD

C

BA

QO

- 9 -

QUESTION 10

BC is a tangent to the circumscribed circle of Δ ABE , and DE / / BC . 10.1. Prove that BD = BE . (4) 10.2. If DTBR is a cyclic quadrilateral, prove that 2BEC − EBC = 180° . (Hint : Let BEC = x and EBC = y .) (7)

[11]

                   

       

RT

E

D

C

B

A

 - 11 -  

GRADE 12 MATHEMATICS PAPER 2 JUNE 2015

DIAGRAM SHEET

NAME : Q 8.1) 8.2.1) 8.2.2)

C BA

O

yx

35° O

E

D

C

B

A

230°

25°

2

11

2 p

D

C

B

A

O

- 12 - Q 9) Q 10)

RT

E

D

C

B

A

TD

C

BA

QO