Mathematics Department EXAMINATIONS-9 NOVEMBER ...maths.stithian.com/Grade 11 Papers/Cornwall...

CORNWALL HILL COLLEGE Mathematics Department

EXAMINATIONS- 9 NOVEMBER 2018

Mathematics - Paper 1

Grade: 11 Time: 3 Hours Marks: 150

Examiner: Mrs. S. Hickling Moderator(s): Mrs. A. Heuer, Mrs. F. van der Merwe and Mrs. T. Meintjies

NAME:

TEACHER:

Instructions:

1. This question paper consists of 27 pages. Please ensure that your paper is complete.

3, 25 and 26 is for any additional working and page 27 is the formula-sheet.

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 necessary working details must be clearly shown.

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

(150)

Grade 11 Mathematics Paper 1

SECTION A [78 MARKS]

QUESTION 1 [20]

a) Solve for x:

(1) 2 + x- x 2 = 0

.t.'"'-.x..-"2.:: 0

( JL - "Z )(:X.. + } ) ::: 0

~ '-...J )<..-::.. Z. or J<-=-1

(2) 2x2 - 4x > 0

(3)

;;x_{x_- :2.) _:::,,... 0 "'

,_ \ I+

4- V2- x = 3x ..--~

-/:z.-~ = 3X-- 4-

;z - JL.

0 - :23X.. + ll.f-

0 = Cx -X - o,-

1 Yr x. - 111- ) 1~/

.?t = /? NA ~

Page2o(27

(2)

(2)

(4)

Grade 11 Mathematics Paper 1 o(27

b) Solve for x, in terms of h, if 2x 2 - 7x - h = 0

7 + !teCJ r f!h

c) If (x- .JS)(zx - 8) = 0

d)

( 1 ) Solve for x if: x E Z

}(_ ~3

(2) Solve for x if: x E Q I

1- ~ )(. - -.5

20 Given p is one of the roots of x + 4p = determine the possible value(s)

X

of p.

= 0 + 4p I

I 'i t:J 'Z. ;zo

I '-......} -

(2)

(2)

(1)

(3)

Gradel1 Mathematics Paper 1

e) Solve for x and y: 3x + y = -1

9x2 = yz

g ~ -I - 3JC. ~

9X"L """-J 2.

= (-I - 3X)

rx~ - I + 6.X.. + ?X.

6Jt. - -!

)(_ - - Y. '-......J ~

-; I

- -! -+ ~

- - ~2. ~

QUESTION 2 [1 0]

a) Solve for x: s-x.sx-2 = 252X 5

'-.....! -)C. ... x.- :L ( 5 ~)~X. . 5 -

-l... II- :>C..- I

5 - 5

4- JC... - I :. - 2.

4-X =- -I

l..

Paqe4of27

(4)

(4)

_, 5


b) ( 1) 3 = 3 (3)

LHS + 3

v3-' - / 'i: + 3

3

RHS

(2) Hence solve for x if j3x+l _ 3x (l)X-2 ---+3=-3X-1 3 (3)

-.X. + '2.. - I

-.x.. ::.

~ .X. -


QUESTION 3 [6]

a) Give the next term in each of the following sequences:

1 2 4 8 _1!!._ "-../

(1) . - . lfo 4 , 7 I 10 I 13 I

xz x 3 x.."" '-.....J (2) x; . . ~7

3 ,

9 ,

b) A horizontal line intersects a piece of string at four points and divides it into five different parts as show below.

The piece of string is intersected in this way by more parallel lines, each of which intersects the string at four points.

Showing all working, determine the number of parts into which the string will be divided when 2014 parallel lines intersect the string in the same way as shown

(1)

(1)

above. (4)

s · 13 . .. .

'-...1 - q. (z 0 I I+ ) + I


QUESTION 4 [8]

A survey was carried out among 1 00 learners about which movies they have watched recently. The results are given below:

• 43 watched Venom (V) • 41 watched Halloween (H) • 50 watched Papillon (P) • 6 watched all 3 movies • 7 watched Venom and Papillon but not Halloween

• 18 watched Halloween and Pap:illon

• 15 watched only Halloween

The above information is represented in the Venn diagram below.

e

a) Write down the values of a, b, c, d and e in the Venn diagram above.

a. -:. 12 b ~ 8 c. ~ z z "'-.....; (5)

b) Calculate the probability that a learner selected at random from this group has not

watched Venom or Halloween or Papillon. (1)

5 :: _!_ '-.....} 100 ilO

Gradell Mathematics Paper 1 Paqe8o(27

c) Calculate the probability that a learner selected at random from this group has

watched at least 2 of these movies.

7+b~ f/+12. /00

QUESTION 5 [12]

33 /00

Consider the sketch below showing the graphs of:

k g(x) =-+ q and f(x) = zx-m + n, with asymptotes at x = 1 and y = -2:

x-p

f(x ) ---!..~"-!"!!'--~-~-~--~-~--=-------- ----.... ------------------------·

I - 2 I I

a) Determine the equation of g(x) by finding the values for k, p and q. "'-..., "-..J

p-I cr=-~

- :2.. d - X-! lc

-:1.. 5ulo 5 (z.,· & ) ' ' = 2 _,

. . q(x) 8

-~ u

X.. _,

(2)

(3)

Grode 11 Mathematics Paper 1 o(27

b) Determine the equation of f(x) . (3)

X.-1"17 ~ d-:. ;z -;L 2 _,

5ulos (-z. , · b ) : '-......~ l, - J. - 2.

8' 'Z.- ,..,

= :2

3 - z -m

m - - I "-.....;

c) Determine the equation of the axis of symmetry of g(x) which has a positive gradient. (2)

rn :::: 1

(j

Cl

d)

CD .

Untfs

e) Write down the range of f(x). (1)

J t--'c-:2 .. oO) u

Grade 11 Mathematics Paper 1 of27

QUESTION 6 [7] y

Given below is a sketch of y = bx .

a) Write down the coordinates of point P . (1)

b) Use the sketch of y = bx to help you determine which of the given graphs best fit

each of the following equations: ~"'

(1) y = 2bx ]) (2)

(2) y = bx -1 B ~~ (2)

(3) y = bx+3 £ ""''-.1 (2)

y y y

B c

y


QUESTION7 [15]

a) Rhino poaching is a serious problem, and has

resulted in rhinos becoming an endangered

species, particularly the black rhino.

Statistics suggest that there were 60 000 black

rhino in 1970 , but that only 4 200

remained in 2011.

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( 1) Determine the annual rate of depreciation as a percentage

that these statistics represent if the rate is compound decay, correct to 2

decimal digits. (4)

'-.....~ '-...J / lf 2. 0 0 - 6 0 0 OCJ U -

~J l/.1.-CJO bOOOO - I (..,

c, = I - 1#:/ ~ "00

L - 0 0(,2.8' .. . . -

r - b, zg l

(2) Using the answer calculated above, determine the number of black rhino there

will be in 2050 if this rate of compound decay continues. ~ ~ 80

(3)

A= 6oooo (/ o. Ob Zfl) ,

:; 33 '-t. 7¥-I

31 A = 1t zoo (, - a. o b z s )

I

:: 33'-t-. 7 4-,


b) When Pumi purchased her new car, the bank offered her a loan at a nominal

rate of 17 % p.a., compounded monthly. What was the effective annual interest

rate that she was required to pay?

I + L

"'-.......; o, 11 )rz. - (t -1- 12..

r ~ 18 3? l.

(3)

c) A school decides that they are going to invest R 60 000 at 12% p.a. compounded

monthly in order to put up floodlights on the Hockey AstroTurf. Three years later the

interest rate drops by 1 % and it was compounded quarterly. Four years after the

first investment, R 1 0 000 is withdrawn to buy rugby poles.

What is the total investment worth 13 years after the first investment was made? (5)

&oooo (/o ooo)

'-....~ A = boooo

I 'l.X J(, ;;;:, I+- - /0000 (, ..,..

Grode 11 Mathematics Paper 1 o(27


SECTION B [72 MARKS]

QUESTION 8 [14]

a) Solve for x: 18

x(x + 2) - 1 + 2 = 9 (6) x +2x+1

"Z. 1'8 x. +zx. -J + X~ 1' 'Z.X ~ I - 9

L~l- k : X t. + ZX

/c. 18 -I k 1- I = 9 . .

LeD : k. +I

k?.. I -1- ~~ -kt. - 'II< ..,.. 8 ~ == 0

0:. g t~c- I ) -- 0

~ k :::. 8 0,.- k. :::. I

X z + ZJt - 8'" - 0 - 'Z. X. -t ZX. - I ::: 0

{x + LJ- )(x - z ) - 0

:X = -4- or )(. - 2.

'-........J

X :: 0 Jrl cr J( ..:: ~2 lf-1


b) Solve for X, without using a calculator: zx+Z + 7-JZX = 2

X. 2 "-..~ ~%. J. X:l + 7x2 ~ - o

.X. ~

f.!. X~ +:. 7 X J. %. ;1. - 0 -u ~ ~t ~ + .:2. ) 0 '+ J1. ;z I 2

:Yoz. ~ ~ -J - ll- or :2 - 2 :y.~ -Z. NA ~ = :2

~ -= -z )L :. 4-

'-......;

c) Prove that the equation 6x2 + 2px- 3x- p = 0 has rational roots for all

rational values of p.

'l.. ~ ~ X -+ X (-zp - 3) - p = 0 ·

'f~ 'Z. - rzo f- 9 -

I , 'Z.

-:;. lrf' 7- !2p r ::: (zp 1- 3)~ '-....J

roofs ra . .f,-o,-, cJ.

(4)

(4)

Gradell Mathematics Paper 1 a(27

QUESTION 9 [14]

a) A farmer wishes to develop a tree plantation. He starts by building a mill and planting trees in a rectangular fashion around it, so that the mill will be the centre of the plantation. The mill takes the space of two trees. Each month he adds another outer shell of trees as demonstrated in the figures.

Determine:

(1) how many trees there are after 4 months. (1)

88"'

(2) how many trees there are after 5 months. ( 1)

/30~

(3) how many trees there are after n months. (4)

/0 2.8 54- 88 130

2.0. ::: 8' 3 Q. + h =- 18 a+- h + c -= 10

c. :::: 0

Grade11 Mathematics Paper 1

(4) how long it will take him to have a forest of 550 trees.

'2. 4-"

lfrl 'l... +

;JI1'Z. +

+ fan

hn -3n

(?.n + 25 X'n n ::. -~ 2.

NA

~ = SSo

550~ - 0

:275 0

- I I ) - 0

t1 - I I cr

, II rno,tJ,s.

o(27

(3)

b) A given quadratic pattern Tn = an2 + bn + c has T2 = 9; T4 = 39 and a

second difference of 4 . Determine the value of the 1st and 3rd terms of the pattern.

(5)

'----" ____. ____... cr-;t y-cr 3'-J

39 -'I- (Cj-Cf) ~ '-!-v u

-;;Jt{ - -11-l.!-0 "-........J -(} 22.

J- '1 - (1-X) - l.f.

22. -9 -C( f-X - ll-

)(. - 0~

.-,-,, :: 0

Grade 11 Mathematics Paper 1 Page 1Bof27

QUESTION 10 [19]

The graphs of f(x) = x 2 - 3x- 10 and g(x) = ax+ b are given below. A and B

are the x-intercepts, C is the y-intercept and D is the turning point of f. The graph

of g passes through B and C.

Find:

a)

b)

The coordinates of points B and C.

u .: X 'Z - ~X -I 0 J o =- {x. -sXx .,.-z,) ~

X=. 5 or )( :. - 2.

The coordinates of the turning point D.

-b X =- za....

~ - {-3)

I

(3)

(2)

-!o

- _lfk:


c) The values of a and b.

(5,· o) (o · -to) I

~

m :::..

d) The values of x for which:

(1) f(x) < g(x)

(2)

X E

f(x) > 0

g(x)-

'-.........1

5) '-......_;

(5 ' otJ) I '

::.

-10 -0

0 - s

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(3)

(1)

(2)

e) PQ is a vertical line between B and C. Find the maximum length of PQ. ( 4) j{' j(x.) ::~X -10.

PQ :. 2JC -Jo - G. 't..- 3)( -Jo) 'l. '-...) -b

-)(. + s:x.. .X. - -- '2..Cl..

- c~z.)~ + .5 (~z.) - -5 - -z. ~ "-...1 "-... ~ :::::. - 2.

Grade 11 Mathematics Paper 1 Paqe20of27

f) The graph of h(x) = x + k touches f once. Find the value of k .

-z... '-.......J .X. - 3 X - 10 = )( -r /c 'l.

)t - 3.JC.. - /0 - )( /c.. : 0

~ - /c. X -'+X /0 - 0 -Gtt) l.

~ L1 - - '+ (,)C.. to - /.:. )

- '" + l,f.O + 4- /c.

5{:, -t LJ.!c- "-..J - 0 -4-k- = -s 6

k. - -Ill- '-..J

QUESTION 11 [14]

a) The following contingency table shows information on the drivers' tests of 100

potential drivers tested at a test centre in Centurion.

Male Female Total Pass 30 47 77 Fail 7 16 23 Total 37 63 100

(1) A driver is randomly selected from the 100 drivers.

(4)

(i) Determine the probability that a female that failed is selected. (1) I b !t:.._ '-....J

/00 ::::. z.s

(ii) Determine the probability that the driver passed, given that it is male.

,30

37

(2)

Grade 11 Mathematics Paper 1 PaqeZ1 o(Z7

(2) Determine, using suitable calculations, whether "being Male" and "Pass" are

independent events. '-....;

(4) 30

f(ffl 17 p) - /00 - O_r3

:-'J 37 P(m) x P(P) 7 7""""'-.J too X /00

= 0 ZS'i!-1

P(mnP) 4= P(m) )C p(p)"-.J

b) Statistics show that Anton scores in 72% of the Cornwall 1st Hockey Team matches.

If he does score in a particular game, the probability that the team will win the match

is 0,86. If he does not score, the probability that they win is only 0,64.

( 1) Represent the information as a tree diagram. (4)

w '5w

s L SL "-.../

NW

N

NL

(2) What is the probability that the Cornwall 1st Hockey Team will win a match? '-o......J '-...J

G>, 7'2. .X 01 8 b) + (o1

Z.8' x o1 bit) (3)

- 0 7 9~l.f '-.J I

Gradell Mathematics Paper 1 Paqe22a(27

QUESTION 12 [11]

a) The Gateway Arch , in St. Louis, in the state of Missouri, can be approximated as a

parabola. Below, a skematical diagram is given on a set of axes (not drawn to scale).

Note: The height of the Gateway Arch is 192 m from the foundation of the pillars and

the legs, which are in the same horizontal plane, are 210 m apart at the foundation.

A (20; - 60) and B represent the base of the arch. C represents the turning point of

the arch.

The x-axis is at sea level. c

A (20; -60) B

( 1) Determine the coordinates of B. (1)

(3 {23o ; - bo)

(2) Determine the coordinates of the point C, the turning point of the parabola. (2)

i 7


(3) Hence, determine the equation of the parabola in the form:

y=a(x-p) 2 +q.

J "" a (x - tZS t + 13:1.~ "Suh.s (z.o; -6o)

-60

. .

a { 2o - rzs)z..

- -112. 1/07_5

-61+ 31,15 (x. - /25)~

~ + 13 z.

1- 132 .

Paqe23o(27

(3)

Grade 11 Mathematics Paper 1 Paqe24 o(27

b) In still waters, a man can row at a speed of 10 km/h. He needs 40 minutes more

to row 16 km upstream than to row 16 km downstream. Determine the speed of

the current. (Let the speed of the current be x km/h.). (5)

D s T

Upsf~Q,., lb 10 -.)C.. II.

/O-X.

Down '"' /0 +X. II:.

/0 .,.. X.

'-.....1 lfe. ~ //:, 'f~ /0- X.. /0 +.X.. bO

/fa lb z -/0 -x.. /0 .,.. A.. .3

LCD: 3(,o-x )6o 1' :x.)

/J-8 Go ~'-'t) - lf8 {io -x) - z (,o-x ){,o +x.)

1f80 + *-B'X - IJ-8'0 +~.x.. - ;;lOO - ~xt.

,]X"L + Cfb X. "ZOO - 0

X.~ + 11-8' .X. /00 ~

0 -(x r soYx 2 ) - 0

X. - -so or- X ~ ')...~

NA

Mathematics Department EXAMINATIONS-9 NOVEMBER ...maths.stithian.com/Grade 11 Papers/Cornwall...

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