# Platinum mathematics grade 10 exam practice book

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2011/09/02 3:16 PM

Mathematics Tests.pdf 1 7/2/2011 4:11:24 PM

MATHEMATICS

PRACTICE  TEST  ONE

Marks:  50

1.

1.1

1.2

2.

2.1

2.2

2.3

3.

3.1

3.2

3.3

4.

4.1

4.2

4.3

4.4

Fred  reads  at  300  words  per  minute.  The  book  he  is  reading  has  an  average  of  450
words  per  page.
Find  an  expression  for  the  number  of  pages  that  Fred  has  read  after  x  hours.

(4)

How  many  pages  would  Fred  have  read  after  3  hours?

(1)

The  next  two  questions  are  based  on  the  expression   y

6 x 2  37 x  35.

Factorise  the  expression.

(2)

Find  the  value  of  y  if  x  =  2.

(1)

For  what  value/s  of  x  will  y  =  0?

(2)

The  sum  of  two  numbers  is  5.  Their  product  is  3.  Find  the  sum  of  the  squares  of  the  two
numbers  by  answering  the  following  questions.
Expand  to  complete  the  following:   ( x  y ) 2

(1)

If  the  two  numbers  mentioned  above  are  x  and  y,  then  write  down  the  equations  for  the
sum  and  the  product  of  the  two  numbers.

(1)

Substitute  the  information  given  above  into  your  answer  for  3.1  and  hence  determine
the  sum  of  the  squares  of  the  two  numbers.  (Hint:  make  sure  to  include  both  sides  of
the  identity.)

(3)

Factorise  the  following  expressions:

2 x  3 xy  6 y  4

(3)

5 x 2  13 x  6

(2)

2 x  x 2

(2)

1  p 6

(3)

Mathematics Tests.pdf 2 7/2/2011 4:11:24 PM

5.

5.1

5.2

5.3

6.

Solve  for  x:

x
3
4

2

x

3

(1  x)(1  x)
7 x 1

(3)

3 x 2  5 x  2

(4)

3431  x

(3)

Study  the  graph  of   f (x)  below  and  answer  the  questions  that  follow.

6.1

6.2

6.3

What  is  the  range  of   f (x) ?
If   f ( x)

(1)

tan x  k ,  find  the  value  of  k.

(2)

For  what  value/s  of  x  is   f (x)  increasing?

2

(2)

Mathematics Tests.pdf 3 7/2/2011 4:11:24 PM

7.

Study  the  graph  below  and  answer  the  questions  that  follow.

7.1

7.2

7.3

7.4

8.

8.1

8.2

8.3

8.4

8.5

What  is  the  period  of   f (x) ?

(1)

Write  down  the  equation  of   f (x).

(2)

What  is  the  maximum  value  of   f (x) ?

(1)

Which  one  of  the  following  statements  is  correct?  (Write  down  only  the  correct  letter.)
a)

f (x)  is  not  symmetrical  about  any  line.

b)

f (x)  is  symmetrical  about  the  x-­axis.

c)

f (x)  is  symmetrical  about  the  y-­axis.

d)

f (x)  is  symmetrical  about  the  line  y  =  x.

(1)

Find  the  missing  term  of  each  of  the  following  sequences:
3;;  ?;;  7;;  9

(1)

6;;  ?;;  24;;  48

(1)

1;;  2;;  4;;  7;;  ?;;  16

(1)

1;;  3;;  7;;  ?;;  21

(1)

p 2 ;;  ?;;

1
1
;;   3
2
q pq

(1)

[TOTAL:  50  marks]

3

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MATHEMATICS

PRACTICE  TEST  TWO

Marks:  50

ax 2  b.

1.

1.1

Consider  a  function  of  the  form   f ( x)

Determine  the  coordinates  of  the  turning  point  of   f (x)  in  terms  of  a  or  b.

(2)

1.2

Depending  on  the  values  of  a  and  b,  the  turning  point  could  be  either  a  maximum  or  a
minimum.  If  the  turning  point  is  a  minimum,  write  down  the  possible  values  of  a  and  b.

(3)

2.

2.1

Consider  the  functions   f ( x)

8
 4  and   g ( x)
x

x 2  4.

2.2

Sketch   f ( x)  and   g ( x) on  the  same  set  of  axes.  Label  all  intercepts  with  the  axes,
asymptotes  and  turning  points.
There  is  one  value  that   g (x)  can  take  on  that   f (x)  cannot.  Write  down  this  value.

3.

Refer  to  the  graph  below  and  answer  the  questions  that  follow.  The  functions  drawn
below  are:   f ( x)



6
 k  and   g ( x)
x

(4)
(1)

x  2.

3.1

3.2

3.3

Find  the  value  of  k.

(2)

Find  the  coordinates  of  point  A.

(3)

Write  down  the  domain  of   f (x).

(2)

4

Mathematics Tests.pdf 5 7/2/2011 4:11:24 PM

3.4

3.5

4.

4.1

4.2

4.3

4.4

5.

5.1

Find  the  coordinates  of  point  B.

(1)

Find  the  y  coordinate  of  point  C  (which  is  directly  above  point  B).

(2)

The  function   f ( x)

2 x 2  2  is  given.

Sketch  the  graph  of   f (x)  showing  all  intercepts  with  the  axes  and  other  important
points.

(5)

What  is  the  range  of   f (x) ?

(2)

For  what  value/s  of  x  is   f ( x) ! 0 ?

(2)

What  will  the  equation  of   f (x)  become  if  the  graph  is  shifted  down  by  3  units?

(1)

Below  are  pairs  of  parallelograms.  If  you  are  only  given  information  about  their
diagonals,  in  which  pair(s)  can  you  distinguish  between  the  two  parallelograms?

a)   a  rhombus  and  a  rectangle
b)   a  square  and  a  rhombus
c)   a  kite  and  a  trapezium
d)   a  rectangle  and  a  square

5.2

(2)

Match  each  definition  with  the  correct  figure.  If  a  definition  applies  to  more  than  one
figure,  then  choose  the  figure  that  it  describes  the  best.  You  may  only  use  each
definition  once.  (Write  the  number  of  the  figure  and  the  letter  of  the  definition  –  you  do
not  have  to  rewrite  the  whole  definition.)

Figure

Definition

(i)

square

A

a  quadrilateral  with  diagonals  that  bisect  at  90q

(ii)

rhombus

B

a  quadrilateral  with  one  pair  of  parallel  sides

(iii)   kite

C

a  quadrilateral  with  a  90q  corner  angle  and  four  equal  sides

(iv)   trapezium

D

(8)

5

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6.

6.1

6.2

6.3

6.4

6.5

For  each  of  the  following,  determine  whether  the  statement  is  true  or  false.  If  false,
correct  the  statement.
Both  pairs  of  opposite  sides  of  a  kite  are  parallel.

(2)

The  diagonals  of  a  rectangle  bisect  at  90q.

(2)

The  adjacent  sides  of  a  rhombus  are  equal.

(2)

A  trapezium  has  two  pairs  of  parallel  sides.

(2)

A  square  is  a  rhombus  with  a  90q  corner  angle.

(2)

[TOTAL:  50  marks]

6

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MATHEMATICS

PRACTICE  TEST  THREE

Marks:  50

1.

Which  of  the  following  accounts  would  the  best  investment?  Assume  that  you  have
R1  000  to  invest  for  3  years.

a)   Zebra  Bank  offers  8%  per  annum  compounded  monthly.
b)   Giraffe  Savings  offers  8,2%  per  annum  compounded  yearly.
c)   Rhino  Investments  offers  8,4%  per  annum  simple  interest.

2.

(5)

The  points  A(x;;1),  B(–1;;4),  C  and  D  are  shown  on  the  Cartesian  plane  below.

2.1

If  the  gradient  of  AB  is  3,  show  that  x  =  –2.

2.2

If  the  gradient  of  AD  is   ,  show  that  D  is  the  point  (0;;2).

(3)

If  D  is  the  midpoint  of  AC,  find  the  coordinates  of  C.

(4)

Determine  whether  'ABC  is  equilateral,  isosceles  or  scalene.  Show  all  of  your  working.

(5)

2.3

2.4

1
2

7

(3)

Mathematics Tests.pdf 8 7/2/2011 4:11:24 PM

3.

Use  the  diagram  below  to  answer  the  questions  that  follow.

3.1

Write  down  an  expression  for:
a)

tan Į

(1)

b)

tan ș

(1)

3.2

3.3

3.4

4.

4.1

4.2

4.3

5.

5.1

5.2

5.3

5.4

5.5

Hence,  if  AC  =  6  units,   Į

AB
BC

22,76q  and   ș

tan Į
.
tanș
39,97q, find  the  length  of  AB.

Use  your  calculator  to  find  the  value  of   sin( 2ș  3Į),  correct  to  two  decimal  places.

(3)
(4)
(1)

the  following  questions.
A  car  salesperson  says,  “I  sold  five  cars  last  week.  That’s  an  average  of  one  car  every
day.  That  means  that  I’m  going  to  sell  20  cars  this  month.”  Do  you  agree  with  his  logic?

(3)

A  study  was  done  to  see  if  a  new  skin  cream  could  make  wrinkles  disappear.  It  was
tested  on  six  women  while  they  were  visiting  a  health  spa  and  over  80  %  reported  that
their  skin  felt  smoother.  Do  you  think  the  results  of  this  study  are  reliable?  Give  at  least

(5)

The  average  life  expectancy  in  a  certain  country  is  around  70  years.  Does  that  mean
that  nobody  will  live  to  be  100?

(2)

Determine  whether  each  of  the  following  statements  is  true  or  false.  If  the  statement  is
false,  explain  or  give  a  counter  example  to  prove  that  the  statement  is  false.
The  diagonals  of  a  trapezium  are  never  equal.

(2)

A  square  is  a  rhombus  with  a  90q  angle.

(2)

A  rhombus  is  the  only  quadrilateral  with  adjacent  sides  that  are  equal.

(2)

The  diagonals  of  a  kite  always  bisect  at  90q.

(2)

The  diagonals  of  a  rhombus  always  bisect  each  other.

(2)

[TOTAL:  50  marks]

8

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MATHEMATICS

PRACTICE  TEST  FOUR

Marks:  50

1.

1.1

Simplify  the  following  expressions  as  far  as  possible:

1.2

x

2.

x  3
3  2 x

(2)

2x 2x  1


3
7

(3)

Bernard  inherited  a  flat  in  England  that  belonged  to  his  grandmother.  He  decided  to  sell
it  and  use  the  money  to  buy  a  house  in  South  Africa.  Below  are  the  exchange  rates  at
the  time  of  the  sale:

Cross  rates

2.1

2.2

2.3

3.

Rand  (R)

Pound  (£)

1  Rand  (R)  =

1

R14,46

1  Pound  (£)  =

£0,0692

1

The  flat  was  sold  for  £240  500.  How  many  Rands  is  this?

(2)

Would  Bernard  want  a  strong  Rand  or  a  weak  Rand?  Give  a  reason  for  your  answer.

(2)

Refer  to  the  table  of  cross  rates.  Describe  the  mathematical  relationship  between  the
two  numbers  14,46  and  0,0692.

(1)

The  diagram  below  shows  squares  of  increasing  sizes.  W ith  each  extra  layer  of  small
squares  we  add,  we  build  a  bigger  square.

In  the  second  layer,  we  add  3  small  squares.  In  the  third  layer,  we  add  5  small  squares.

3.1

How  many  tiles  will  there  be  in  total  if  we  have  n  layers  of  small  squares?

9

(2)

Mathematics Tests.pdf 10 7/2/2011 4:11:24 PM

3.2

3.3

3.4

How  many  small  squares  will  be  added  on  in  layer  5?

(1)

Write  down  an  expression  for  the  number  of  tiles  added  on  in  layer  n.

(3)

Study  the  pattern  carefully  and  use  the  relationship  between  the  layers  and  the  whole
area  to  find  the  value  of  the  following  sum  to  8  000  terms:

1  +  3  +  5  +  7  ...

3.5

4.

(2)

Use  your  answer  to  3.4  to  find  the  value  of  the  following  sum  to  8  000  terms:
2  +  4  +  6  +  8  ...

(2)

Use  the  figure  below  to  answer  the  questions  that  follow.

4.1

4.2

4.3

4.4

Find  the  midpoint  of  AC.

(2)

Use  midpoints  to  prove  that  ABCD  is  a  parallelogram.

(3)

Prove  that  ABCD  is  NOT  a  rhombus  in  two  different  ways:
a)   using  sides

(3)

b)   using  diagonals

(3)

Prove  that  ABCD  is  not  a  rectangle.

(4)

10

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5.

Your  friend  Nandi  is  working  on  a  homework  exercise.  She  is  getting  very  frustrated
because  her  answers  do  not  seem  to  make  any  sense.  In  the  two  triangles  below,  she
is  trying  to  solve  for  x.  Explain  why  her  answers  do  not  make  sense  in  each  case.

6.

6.1

6.2

7.

7.1

7.2

(5)

Your  favourite  soccer  team  is  changing  its  kit.  The  new  kit  will  be  a  striped  shirt  and
plain  shorts.  The  team  colours  are  blue  and  white.  The  stripes  and  the  background
colour  of  the  shirt  must  be  different  (i.e.  white  with  blue  stripes  or  blue  with  white
stripes).
Write  down  the  different  possible  colour  combinations  for  the  team  kit.

(2)

What  is  the  probability  that  the  stripes  on  the  shirt  and  the  shorts  will  be  the  same
colour?

(3)

For  two  events,  A  and  B,  the  probability  of  both  occurring  is  0,2  and  the  probability  of
neither  occurring  is  0,1.
If  P(A)  =  0,6,  use  a  Venn  diagram  to  find  P(B).

(3)

Find  P(A  or  B).

(2)
[TOTAL:  50  marks]

11

Mathematics Tests.pdf 12 7/2/2011 4:11:24 PM

MATHEMATICS

PRACTICE  TEST  ONE  MEMORANDUM

1.1

1  hour   =   60  minutes

?  in  one  hour,  Fred  reads  300  u  60  =  18  000  words.  9

18  000
Pages  per
9
=
hour
450

=   40  9

?  pages  after  x  hours   =   40x  9

(4)

Pages
=   40(3)

1.2

=   120  9

(1)

2.1

RHS   =   6 x 2  37 x  35
=   (6 x  5)( x  7)  99

(2)

y   =   6 x 2  37 x  35

2.2

substitute  x  =  2

=   6(2) 2  37(2)  35

=   –85  9

(1)

2.3

0   =   (6 x  5)( x  7)
?  x     =   

5
9or   x
6

7  9

(2)

12

Mathematics Tests.pdf 13 7/2/2011 4:11:24 PM

( x  y ) 2   =   x 2  2 xy  y 2  9

3.1

xy   =   3  9

(1)

( x  y ) 2   =   x 2  2 xy  y 2

3.3

2
2
5 2   =   x  2(3)  y  99

x  y   =   5

3.2

(1)

?   x 2  y 2   =   19  9

(3)

2 x  3 xy  6 y  4   =   x(2  3 y )  2(3 y  2)  9

4.1

=   (2  3 y )( x  2)  99

(3)

4.2

5 x 2  13 x  6   =   (5 x  2)( x  3)  99

(2)

2 x  x 2   =   x(2  x)  99

4.3

4.4

(2)

1  p 6   =   (1  p 3 )(1  p 3 )  9

=   (1  p )(1  p  p 2 )(1  p )(1  p  p 2 )  99

13

(3)

Mathematics Tests.pdf 14 7/2/2011 4:11:24 PM

5.1

x
x
 3   =   2 
4
3

x
x
 1   =   
4
3

3 x  12   =    4 x  99

7 x   =   –12

?   x   =   

12
9
7

(3)

5.2

(1  x)(1  x)   =   3 x 2  5 x  2

1  x 2   =   3 x 2  5 x  2

0   =   4 x 2  5 x  1  9

0   =   (4 x  1)( x  1)  9

?  x  =   

1
9   or   x  =  –1  9
4

(4)

5.3

7 x  1   =   3431  x

3 1 x
7 x  1   =   (7 )  9

7 x  1   =   73  3 x

?  x  –  1   =   3  –  3x  9

4x   =   4

?  x   =   1  9

(3)

14

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6.1

y  R  9

(1)

6.2

The  tangent  graph  has  been  shifted  up  by  2  units.

–k   =   2  9

?  k   =   –2  9

(2)

 90q  x  90q   or   90q  x  270q  99

6.3

In  other  words,  all  values  of  x  between  –90q  and  270q,  except  for  –90q,  90q  and  270q.

(2)

7.1

360q  9

(1)

7.2

y   =   3 cos x  1  99

(2)

7.3

4  9

(1)

7.4

c)  9

(1)

8.1

5  9

8.2

12  9

8.3

11  9

8.4

13  9

8.5

(1)

(multiply  by  2  each  time)

(1)

(1)

(1)

p
9
q

(multiply  by

1
each  time)
pq

[TOTAL:  50  marks]

15

(1)

Mathematics Tests.pdf 16 7/2/2011 4:11:25 PM

MATHEMATICS

PRACTICE  TEST  TWO  MEMORANDUM

1.1

Turning  point  occurs  at  x  =  0,  and  when  x  =  0,  y  =  b.

Thus,  the  turning  point  is  (0;;b).  99

(2)

1.2

If  the  turning  point  is  a  minimum,  then  the  parabola  must  be  U  shaped.  This  means
that  the  coefficient  of  x2  must  be  positive.  There  is  no  restriction  on  the  value  of  b.

a  >  0  99

b  R  9

(3)

2.1

9999

(4)

2.2

–  4  9

(1)

16

Mathematics Tests.pdf 17 7/2/2011 4:11:25 PM

3.1

Point  D   =   (0;;2)

(y-­intercept  of  the  line  y  =  x  +  2)

The  hyperbola  has  been  shifted  up  by  2  units  because  y  =  2  is  now  its  asymptote.
?  k   =   2  99

(2)

3.2

A  is  the  x-­intercept  of  the  hyperbola  where  y  =  0.

y   =   

6
 2
x

0   =   

6
 2  9
x

6
=   2
x

6   =   2x

?  x   =   3  9

Thus,  A  is  the  point  (3;;0).  9

(3)

3.3

Domain:   x  R, x z 0  99

(2)

3.4

At  B,  y  =  0,  so  substitute  into  y  =  x  +  2.

0   =   x  +  2

?  x   =   –2

Thus  B   is   the  point  (–2;;0).  9

(1)

3.5

Point  C  will  have  the  same  x-­value  as  point  B  because  it  is  directly  above  it.  Since  we

know  the  x-­value,  we  can  substitute  into  the  equation  of  the  hyperbola  to  find  y.

y   =   

=   

6
 2
x

6
 2  9    =    5  9
2

(2)

17

Mathematics Tests.pdf 18 7/2/2011 4:11:25 PM

4.1

99999

y d 2  99,   y  R

4.2

(5)

(2)

4.3

 1  x  1  99,   x  R

(2)

y   =    2 x 2  2  3

4.4

=    2 x 2  1  9

(1)

5.1

(a)  9  and  (d)  9

(2)

5.2

(i)

C  99

(ii)

A  99

(iii)

D  99

(iv)

B  99

(8)

18

Mathematics Tests.pdf 19 7/2/2011 4:11:25 PM

6.1

False,  both  pairs  of  adjacent  sides  of  a  kite  are  equal.  99

(2)

6.2

False,  the  diagonals  of  a  rectangle  bisect  each  other,  but  not  necessarily  at  90q.  99

(2)

6.3

True  99

(2)

6.4

False,  a  trapezium  has  one  pair  of  parallel  sides.  99

(2)

6.5

True  99

(2)

[TOTAL:  50  marks]

19

Mathematics Tests.pdf 20 7/2/2011 4:11:25 PM

MATHEMATICS

PRACTICE  TEST  THREE  MEMORANDUM

1.

The  best  investment  will  be  the  one  that  has  the  highest  value  after  three  years.

Zebra  Bank:

A   =   1  000(1 

0,08 36
)  9
12

=   R1  270,24  9

Giraffe  Savings:

A   =   1  000(1  +  0,082)3

=   R1  266,72  9

Rhino  Investments:

A   =   1  000(1  +  (0,084  u  3))

=   R1  252  9

Zebra  Bank  is  the  best  investment.  9

(5)

20

Mathematics Tests.pdf 21 7/2/2011 4:11:25 PM

2.1

mAB   =

1 4

x  (1)

3
99
x 1

3   =

3x  +  3   =   –3

3x   =   –6

x   =   –2  9

(3)

2.2

y   =

1
x  c 9
2

1   =

1
 2
 c  9
2

2   =   c

Substitute  in  point  A(–2;;1).

Since  D  is  the  y-­intercept  of  AD,  D  must  be  the  point  (0;;2).  9

(3)

2.3

Let  C  be  (x;;y).

x2
=   0  9
2

?  x   =   2  9

y 1
=   2  9
2

?  y   =   3  9

?  C  is  the  point  (2;;3).

(4)

21

x

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