RONDEBOSCH BOYS’ HIGH SCHOOL

SENIOR CERTIFICATE

GRADE 12

MATHEMATICS PAPER 2
6 SEPTEMBER 2017

MARKS: 150 EXAMINERS: P GHIGNONE

D GELDENHUYS
M VAN ZYL
S VERSTER

TIME: 3 HOURS MODERATOR: S CARLETTI

This question paper consists of 13 pages INCLUDING a formula sheet.

Mathematics/P2 RBHSSeptember 2017

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of 11 questions.

2. Answer ALL the questions in the ANSWER BOOK PROVIDED. Indicate clearly
where EXTRA SPACE has been used.

3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.

4. Answers only will NOT necessarily be awarded full marks.

5. You may use an approved scientific calculator (non-programmable and non-

graphical), unless stated otherwise.

6. If necessary, round off answers to TWO decimal places, unless stated otherwise.

7. Diagrams are NOT necessarily drawn to scale.

8. An information sheet with formulae is included at the end of the question paper.

9. Write neatly and legibly.

Page 2 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 1

The ogive below represents the Grade 12 Life Orientation project marks.

Cumulative Frequency

Life Orientation Marks

Interval Frequency Cumulative Frequency

0 ≤ x<10 a 2
10 ≤ x <20 5 7
20 ≤ x <30 6 b
30 ≤ x < 40 10 23
40 ≤ x< 50 13 36
50 ≤ x <60 c 71
60 ≤ x <70 42 113
70 ≤ x <80 31 144
80 ≤ x< 90 28 172
90 ≤ x<100 8 d

1.1 Determine the values of a−d from the frequency table above. (2)

1.2 Use the ogive to determine the values of the lower quartile, median and upper
quartile. (3)

1.3 If the minimum mark was 8 % and the maximum mark was 98 %, draw a box-
and-whisker plot to represent this data. (3)

1.4 The marks then go through moderation and it is decided that all the marks will be
raised by 2 %. If the mean is p and the standard deviation is q, write in terms of
p and/or q:

1.4.1 the moderated mean. (1)

1.4.2 the moderated standard deviation. (1)

[10]

Page 3 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 2

2.1 A parachutist jumps out of a helicopter and his height was recorded at various times
after his parachute was released. The following table gives the results where y
represents his height above the ground (in metres) and t represents the time (in seconds)
after he opened his parachute.

t (s) 2 3 4 5 6 7 8
y (m) 500 300 200 120 70 40 20

2.1.1 Use the data to draw a scatter plot. (2)

2.1.2 Determine the equation of the least squares regression line for this data. (3)

2.1.3 Write down the correlation coefficient and comment on the strength of the
relationship. (2)

2.2 Five numbers in ascending order are:

2 ; x ; 7 ; y ; 18
They have a mean of 9 and an interquartile range of 12. Determine the values of
x and y . (4)
[11]

Page 4 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 3

D(1 ;−5), E(0; 4) and F (−4 ;−1) are the vertices of ∆≝. M is the midpoint of DE and
D^
E F=θ .

E (0;4)


M
F (-4;-1)

D (1;-5)

3.1 Determine the co-ordinates of M . (2)

3.2 Show that EF=DF. (4)

3.3 Determine the co-ordinates of G such that EGDF, is a rhombus. (2)

3.4 Determine the equation of line ED in the form y=mx+c. (2)

3.5 Calculate the magnitude of angle θ. (5)

[15]

Page 5 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 4

4.1 The circle with centre C (2;−3) passes through point K (6 ;−1) and point S, which
lies on the y−¿ axis. M (4 ; e) is a point such that MK is a tangent to the circle at K.

M (4;e)
S
x
K (6;-1)

C (2;-3)

4.1.1 Determine the equation of the circle in the form:

( x−a )2+ ( y −b )2=r 2 (4)

4.1.2 Determine the equation of tangent MK. (3)

4.1.3 Determine the value of e. (2)

4.1.4 Hence, prove that MS is a tangent to the circle. (6)

4.2 Given that x 2+ y 2−2 px +4 py + 4=0 is the equation of a circle.

4.2.1 Determine the co-ordinates of the centre of the circle and its radius in terms
of p. (4)

4.2.2 If ( sin α + p ; cos α −2 p ) is a point on the circle for all values of α,

determine the value(s) of p. (3)

4.2.3 For which values of p does the equation NOT represent a circle? (3)
[25]

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Mathematics/P2 RBHSSeptember 2017

QUESTION 5

5.1 Given sin 25 °=u. Determine, without the use of a calculator:

5.1.1 tan25 ° (3)

5.1.2 sin 130 ° (3)

5.2 Evaluate without the use of a calculator. Show all working.

sin 2 (−x ) +cos (360 °− x ) . cos ⁡( x−180° )

(6)
cos2 x+ cos ( 90 ° + x ) . sin ⁡( 180 °− x)

5.3 Determine the general solution of the equation:

sin 2 θ−4 sin2 θ=0

(6)

sin 3 P cos 3 P
5.4 Simplify in full: − (4)
sin P cos P

1 1
5.5 cos ( A+ B ) = and cos ( A−B )= , find without the use of a calculator, the
3 2
value of tan A . tan B (4)
[26]

Page 7 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 6

Given: f ( x )=−3sin 2 x

6.1 Determine the period of f ( x ) . (1)

6.2 Sketch g ( x )=−3 tan x for x ∈ [ −90 ° ; 180 ° ] . (3)

6.3 Determine the values of x for which −f ( x ) . g ( x ) <0 if x ∈ [ −90 ° ; 90 ° ] . (2)

[6]

QUESTION 7

In quadrilateral ABCD, AB=BC=x and CD=DA =2 x .

x 2x

B D
x 2x

7.1 Prove that: AC=x √ 2¿ ¿ (3)

7.2 Hence, prove that: 4 ¿ (4)

[7]
Page 8 of 14
Mathematics/P2 RBHSSeptember 2017

ALL STEPS AND REASONS NEED TO BE SHOWN FOR QUESTIONS 8 – 11.

QUESTION 8

8.1 In the figure, S is the centre of the circle PQR . Prove the theorem that states:
Q S^ R=2 ×Q P ^ R.

Q (5)

8.2 In the diagram O is the centre of the

^ 2 =p
^ Z . XZis joined and W
circle and WXYZ is a cyclic quadrilateral such that OW bisects X W
.

X
2 1

Y
1

O
1

2 1
1 p 3
2 Z
W

8.2.1 Give two other angles also equal to p . (3)

8.2.2 Give Y^ 1 in terms of p. (2)

8.2.3 Find ^
X 2 in terms of p . (3)

8.2.4 Can WX be a tangent to the circle through XYZ ? Give a reason. (2)
[15]
Page 9 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 9

In the figure, AB is a diameter of the circle ABED and AC is a tangent to the circle at A. BDF
and BEC are straight lines. ^
A1=x.

A
3 1
2
x

1 F
2
1
2 D

2
1
2 3 1
C
B E

9.1 Give a reason why ^B1=x . (1)

9.2 Express ^
E1 in terms of x. (4)

9.3 Prove that CFDE is a cyclic quadrilateral. (4)

[9]

Page 10 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 10

In the diagram, NT is the perpendicular bisector of side MP of ∆ MNP . Q is a point on NP

such that NQ : NP=3 :5. MQ and NT intersect at S . NT is drawn parallel to QR .

S
R

P
N Q

10.1 Prove that ∆ PQR∨¿∨∆ PNT .

(3)

10.2 Determine:

QR
10.2.1 (2)
NT

MT
10.2.2 (4)
MR

Area ∆ STM
10.2.3 (4)
Area ∆ QRM
[13]

Page 11 of 14
Mathematics/P2 RBHSSeptember 2017

QUESTION 11

In the diagram, A , B , C , Dand E lie on the circle with centre O. FE is a tangent to the circle at
E and AP=PD . BOPE and APD are straight lines.

B
A
1
2
3

1 O
4 2
3
F P

1 2
3
1 2
E 4
D

Prove that:

11.1 AD∨¿ FE (4)

11.2 E D 2=PE . BE (6)

D P2
11.3 BE= + PE (3)
PE
[13]

[TOTAL: 150 MARKS]

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Mathematics/P2 RBHSSeptember 2017
INFORMATION SHEET: MATHEMATICS

−b ± √ b2−4 ac
x=
2a

A=P ( 1+¿ ) A=P ( 1−¿ ) A=P ( 1−i )n A=P ( 1+i )n

n n
n ( n+1 ) n
∑ 1=n ∑ i= T n=a+ ( n−1 ) d Sn= ( 2 a+ ( n−1 ) d )
i=1 i=1 2 2

a ( r n−1 ) a
T n=a r n−1 Sn= ; r ≠1 S∞ = ;−1<r <1
r−1 1−r

x [ ( 1+i )n−1 ] x [ 1−( 1+ i )−n ]

F= P=
i i

f ( x +h )−f ( x )
f ' ( x )=lim
h→ 0 h

2 2
√
d= ( x 2−x 1 ) + ( y 2− y1 )
M ( x +2 x ; y +2 y )
1 2 1 2

y 2− y 1
y=mx+c y− y1 =m ( x−x 1 ) m= m=tan θ
x 2−x 1

( x−a )2+ ( y −b )2=r 2

In ΔABC:
a b c 1
= = a 2=b2 +c 2−2 bc .cos A area ∆ ABC= ab . sin C
sin A sin B sin C 2

sin ( α + β )=sin α . cos β +cos α .sin β sin ( α −β )=sin α .cos β−cos α . sin β

cos ( α + β )=cos α . cos β−sin α . sin β cos ( α−β ) =cos α . cos β +sin α . sin β

cos2 α −sin 2 α

{
cos 2 α = 1−2 sin 2 α
2 cos2 α −1
sin 2 α =2sin α . cos α

( x ; y ) → ( x cos θ− y sinθ ; y cos θ+ x sin θ )

n
2

x́=
∑ fx x −x́ )
2 ∑( i
σ = i=1
n
ń

n(A)
P ( A )= P ( A∨B )=P ( A ) + P ( B )−P ( A∧B )
n( S )

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Mathematics/P2 RBHSSeptember 2017

^y =a+bx b=
∑ ( x−x́ )( y− ý )
∑ ( x−x́ )2

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