CORNWALL HILL COLLEGE

Mathematics Department

EXAMINATIONS- 21 JULY 2014

Mathematics- Paper 1

Grade: 12 Time: 3 Hours

Marks: 150

Examiner: Mrs. S. Hickling

Moderator(s): Mrs. M. van Niekerk, Mrs. T. Knoetze and Mr. P. van Schalkwyk

Instructions:

1. The question paper consists of 9 pages and 12 Questions. Please check that your paper

is complete.

2. Read the questions carefully.

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

4. All the necessary working details must be clearly shown.

5. Approved calculators may be used except where otherwise stated.

6. Answers to be rounded off to 2 decimal digits where necessary.

7. GOOD LUCK!

Grade 12 Mathematics - Paper 1 21 Julv2014 Page 2 of9

SECTION A [77 MARKS]

QUESTION 1 [18]

a) Solve for x:

2 I - -

(1} (x 3 - 4)(x 3 + 2) = 0

{2) x 2 + 1

~0 X+ 1

{3} 22X+ I _ 3(22x - l) + 4x = 12

(4} x 2 - 4x - 1 = 2

x 2 -4x- 2

b) There are four solutions to the equation (x2- 5)(2x 2

- 9x + 9) = 0

Solve for x if:

1} X E Z

2) X E Ql

c) The roots of a quadratic equation are 3 ± .J12 - 3a 2 • Determine the value(s) of a

for which the roots will be equal.

QUESTION 2 [16]

a) Consider the number sequence 2; 5; 2; 9; 2; 13; 2; 17; ...

(1} Write down the next two terms of the sequence, given that the pattern

continues.

(2} Calculate the sum of the first 100 terms of the sequence.

b) 3 645 135

The seventh term of a geometric sequence is and the fourth term is 64 8'

determine the first term .

(2}

{1}

(4}

(5}

(2)

{2)

(2)

(1)

(4)

(4)

Grade 12 Mathematics - Paper 1 21 Ju/y2014 Page3 of9

c) The sum of 5 + 15 + 45 + ... to n terms is 605. Determine the value of n. {3)

d) Show that :

2k- 3

5- 2k

QUESTION 3 [27)

a) Given: 4

f(x) = -- + 1 X

- -- - - ---~-------------- -

~----6------------~--------~~------~X 0

f

{1) Give the equation of the asymptote through point B.

{2) Determine the coordinates of point A, the point where the straight line

g (x ) = - x + 4 intercepts the hyperbola.

(3) Calculate the length of AB when B is the reflection of point A in the line y = - x .

{4)

{1)

{4)

Leave the answer in surd form . (3)

{4) Give the equation of h if h is the translation of f by two units to the left. {1}

(5} Hence give an equation for any of the axis of symmetry for h. (1}

(6) Determine the values of x for which f(x) ~ g (x) (2)

Grade 12 Mathematics - Paper 1 21 July 2014

b) The sketch shows f(x) = ax2 + bx + c and g(x) =e . Both pass through the point

(1; 3).

(1) Determine the values of a, b and c.

(2) Determine the value of k .

(3) Determine the range of f (x ).

(4) Consider the graph of f(x).

(i) Explain why .f(x) does not have an inverse.

(ii) What are the values of xfor which f(x) will have an inverse?

c) The graph of f(x) = log 1 x and h (x) = -3 are given below: 3

y

X

h

Page 4 of9

(4)

(1)

(2)

(1)

(1)

Grade 12 Mathematics - Paper 1 21 July 2014

{1} Determine the coordinates of:

{i} A

{ii} B

{2} Use the graph to solve for x if log 1 x ~ - 3 3

Page 5 of9

{1)

{1)

{2)

{3) Determine the equation of f -1 (x) . {1)

{4) Determine the new equation of f (x) I if j(x) is reflected about the x-axis. {1)

QUESTION 4 [10]

a) Given : f(x) = x 2 - 3x I find f 1 (x) from first principles.

b) Find dy in each of the following: dx

{1) r 4 3 y = 2....; x - -x· + 7r 9

(2) 1

8x 3 - 2xy + y - 1 = 0 ; x # -

2

QUESTION 5 [6]

A survey was conducted among the 640 soccer

players at the Soccer World Cup about their preferences

for Adidas1 Nike and Puma soccer boots.

The findings were:

196 prefer Adidas -.....A ... 256 prefer Nike

284 prefer Puma

36 prefer all three types

60 prefer Puma and Adidas

43 prefer Nike and Puma

.............. adidas

64 do not prefer any of these three types

x prefer Nike and Adidas but not Puma.

Draw a Venn diagram to summarize the information and determine

how many soccer players prefer Nike and Adidas but not Puma.

{4)

{2)

{4)

~ (6)

Grade 12 Mathematics - Paper 1 21 Julv2014 Page 6 of9

SECTION B [73 MARKS]

QUESTION 6 [8]

a) (1) Factorise: a 2 + Sa + 6 3 I -1 - - -

a 2 + 5a 2 + 6a 2

a 2 - 4 (2) Hence, simplify the following:

b) Find the value(s) of k for which the nature of the roots of the following equations will

be the same:

x 2 - Sx + 7 = 0 and kx 2 + kx + 2 = 0

QUESTION 7 [14)

A new car currently costs R250 000.

a) The value of the vehicle depreciates at 18 % p.a. on a reducing balance basis.

In which year will the car be valued at 30% of its original value?

b) What will a new car cost in 5 years time if inflation is calculated at 8% p.a.?

c) A man wants to start saving so that he can afford to buy a car in 5 years' time.

He opens a savings account with an interest rate of 9,5% p.a. compounded monthly.

What must his monthly payments into the savings account be, so that he can afford

to buy the car in 5 years time?

(1)

(3)

(4)

(5)

(2)

(4)

d) Calculate the effective annual interest rate as a percentage and correct to 2 decimal

places for the interest rate in 7 c). (3)

Grade 12 Mathematics - Paper 1 21 July 2014

QUESTION 8 [16]

The shown below is defined by the equation: f(x) = x 3 + ax2 + bx + 12. The point

S (1; O) is an x- intercept and T (4; -36) and Q are stationary points.

T (4;- 36)

a) Show that a = - 5 and b = -8.

b) Find the co-ordinates of Q.

c) Find the co-ordinates of P and U.

d) Determine the equation of a tangent to j(x) at point P.

QUESTION 9 [8]

Page 7of9

(7)

(4)

(2)

(3)

Two particles are projected simultaneously towards each other from the opposite ends of a

straight tube, 1 000 em long. Particle A travels 51 em in the first second, 49 em in the second

second, 47 em in the third second, etc. Particle B travels 25 em in the first second, 24 em in the

second second, 23 em in the third second, etc.

1000 em

a) How far does particle A travel in t seconds?

b) Determine how long it takes the particles to meet.

(3)

(5)

Grade 12 Mathematics- Paper 1 21 July 2014

QUESTION 10 [10]

a) If the letters of the word SOCCER are arranged

randomly into a "word",

(1) How many 6 letter words can be formed?

(2) What is the probability that the letters CC

are next to each other?

b) In the Soccer World Cup the players are being

tested for the use of illegal substances. Of the

640 soccer players participating in the World

Cup 50 are using an illegal substance.

A test for an illegal substance is 96% accurate

if the substance is present in the bloodstream

and 99 %accurate when it is not in the

bloodstream. What % of the soccer players

will test positive for an illegal substance?

(Make use of a tree diagram to help you).

qUESTION 11 [10]

A soccer ball is kicked in the air. Its height h in metres,

t seconds after it has been kicked, is given by the

1 formula h=20(t 2 --t 3

). 3

a) What is the velocity of the ball after 1,5 seconds?

b) For what value of t is the velocity a maximum?

FIFA WORLD CUP 6 rASit

c) For what value of t is the height of the ball a maximum? Calculate the

greatest height from the ground.

Page 8 of9

(2)

(3)

(5)

(3)

(3)

(4)

Grade 12 Mathematics - Paper 1 21 July 2014

QUESTION 12 [7]

Given : 1

f(x) = - . The tangent at D where x = a is drawn. X

B

f

0 a c

a) Show that the equation of the tangent at D is x + a 2 y = 2a.

b) Calculate the area of ~ OBC.

Page 9 of9

(4)

(3)

(:x_ ~ ){x_ /,-3 2 ) 4- -t - 0

7,--3 ?.3 = -2 2 - 4- CJy-' X

a) (,)

+ 8 ~

.l:__ '-..! X- = = - B'

(z.) X-z.. r I

{z.)

)(. + I $ 0

JC-tt <. 0

(t)

(3) 2.X. + I 3 (z 2X.-,) JL ;z -t 4- - /2...

2x.. r' 2X -1 ;).zx.. ::2. - 3 • 2 + - /2

'-....; :<2x..(z' ~ -I ) - 3X z.. 1- l =- /2

;). 'Z.)( (t fz.) - /2 2

zx = ~

ZX- = 3 . . x_ - .3/'2.

2. {~.~-)

4 Jc.. - I -

- X..'- - 4- JC

t. - I ~

t."Z. - 31c- f2... -t, (fc- - 3) -:: 0

..:xz.-4X. - 0 '-..J or z ..X:. - 4X - 3 = 0

X (JC_ - 4 ) ~

X ::: Cl or

::: 0

X= 4

4- :t I ~s

:= 4- b5 I or -o ~5

I

(s)

b) (x. *'-- - s-)(zx~ ·- Cf x -1- <=t) - a

S )(zx. ~

3) (x, "t. - - 3 )(;._ - - a

.)L :::. t:.{S or .}(_ = ~ "2.... or X.== 3

ct ) .X.. =- 3"-...J C-z.)

(z.) X - + rs y ~ ~

(~) 2, 2. 4- .

c) ~ - 0 'Z.. '-...J

~~ - 3 q - 0

4- - Q'Z.

+ ~ '-....J ('-) - a...

a} (i) ::z I

2/ ~ c,)

~

(z) s,oo = 2 :X so + 50 [z 2. 2 X 5 + 4-~ (4- )]

-......; - )OCJ + St5o ~

- 5:1.50 C4) - ~

arl:J== 3/:,lfS

Q) 3 i3S b) b4- ar ::::

8 @ 3b4-5

ar~ "'-..J bl+

ar 3 -135

-.........; g-

r3 ;J.7 - -

8

r ~= 3 ; T . .

SUbs ,,fo (]) : (~)3 }35 a =

~ -....;

(4-) 0- :: 5

c)

o)

s + IS + ~5 + ... .

'-.....! 5 (3 11 - I ) '-....;

b05 = 3 - I

;J '+ 2.. = 3" - I

24-3 .::: 3" 35 = 311

'-......J , .. n :::: 5

~ (;:- ~~)11 = (;: --%.) , .... ,

.::;

Q v.v:rf 1 o IV 3

(, ) d :;::.

(~:- - 3~) "-.!

I - (I:- 3/~)

;;J:_ - 3

z.

;2/:..-3 5 - "Ak..

"'-...J

C"2- 7.1

( '-..J

'-....J (z) -4-

.x_ -1-( - - X. -r 4-"l:..

-4- r- x = -x rt..;..x.. -z. ~ x. -3x -4- -a

( )( - 4- X)(_ T I ) - 0

k. -=. 4 or J(.

J

5 11 = bos

(3)

3

f- (.c- ~) + .. .

c lJ.)

(t)

~ ::: -- I

-:::: - (-I) t '1--::. 5

(t~)

(3) A (-t, · s) /3 (-?; I)

A/3 = / (-; .,. 5) -z. -r (s - 1 )t. "--'

= 4-12 ~ (3)

-4 '-....! (,) (4-) ;, (x.) - + ( - :X..+Z..

(s) cJ .s Gc.+-z) r 1 or :! = - (x. t 'Z.) ..,. J

::..J:t3 ~ (I) =-..l:_ - I

~ L4- ;~ ) (t:) [-,,. 0) (c..)

~ b) (J) :J ~ a (x - zXx. - lf)

"S CAioS (o ,· B)

8' - a (o - z ¥o - 4)

( - {A;~

. J J (x_ - -z ){X - 4-) .. ;::.

-.::. 2.. X, - b)(_ t- 8 -........;

a := J h ~ -b c .;:. g ~ Ct+) . .

(z) fiuhs (, '3) ,,.,fo : /f ::kx.

I

3 ::: /<. I """-J (u

(3) . 'Z...

x.:::. 3 subs Jnfo fl -=- :X - b J(_ r 8

-::.{3)'2--to(3) t 8' "-..1

=. - I ~

. . g ~ -I c~J

(If.) (t) r· _,(x) '-......1 I . - on~ fc:J mqr;(/ ver+ / c a) I,,., e.- no

I

(1) fu11 c hc

(z) '-....1 ~ 3 X?- 3 or X.

Ct)

c) (J) (j) A(,/ o) '-....J (,)

Qi) B(;J.I,' -3)~ ~ :::: fay~. X

~ c ,) -3 ::::. /o(/"3 X

.. ..):_ :::. c~3)-3

'-....J ~ :::. 27

(z) (o ,· z) (-z)

(3) r -r (-x_) ==

(~3 )X. '-....J (,)

(tr) -!(x) = - Jc;J I. x_ v..3 (I) loy3 X ~

-

Q v..sv:d 1 o JIV 4- Go]

a) r 1 !': ) ;,,.:, l:t. :: n-7 o

_. /ovl - h-7o

h

- zx. -3 -.........; (4-)

h) (,) J = 2. {:x_ - ~ ):3 +-IT ~'z.. ~Cf x...3 -= ;;;X - + 7T

;&__ '-.../ /. ~ (zj

--z_ .!L x.'Z. dx... = L - 3

c~) .3

8..X. - zd -f-)j -I := 0

21 - -;J:Xd - I - 8 .)(_...3

~:! (, - 2X) 3 - I - 8X "-...J (i - 2.l:: 2(/ 7 :LX. f ~X~) c:f -( 1- 2x)

:J - I + '2 ):_ f-~ "-....J

4-X

~Jic_ "'-.;

(4-) - 2 t g;x_

b4-0

/3fo - X + X. t :2/3 - X + 3 6 + 7 1- Z lf. -t '217 + b lf

+- b 17

57 = .X..

a)

b)

a-z. + !3a. t- b '-......J

= (a + 3 )( q -f z. )

(z.)

= c;; ~?, c ct t 3 J '-....J

(a..- z)

X-7... -SX+ 7 =o

A = h -z. - '+q c.

~ (: 5 ) 7.. -q. (, )(7 ) ~ -3 '-......;

Roofs no 11- rt. a.)

, . . < 0

1:: X.~ + i..x. -t 2 == 0

t1 ~ f-7..- (4-Y!c)z. ~ ~~ - 8.1:.

'-...../

( 4-)

0UJVJf /0 Yt.; 1 a~_]

a) A - p (, - G )n ~ '-.....J (t o 1 t8 )

11 75 000 - ';2.5'0 000 -

75ooo -zsoooo

- (o,8z.)""

"'-...J 1!5

- / ~ o,8''Z. 2.5 0

'-.....J n - fo,Ob6'6 .. ~ .

(s)

b) A - P( 1 -f c ), ""-..1

- ;;zsoooo (t + o, 0 g) .5

:= 1<. 367 332. 0:(. ~ (~) )

'-...J

c) '-......1 ~(.[(; +

0 1 0 'fS ) &o - I] 3b7 33Z1

oZ.. - t-z -o 1 oC(S

JZ.

'-.....; ~ (q,) .)(_ ..- f<. 4 a--a6 6 I - J

d) I + L - {t + ·i )~ . '-.....)

(, a,ocrs- )'z. -t (..) -;:. -+ rz ..... • (, oocrs-)'7... u - -1- - I 1<-

~ . u -;:. o, 0 'f Cf'Zlf .•••

~ (3) ~ 1 'G % p. a..

a)

b)

5 uhs

""""' 'Z. m = 3X +~q)c. +b

= 3(4-)z. t ~a..(y.) 7- b 0 '-.......~ ::. 4-8' + ~a.. -t b

( , . 0) : J

'-.....1 0 -

- 13 '-...] :=.

I -t a. a + b

. . -lj.<;r - g-CL = -{3 -a..

-7 o.._ = 35 a_ '-...../ = -s

5ubs Jn t C!J @ :

-J3 -= -5 +b '-.....;

-8 -= b

3 "2. X -5x. -&x. + /2. .

-z. "-...1

""" 3X. - /0 )(._ - 8' - 0

(3 }(_ 1- 2. )(x. lf. ) - a '-........J

(!)

-th + IZ

----®

(7)

.X -~3 X 4 =. cr -

G) J:.. 3

- 5 JC 2

- ~ X -t I "2... ::= d

(jC. - I ) ( JC ~ - '+ J C... - I 'Z. ) := 0

{ x. - 1 ) ( :x_ - tD )(:x- + z ) = o

~ .·. P{-2./ a)

}(_ ::= .... 2..

(zj

d) m fo,., -=

==

-

tJ - 0 -

J '-......; -

. Q UJZ/.J f 1 oYLJ 9

A: 8:

ff!

~5 .

I

~

3JG - /0 J( - g'

3 (-?.. )-z_ - /0 (-z,) - 8

2 4- "'-..1

;llf Gc.. + ~

z)

2. 4-:JC -t 4-8. (3)

[8']

4'~ . J

4-- 7 . . .. I

2.4 I

23 . J • • •

S""n - ~ {ia t- (n- ,) c!] ~ ~

a) St :::: ~ {'-x5 t + (t- 1 )(-G)]

= . ;_ (io~ - ;l.t -t 2 J e fi'LC - 6 '2. '-.....1

- ; [so - 6 t- 1 ]

=

5?. t - c '2...

to4- t - zt-z.. a

fil t t}- ~ '2.. '2...

+ z. I ooo +-'-LJ = Zooo

'-.....J 15St + zooo

(3)

0 8'o Xt ;;2 5 )

G = ~o

- 2 6, {Q

1\JA.

. or- c = ~5 s '-........J (s)

Q Wl/J f /0/1.; Jo Cio]

q) (, ) b! '-J

Cz) ~ ~ '-.) - 3b0

('2.) '-......J ,

5 I ~ "--.J ;;z.. . (3) '-.....J 3bO

=- 3

T+ f T+

~ b) p

~ T- f T-

f I T-1-

cs)

a) h = ~a {t ~ - ~ t 3) = ;}a t ~ - ~0 t 3

dh '-.......; de ~ 40 6 - ::zotz.

= lfCJ (t,s )'-J - :20 (r, s)'2.. IS rn/s ~

b)

G)

b)

~ - 0

4-o = 4-o -t I '-.....1 = t

~ ~ 4-0 c - ~ 0 t '- -

;2.0 t. ( 2 - t ) -0

0

t ::: 0

NA or- '-......J t - 2.

. ·. A ~o (z~ - ~ (~)3 ) = ;;2 b t m .'-........J

'-...; .'. D(a/ Ya.)

-z m = -1 x

'-..J

J - ~ -CJ.-.

aJ_ a. -

-qJ q_

'-...) aJ -t )(_ -

Jnfo

- I ( 0'2. )C.. -

- t (:x_ - q) - Jc._ -t-a..

za..

a)

(4)

qz(o) -t )(. =- za... X.. = 2a_

Qr-€a.- ::: ~z (zq )(~) !:= 2.. ~

(lf)