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CORNWALL HILL COLLEGE Mathematics Department
EXAMINATIONS- 28 JULY 2014
Mathematics Paper 2
Grade: 12 Time: 3 Hours
Marks: 150
Examiner: Mrs T. Knoetze
Moderator(s): Mrs. M. van Niekerk, Mrs. S. Hickling and Mr. P. van Schalkwyk
Instructions:
1. This question paper consists of 24 pages. Please ensure that your paper is complete.
2. Answer ALL the questions on the question paper and hand this in at the end of the
examination.
3. You may use an approved non-programmable and non-graphical calculator, unless
otherwise stated.
4. Round off your answers to two decimal digits where necessary.
5. All the necessa ry working details must be clearly shown.
6. It is in your own interest to write legibly and to present your work neatly.
Q1 (8)
Q2 (26)
Q3
(23)
Q4 (18)
QS
(10) QG (17)
Q7 (24)
Q8 (6)
Q9 (6)
Q10 (12)
TOTAL
Grade12 Mathematics Paper 2 Paqe2of24
SECTION A [75}
QUESTION 1 [8}
In preparation for the Soccer World cup the Brazilian coaching team looked at the amount of goals scored by Brazil in the last nine tournaments.
(a) Fill in the missing amounts in the cumulative frequency table below.
Tournament year Goals scored Cumulative frequency
1978 10 10
1982 15 25
1986 10 35
1990 4 39
1994 11 so 1998 I'+ b'+ 2002 18 82
2006 10 92
2010 9 101 ~
............,
(2)
(b) Determine the average number of goals scored in the last nine tournaments by Brazil. (2)
(c)
/CJI ~ r¥J
Determine the standard deviation correct to two decimal places. ~ o- :: 3 7 'f '-....,; ~
(2)
(d) In how many tournaments do the goals scored, lie within one standard deviation of the
mean? (2)
{V,z?... -3,71 If, 22. + 3, 79] 1
15. OJ 7 '-....1 l"rJ '
[_ 7, '+-3 I
7 +ourname.nfs. '-...! R
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Grade12 Mathematics Paper 2 Paqe3o(24
QUESTION 2 [26]
(a)
(b)
If cos 65° = m , determine the following expressions in terms of m .
(1) cos 115° (1)
= coa (,8oq- b5 o)
= - COQ 650 ~ = - rY)
(2) sin 25° (1) ::: ~,,..., f=to •- b5°)
Cl ::: roD bS :: YYJ ~ ~
(3) sin 130° ........... di (4)
= s /;') ~ (6so) . CC/.J b 5 ° ~ 19 : ~ Stn b5~
'::: ;2. m J1-m2. "'A '-....Jt9
Given: psinfJ- q = 0, and 90° :5 f3 :5 270° , with p > 0 and q > 0.
Use a sketch to evaluate, in terms of p and q:
(1)
'-........! n-,
5 Ll = _:t:_.:: ~ 1n o r
=
cos f3 (4)
= ~='~e;~:-:1~'~-~--~c~~-----------------------------p
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Grade12 Mathematics Paper 2 Paqe4of24
(2} cos zp '-..J A = 1 ~ s tn "t. t3
(2}
- I - :L{j-)z_
- I - D1. 7 '--! Cfi
(c) 2 sin(180°+A) sin(90°-A)
Prove that: 4 . 4 = tan(360° - 2A) cos A- sm A
(6}
~Jq ~A z (-s,~ A ) (('cv:JA)
- :z.s IY1 A (' 00 A
= - +Qn zA
RHS fan zA. '-' A
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Grade12 Mathematics Paper 2
(d) Given f(x) = 2 cos(x + 45°) and g(x) = tanx
Determine:
(1)
(2)
The amplitude of f(x)
~ '-.1 r:}
The period of g(x)
180 o'-..J A
PageS of24
(1)
(1)
(3) Sketch the graphs of f(x) and g(x) for the interval x E [ -90°; 225°] on the (4) grid below. '-...l s t,~ Pt
cos
I ___ ---- -----'-1---ql'V\p--1,+~ £ --- ---- -~
~--
,-- -
_! -- -
(4)
+a" -......., shr:rpe
1- - ---- -- - -
, '-J crs0mpf~t-e.a ~
- ---~1- ----- _ __ J
~2 .. -- -- ------~- - -
- -- - - - --- - - -- - 1 t --- +--- ------- - - - - - _ __. ________ ---- - - - -- ~
' 1
L __ L ___ __ __ _ _ ------ ~ --- --- ____ __ i
'l.l
Indicate, using thickened lines, the intervals on the horizontal (x)-axis where
f(x) ~ g(x) for the interval x E (-90°; 225°). (2)
'-....J 1",._/-eyy~ls
-.....; dc:JI-S
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Grade 12 Mathematics Paper 2 Paqe6of24
QUESTION 3 [23)
(a) Given b. ABC with vertices A( -5; 3), B( -2; -2) and C(2; 1).
A( -5; 3)
C(2;1)
B(-2; -2)
Determine:
(1) The length of AC, in simplest surd form. (2)
Ac ~ I (2- f s ) ) z .,. ( ' - 3 ) 2.
:= J 53 'v fl J
(2) The midpoint of AC. (2)
e:-s ;~ 3+1 j I
~
:::: ( -; ;l ) J ,.
~ -........!
/.1
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Grade12 Mathematics Paper 2 Page 7o(24
(3) The equation ofthe straight line passing through C, parallel to AB.
-2-~; ( ) mFIB = ~'J: (- J cz,. I
(3)
5
'j- y, = ,q rn (x - :x,) ~ "Y) ' """-1
j
(4)
.
(5)
- I = -s 3 (:x_ - z)
- .§. X. I :=. 3 T- 4-3 '-.......J f} J - .2. X- ~
J - 0 + 3
The size of sAc. +cu"'l G -
. - 3
G - "' - 5Cf o3 -+ tsro· I
=
- BAc • • ::: /bY. os - /20 9b
• ::: '+3. 0? '-...1 /Y)
The co-ordinate of D if ABCD is a parallelogram.
'-..1 fl -.....J A
I '
(5)
f - .'1 :t - (-s)
e Cl tl = - J s 9 s -t /rto I
= lblf 05 0
(2)
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Grade12 Mathematics Paper 2 Page 8 o(24
(b) Sketched is the circle with the equation x 2 + 12x + y 2- 4y = - 15 passing through point
( -2; 5) .
Determine the centre and the length of the radius ofthe circle. (5)
x"l_ + 12x. t (b)-z. + Cf't._ 4-J r fz)'l. = -Js + {b)'t.-r(-zJ ( '-..J , J~ ( "A 'l.. ~!} x t b r l/ - -z ) = c25
(1)
cJ ,
r=
(2) Determine the equation ofthe t angent to t he circle passing t hrough point ( - 2; 5) .(4) 5 - Z q. 'v
m,qcJ, u-o = -2- &6J mfcmg~n+ :: - 3 , 3 ~,q
5
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Gradel2 Mathematics Paper 2 Paqe9of24
QUESTION 4 {18]
(a) Prove the theorem that the angle between a tangent to a circle and a chord drawn from the point of contact is equal to an angle in the alternate segment. (S)
F
D
A c Given: Tan(J-t n r 1913 G ~11 Required to prove: ~2: ~ Proof :
~ --\ 0 -......JA f oYI _j_ !3 l + IB :z.. = 'fo rCi ciiU<J
• • E, + £z. : 9o ~A L ,_ r~mJ Ctr cle
E, = 8 l ~.,q L5 IY\ S Q'm e. s~(} h1 en?-
8 .: £ ~ 11
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Grade12 Mathematics Paper 2 Paqe10of24
(b) In the diagram below 0 is the centre of the circle and sfp = 90°. If R = 43° calculate the sizes of each of the following angles. No reasons required.
R
A
(1) 01
"""" (1)
~ ;2 1<.. 0 !:.
.:::: Bb · '-JA
A
(2) s (1) A ....-'\ 0 A s ::: r<. 4-3 ~ =
e a /8'0 - 8'6
(1)
2..
.. "' A
A
(4) p3 (1) ..,.,
C!J 0 /\
f3. :: /80 'l_o s = l./-7. ~ A
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Grade12 Mathematics Paper 2 Paqe11 of24
(c) Refer to the diagram below.
s RT is a tangent to the circle at T. Lines MN and NT are produced to RandS respectively, so that
RS = RT. Sand M are joined. Let T4 = x. Supply proper reasons.
(1) Give 3 other angles that are equal to x. (6)
Ty. = T, .=. X. """'-J
/\ A '-J y,, .:: m3 :. -:;c_
I ,.... ""' /'.
.:t ~ Ti -=- 5, + S:z.. :. ~ tsos 6 · f<S = RT I
(2) Prove that RSTM is a cyclic quadrilateral (3)
prov~c/ ahove..
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Grade 12 Mathematics Paper 2 Page 12o(24
SECTION B [75]
QUESTION 5 [10]
The all-t ime ran kings in World Cup tournaments of fifteen count ries are depicted below. It indicates t he matches played {MP) by that count ry and the amount of goals scored {GS).
u
..c ro ::J u ro c. ·;:: c E
C!J '+- >-'.j:j a:: "0 0 <( c ro
::J c C!J u "0 ro C!J ::J c - ~ ro u c ~ c "0 C!J '50 N Q) u "'tib c >- ·x ro 'Vi ..... C!J
t:lO t:lO ro ·- C!J ro C!J Ill ::J ro ::J
~ C!J .... ~ N c .... ro ~
0 ::J a ~ J_ .... co a::l u u UJ u... ..... 0.. 0::: ::::>
MP 70 36 97 29 33 59 54 80 49 31 37 9 56 46 47
GS 123 46 210 34 47 77 96 126 52 44 64 11 88 74 76
The scatter plot representing the relationship between t hese two sets of data is show below.
220 y
210
200
190
180 v / 170 / 160
/ v 150
./ 140 L~-'
130 / 120 .L_'r"'"
110 .L_ 100 . /'r"'" 90 L_ 80 .y . 70 / . 60 / . 50 y • 40
/-30 / 20 /" 10
" / ........ 10 20 30 40 50 60 70 80 90 100
IY7QA·cheo 12.lav.e d . • (J
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Grade 12 Mathematics Paper 2 Paqe13of24
(a) Determine the equation of the regression line, give your answer correct to four decimal places. (2)
(b) Discuss with reasons the correlation of this data. (2}
(c) Germany has played in 99 world cup matches, predict the number of goals they should have scored. (1}
= /8 ~ b8
/83 (/oCL/s '-.1 "'
(d) Is your prediction in (c) considered interpolation or extrapolation? Explain. (2} , '-.1"1
Tnferpo/a)-to~
(e) Sketch the line of best fit on the given scatterplot showing at least 2 significant points. (3)
..........., 19 ';;{ ;,,~
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Grade 12 Mathematics Paper 2 Page 14of24
QUESTION 6 [17]
P(4; 5), Q(2; -1) and R(10; 3) are given points in the figure alongside. QP is produced to T and TR j_ QR. The circle with centre P passes through K, the midpoint of PQ.
y
R(10; 3)
Q(2; -1)
(a) Determine: (1) the co-ordinates of K. (2)
;:: ( Y:.~Z. _, rs ) z_ I '-
' ........,,q -......J~ jC_ (3 2)
(2) the equation of the circle. (3)
(x- q )~ + (Cf- b)~ 'Z 'J19 - r
{:x_- 4)'L + (:!- 5 )~ =: r7.
Subs (=r ,· z)
(3 - Lf-)L + (2- s) ~ rz '-J, -/0 - r~ '-...1~
{JL - 4- ) L T 0 -5 ) z.. - I 0
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Grade12
(3}
c!i- y,
J -s
Mathematics Paper 2
the equation of the line QP . -1 - 5
= 3 '-.J 19
- IY1 (z. -X 1 )
:. 3 (:L - 4 ) ~
= 3X.. -7 -......J,<j
11?
Page15o(24
(3}
(b) Show by calculation that T is the point (6; 11) if the equation of TR is y = -2x + 23.
(c)
</ = - ZX -!- Z 3 <!:/ ;: 3 X - 7 (4} or ~
'-....;
3X -7 '=- -2 J(. + 23 [;X- .::. 30
.):__ :. b ...._{:JJ
(/ =- 3 (b) - 7 ::. 1/ '-J IYJ
l ( b ; I/) '-.1 Ct:}
Determine (in surd form) the length of the tangent TF .
FT-z...::: PT-z.- fF' . 11 ...........,; rry
= ·(liQ) T_ ( iio ) z. 4=0 - IO
Fr = 130
(5}
PT =- /(lf-h)'l+ &-n)' = JiiQ ~~
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Gradel2 Mathematics Paper 2 Paqel6o(24
QUESTION 7 [24]
(a)
(b)
2+cos{3-cos2{3 Simplify:
3sin{3-sin2B '-' 19 (4)
= ;l + CoO f3 - (;z CcY:;t./3 - I )
(, t- cao(3 ¥".3 - :z coo 8) '-....J4 = ~~--~~----~~========~~-----------------------------5,;,6 (3- ::z ca-::;B) ~ 19
{, + cm8) ---~~~~~,~~---------------------------------
sin(3
(1) Determine the genera l solution of:
cos 30° cos 2x - sin 30° sin 2x = cos 3x (5)
D 30 + ZJ<_ t. 3X. + /c · 3b0° ' k C. 2
t9
30 ° + zx ~ 3 )(_ + k. . 3 b 0 ° • -X.. ::: -30
X. '-./ :::. 30° I'll
30 + 2X.. -= - '3 X.. t- J.: '3bO •
fiX... = -3o· t 1:: • 5bo·
:x-~ : - ~ • T /: • 7:Z 0
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Grade12 Mathematics Paper 2
(c)
(2) Hence determine x if x E [-90°; 90°]
)t. = • 0
-b . 30
Prove that:
L.HS =
=
sin4x
sinx 8cos3 x - 4cosx
5JY1 z (z x_) s ,, X.
'-1- CO".? X ( 2 CoO ~X. - 1 ) ;>
Paqe17of24
{2)
{5)
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Grade12 Mathematics Paper 2
(d) Christ the Redeemer is a statue of Jesus Christ in Rio de Janeiro, Brazil; and was considered the largest Art Deco statue in the world from 1931 until 2010.
It is located at the peak of the Corcovado mountain overlooking the city. A symbol of Brazilian Christianity, the statue has become an icon for Rio de Janeiro and Brazil.
As part of the build up to the soccer world cup, Messi and Ronalda decided to have a competition to see who could kick a soccer ball into the hands of the statue. Both players got to choose where they would kick the ball from. The diagram below is an illustration of their positions.
Page 18of24
The base of the statue, Messi (A) and Ronalda (C) are all in the same horizontal plane. The angle of elevation from Messi (A) to the left hand (D) of the statue is a. The angle of elevation from Ronalda (C) to the left hand (D) of the statue is (90° -a). The distance the ball will travel from Messi (A) to the left hand is Zx metres and from Ronalda (C) to
the left hand is x metres. ABC= 120°.
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Grade12 Mathematics Paper 2 Paqe19o(24
{1) Show that the distance between Messi (A) and B is given by
{2)
AB = 2x cos a and that the distance between Ronaldo (C) and B is given by
BC = x sin a.
Tn 61i8D :Tn 6 BeD
A8 Be
{2)
"' .;1 X c co o( """" 19 t3 G = JC. s I,.; o<. "'.J 19
---------------=~--------~----~-----
Show that the distance between Messi and Ronaldo is given by
AC = x-../1 + 3 cos2 a+ sin 2a. '-.....}roy ::,41~ ~ Sqhs ~ {6)
'2.
f-JC::
+
+ ;J Stnc< coc:;o<.)
I - c rY.J rz. o<.
A6 ___ -__ ~Jc~!~~'~+~3~c~ro-~~~--~t~S~'n_·~~~~~--------------
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Grade12 Mathematics Paper 2 Paqe20of24
QUESTION 8 [6]
Below is the pyramid of The Pyramid of King Khafra, at Giza. It is made out of a square base and 4
equilateral triangles. E
(a)
(b)
Show that the height of the pyramid is 162,6 m.
Ac"Z.. ;;230 ~
230~ ::: r ::: /05 8'00
Ac = 325 27 '-.....1 19 I
AF = FC = I b "Z. 1:,3
EF :23 0 "l.. - /b"Z 63
- :1. b '+51 4-8
E/- = j/,2, faY. ~ A .
El- = /b "2 ~ ""'-....J A
Determine the volume of the pyramid if:
'-......J IYJ
~
V = ~(Area of base) x Height. 3 11]
'-...j
v ~ ~.3 (2 3 0 X 2 3 0 ) )' I b z I b
(4)
(2)
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Grade 12 Mathematics Paper 2 Paqe21 o(24
QUESTION 9 [6)
(a) Complete the following sentence:
If a line cuts two sides of a triangle proportionally, then ...... (1}
-thd j,,;-e // fo fn~
(b) In the figure K is a point on AB, such that AK: KB = 3 : 2. Lis a point on AD such that
AL : LD = 3 : 1. KX = XB.
B
c
Prove that: BD 1 -= -BC 2
A (5}
AK 3
K./3 :2..
htd K.X = XB (/1ven
fJK .3 '-......./ kX J
Jocd AL 3 #' v~n ~
LD I
,. 17 }(. AL "-.....J 3 • I
K..X LD I
K.L LL XD '-......; ''"~ dl~.f,dQLJ '~ild.e,o I"" I" OJ), • I ..-,
BD BX -..........; J /,F..L II fo on e. s;ol~ ~tJ . '
DC Xh I
BD I '-...J -Be ;t
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Grade12 Mathematics Paper 2 Paqe22o(24
QUESTION 10 [12]
Refer to the diagram below.
Two circles intersect at A and B. Chords AS and BS of the larger ci rcle meet the smaller circle at D and C respectively. CD produced meets the larger circle at T. AT and TS are joined.
Prove that ~ + T2 = D4 (3)
d /80 0
+ ::; 0
0 + ..-'\ B = !8"00
A..
T. + T = D
hr.A.f /\
'"""' ~ D. ~ D'f. r
-"'\ '/)!.!-r, + Tz =
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LAC/d. •
~cu:J. .
Grade 12 Mathematics Paper 2 Paqe 23 af24
Qb} Prove !J.STD Ill !J.SAT {3}
Tn !::. 5rD one/ 6 5fJT
') " ;"\
S' , ::. s , '-1 ("om mon i
"' z.J """ """ '[)'+ :. Ti + T? ~ Pro v f d " A I
3) . 7'l. = A, /1"1 f- L. a-6 A .. . .
STD 1!1 SAT "-.J AAA • • 4 6
(c) Prove tJ. SDC Ill tJ. SBA (3}
In 6 50C- CIY\ d !::.. 5'8/-)
1) "" /\ s:2. ::It 5 :z... ~ Commo--
z.) A "' Lc5 c/1.. ch c 2 u t:f c/ 0.'3 :::: B ............ l e~f
"" A
3) C:z.. ... f1'2.. tnf L ~ t:J
. . l::. SDC Ill 6 58 A ~ AA!i
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Grade 12 Mathematics Paper 2
( cJ) TS2 = BS. CS
~ SID II/ £3 SAT
ST SD SA ST
. '. rs oz. = so · sA "'-v
A SDC 1/1 6 SBA SD sc 56
- SO· SA = 5C · SB'-......1
' . . Ts'- -= sc · S(3
= cs . 85 "'-...1
Paqe24of24
{3)