CAPS
Z
J
P
[
H
T
L
O
[
4H
m
a
Ex a c t i ce
Pr
Platinum_Exam Practice Book_COVER_Mathematics_Gr10.indd 2
B
k
o
o
10
e
d
a
r
G
2011/09/02 3:16 PM
Mathematics Tests.pdf 1 7/2/2011 4:11:24 PM
Grade 10
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
Mathematics Tests.pdf 4 7/2/2011 4:11:24 PM
Grade 10
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)
Use your knowledge of quadrilaterals to answer the following questions.
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
a quadrilateral with equal adjacent sides
(8)
5
Mathematics Tests.pdf 6 7/2/2011 4:11:24 PM
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
Mathematics Tests.pdf 7 7/2/2011 4:11:24 PM
Grade 10
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
Use your answers in 3.1 to prove that
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)
Sometimes statistics can be misleading. Use your understanding of statistics to answer
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?
Give a reason for your answer.
(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
two reasons for your answer.
(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
Mathematics Tests.pdf 9 7/2/2011 4:11:24 PM
Grade 10
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
Mathematics Tests.pdf 11 7/2/2011 4:11:24 PM
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
Grade 10
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)
read
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
Mathematics Tests.pdf 15 7/2/2011 4:11:24 PM
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
(add on 2 each time)
(1)
(multiply by 2 each time)
(1)
(add 1, add 2, add 3, add 4 ...)
(1)
(add 2, add 4, add 6 ...)
(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
Grade 10
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
Grade 10
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
Equation of AD:
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
Or answer by inspection.
(3)
2.3
Let C be (x;;y).
x2
= 0 9
2
? x = 2 9
y 1
= 2 9
2
? y = 3 9
? C is the point (2;;3).
Or answer by inspection.
(4)
21