Practice Book. Practice. Practice Book

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Exam

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Grade 10 MATHEMATICS

PRACTICE TEST ONE

Marks: 50

1. Fred reads at 300 words per minute. The book he is reading has an average of 450 words per page.

1.1 Find an expression for the number of pages that Fred has read after x hours. (4)

1.2 How many pages would Fred have read after 3 hours? (1)

2. The next two questions are based on the expression

_y

6 _x

2

37 _x

35 .

2.1 Factorise the expression. (2)

2.2 Find the value of y if x = 2. (1)

2.3 For what value/s of x will y = 0? (2)

3. 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.

3.1 Expand to complete the following:

(

x

y

)

2 (1)

3.2 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)

3.3 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)

4. Factorise the following expressions: 4.1 2x3xy6y4 (3) 4.2 5_x2 _x13 6 ₍₂₎ 4.3 ₂_x_x2 ₍₂₎ 4.4

1 p

6 (3)

(3)

5. Solve for x: 5.1 3 2 3 4 x x (3) 5.2

(

1 _x

)(

1 _x

)

3 _x

2

5 _x

2

₍₄₎ 5.3 ₇x1 ₃₄₃1x ₍₃₎

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

6.1 What is the range of f(x)? (1)

6.2 If f(x) tanxk, find the value of k. (2)

6.3 For what value/s of x is f(x) increasing? (2)

(4)

7. Study the graph below and answer the questions that follow.

7.1 What is the period of f(x)? (1)

7.2 Write down the equation of f(x). (2)

7.3 What is the maximum value of f(x)? (1)

7.4 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)

8. Find the missing term of each of the following sequences: 8.1 3;; ?;; 7;; 9 (1) 8.2 6;; ?;; 24;; 48 (1) 8.3 1;; 2;; 4;; 7;; ?;; 16 (1) 8.4 1;; 3;; 7;; ?;; 21 (1) 8.5 3 2 2 _;;_?;;1 _;; 1 pq q p (1) [TOTAL: 50 marks]

(5)

PRACTICE TEST TWO

Marks: 50

1. Consider a function of the form

_f

(

_x

)

_ax

2

_b

.

1.1 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. Consider the functions ( ) 84

x x

f and

_g

(

_x

)

_x

2

4 .

2.1 Sketch f(x )and g(x)on the same set of axes. Label all intercepts with the axes,

asymptotes and turning points. (4) 2.2 There is one value that g(x) can take on that f(x) cannot. Write down this value. (1)

3. Refer to the graph below and answer the questions that follow. The functions drawn below are: k x x f( ) 6 and g(x) x2.

3.1 Find the value of k. (2)

3.2 Find the coordinates of point A. (3)

3.3 Write down the domain of f(x). (2)

(6)

3.4 Find the coordinates of point B. (1)

3.5 Find the y coordinate of point C (which is directly above point B). (2)

4. The function

_f

(

_x

)

2 _x

2

_{is given.}

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

points. (5)

4.2 What is the range of f(x)? (2)

4.3 For what value/s of x is f(x)!0? (2)

4.4 What will the equation of f(x) become if the graph is shifted down by 3 units? (1)

5. Use your knowledge of quadrilaterals to answer the following questions.

5.1 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 (2)

5.2 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)

(7)

6. For each of the following, determine whether the statement is true or false. If false, correct the statement.

6.1 Both pairs of opposite sides of a kite are parallel. (2)

6.2 The diagonals of a rectangle bisect at 90q. (2)

6.3 The adjacent sides of a rhombus are equal. (2)

6.4 A trapezium has two pairs of parallel sides. (2)

6.5 A square is a rhombus with a 90q corner angle. (2)

(8)

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. (5)

2. 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. (3)

2.2 If the gradient of AD is , 2 1

show that D is the point (0;;2). (3)

2.3 If D is the midpoint of AC, find the coordinates of C. (4)

2.4 Determine whether 'ABC is equilateral, isosceles or scalene. Show all of your working. (5)

(9)

3. Use the diagram below to answer the questions that follow.

3.1 Write down an expression for:

a) tanĮ (1) b) tanș (1)

3.2 Use your answers in 3.1 to prove that . tanș Į tan BC AB (3)

3.3 Hence, if AC = 6 units, Į 22,76q and ș 39,97q,find the length of AB. (4)

3.4 Use your calculator to find the value of sin(2ș3Į), correct to two decimal places. (1)

4. Sometimes statistics can be misleading. Use your understanding of statistics to answer the following questions.

4.1 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)

4.2 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)

4.3 The average life expectancy in a certain country is around 70 years. Does that mean

that nobody will live to be 100? (2)

5. 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.

5.1 The diagonals of a trapezium are never equal. (2)

5.2 A square is a rhombus with a 90q angle. (2)

5.3 A rhombus is the only quadrilateral with adjacent sides that are equal. (2)

5.4 The diagonals of a kite always bisect at 90q. (2)

5.5 The diagonals of a rhombus always bisect each other. (2)

(10)

PRACTICE TEST FOUR

Marks: 50

1. Simplify the following expressions as far as possible: 1.1

x

3

2 x

(2) 1.2 7 1 2 3 2 x x x (3)

2. 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 Rand (R) Pound (£)

1 Rand (R) = 1 R14,46 1 Pound (£) = £0,0692 1

2.1 The flat was sold for £240 500. How many Rands is this? (2)

2.2 Would Bernard want a strong Rand or a weak Rand? Give a reason for your answer. (2)

2.3 Refer to the table of cross rates. Describe the mathematical relationship between the

two numbers 14,46 and 0,0692. (1)

3. The diagram below shows squares of increasing sizes. With 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? (2)

(11)

3.2 How many small squares will be added on in layer 5? (1)

3.3 Write down an expression for the number of tiles added on in layer n. (3)

3.4 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 ... (2)

3.5 Use your answer to 3.4 to find the value of the following sum to 8 000 terms:

2 + 4 + 6 + 8 ... (2)

4. Use the figure below to answer the questions that follow.

4.1 Find the midpoint of AC. (2)

4.2 Use midpoints to prove that ABCD is a parallelogram. (3)

4.3 Prove that ABCD is NOT a rhombus in two different ways:

a) using sides (3) b) using diagonals (3)

4.4 Prove that ABCD is not a rectangle. (4)

(12)

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.

(5)

6. 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).

6.1 Write down the different possible colour combinations for the team kit. (2)

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

colour? (3)

7. For two events, A and B, the probability of both occurring is 0,2 and the probability of neither occurring is 0,1.

7.1 If P(A) = 0,6, use a Venn diagram to find P(B). (3)

7.2 Find P(A or B). (2)

[TOTAL: 50 marks]

(13)

PRACTICE TEST ONE MEMORANDUM

1.1 1 hour = 60 minutes ? in one hour, Fred reads 300 u 60 = 18 000 words. 9 Pages per _hour =

450 000 18 9 = 40 9 ? pages after x hours = 40x 9 (4)

1.2 Pages read = 40(3) = 120 9 (1) 2.1 RHS = ₆_x2 ₃₇_x₃₅ = (6x5)(x7) 99 (2) 2.2 y = ₆ 2 ₃₇ ₃₅ x x substitute x = 2 =

6 (

2 )

2

37 (

2 )

35

= –85 9 (1) 2.3 0 = (6x5)(x7) ? x = 6 5 9or x 7 9 (2)

(14)

3.1

₍

_x

_y

₎

2₌

_x

2

₂

_xy

_y

2₉ ₍₁₎ 3.2 xy = 5 xy = 3 9 (1) 3.3

₍

_x

_y

₎

2₌

_x

2

₂

_xy

_y

2 ₅2₌

_x

2

₂

₍

₃

₎

_y

2₉₉ ?

_x

2

_y

2_{= 19 9} ₍₃₎ 4.1 2x3xy6y4 = x(23y)2(3y2) 9 = (23y)(x2) 99 (3) 4.2

5 _x

2

_x

13

6

₌ ₍₅_x₂₎₍_x₃₎₉₉ ₍₂₎ 4.3 ₂_x_x2₌ _x₍₂_x₎₉₉ ₍₂₎ 4.4

_{1 p}

6₌

₍

₁

_p

3

₎₍

₁

_p

3

₎

₉ =

(

1 _p

)(

1 _p

_p

2

)(

1 _p

)(

1 _p

_p

2

)

₉₉ ₍₃₎

(15)

5.1 3 4 x = 3 2 x 1 4 x = 3 x 3x12 = 4x 99 7x = –12 ? x = 7 12 9 (3) 5.2 (1x)(1x) = ₃_x2 _x₅ ₂ _{1 x} 2₌ ₃_x2 _x₅ ₂ 0 = 4_x2 _x5 1₉ 0 = (4x1)(x1) 9 ? x = 4 1 9 or x = –1 9 (4) 5.3 ₇x1₌ ₃₄₃1x ₇x1₌

₍

₇

3

₎

1x₉ ₇x1₌ ₇33x ? x – 1 = 3 – 3x 9 4x = 4 ? x = 1 9 (3)

(16)

6.1 yR 9 (1)

6.2 The tangent graph has been shifted up by 2 units. –k = 2 9 ? k = –2 9 (2)

6.3 90q x90q or 90q x270q 99 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 = 3cosx1 99 (2) 7.3 4 9 (1) 7.4 c) 9 (1)

8.1 5 9 (add on 2 each time) (1)

8.2 12 9 (multiply by 2 each time) (1)

8.3 11 9 (add 1, add 2, add 3, add 4 ...) (1)

8.4 13 9 (add 2, add 4, add 6 ...) (1)

8.5 q p 9 (multiply by pq 1 each time) (1) [TOTAL: 50 marks]

(17)

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 bR 9 (3) 2.1 9999 (4) 2.2 – 4 9 (1)

(18)

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 x 9 x 6 = 2 6 = 2x ? x = 3 9 Thus, A is the point (3;;0). 9 (3)

3.3 Domain: xR,xz0 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 = 2 2 6 9 = 5 9 (2)

(19)

4.1 99999 (5) 4.2 yd2 99, yR (2) 4.3 1 x1 99, xR (2) 4.4 y = _x₂ 2 ₂₃ = _x₂ 2 ₁₉ ₍₁₎ 5.1 (a) 9 and (d) 9 (2) 5.2 (i) C 99 (ii) A 99 (iii) D 99 (iv) B 99 (8)

(20)

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]

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PRACTICE TEST THREE MEMORANDUM

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

Zebra Bank: A = )36 12 08 , 0 1 ( 000 1 9 = 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)

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2.1 mAB = ) 1 ( 4 1 x 3 = 1 3 x 99 3x + 3 = –3 3x = –6 x = –2 9 (3)

2.2 Equation of AD:

y = xc

2

1 ₉ _{Substitute in point A(–2;;1).}

1 =

2 c 2 1 9 2 = c

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). 2 2 x = 0 9 ? x = 2 9 2 1 y = 2 9 ? y = 3 9

? C is the point (2;;3). Or answer by inspection. (4)

(23)

2.4 AB =

₍

₄

₁

₎

2

₍

₁

₍

₂

₎₎

2 =

₁₀

9 BC =

₍

₄

₃

₎

2

₍

₁

₂

₎

2 =

₁₀

9 AC =

₍

₃

₁

₎

2

₍

₂

₍

₂

₎₎

2 =

₂₀

9

'ABC is an isosceles 9 triangle because it has two equal sides. 9 (5)

(24)

3.1 a) BD AB Į tan 9 (1) b) BD BC ș tan 9 (1)

3.2 AB = BD.tanĮ and BC = BD.tanș (from 3.1)

? BC AB = ș tan BD Į tan . BD . 99 = ș tan Į tan ₉ (3) 3.3 AC = AB + BC ? BC = AC – AB = 6 – AB 9 BC AB = ș tan Į tan AB 6 AB = q q 97 , 39 tan 76 , 22 tan 9 AB 6 AB = 0,5 9 2AB = 6 – AB 3AB = 6 ? AB = 2 units 9 (4) 3.4 0,20 9 (1)

(25)

4.1

No, 9 an average is not guaranteed to persist. 9 If he were to take his yearly average and apply that to a given week it might be more reliable, but to use a single week’s average to try to predict future performance is not wise. In the short run almost anything can happen – one could have a good or bad week. It does not make sense to base statistics on a few short-term observations. 9

(3) 4.2

No, 9 the results are not completely reliable. Firstly, testing the product while the women are at a spa is misleading. The results of the spa treatment can not easily be separated from the results of the face cream. 99 Secondly, there are too few

people in the test group to make any deductions. 99 What seems true for six people may not apply on a larger scale. The women might also have responded positively for emotional and psychological reasons.

(5) 4.3

No, 9 some people die very young and some people die very old. The highs and the lows balance out. An average does not describe every value in the range. 9

(2)

(26)

5.1 False. Diagonals can be equal if opposite sides are equal. See below. 9

9 (2) 5.2 True 99 (2)

5.3 False, a kite and a square also have adjacent sides that are equal. 99 (2)

5.4 False, diagonals do not necessarily bisect – see below. 9

9 (2) 5.5 True 99 (2) [TOTAL: 50 marks]

(27)

PRACTICE TEST FOUR MEMORANDUM

1.1 (x3)(32x) = ₉₃_x₂_x2₉₉ ₍₂₎ 1.2 7 1 2 3 2 x x x = 21 ) 1 2 ( 3 ) 2 ( 7 21x x x 99 = 21 3 6 14 21x x x = 21 3 x ₉ (3) 2.1 £240 500 = R240 500 u 14,46 9 = R3 477 630 9 (2) 2.2

Bernard would want a weak Rand relative to the Pound. 9 This would mean that he would receive more Rands for each Pound that he earned on the sale. 9

(2)

2.3 An inverse or reciprocal relationship (9) exists between the two rates. Mathematically: 0,0692 46 , 14 1 and 14,46 0692 , 0 1

(9) (either description will earn 1 mark) (1)

(28)

3.1 n2₉₉ ₍₂₎ 3.2 9 9 (1) 3.3 Tiles added = 2n – 1 999 (3) 3.4

With each layer we add on, we make a bigger square. This means that the sum of n layers (odd numbers) is n2_{. This tells us that the sum of n odd numbers is n}2_.

Sum of 8 000 odd

numbers = 8 0002 9 = 64 000 000 9 (2)

3.5

This is almost the same as the sequence in 3.4, except each term is 1 larger. This means that the whole sum will be a total of 8 000 larger. Sum to 8 000 = 64 000 000 + 8 000 9 = 64 008 000 9 Note: A general term for the sum of this sequence would be S_S n = n2 + n, or

n = n(n + 1). (2)

(29)

4.1 Midpoint AC = ¸ ¹ · ¨ © § 2 5 3 ;; 2 2 1 9 = ¸ ¹ · ¨ © § 1;; 2 1 9 (2) 4.2 Midpoint BD = ¸ ¹ · ¨ © § 2 3 1 ;; 2 3 4 9 = ¸ ¹ · ¨ © § 1;; 2 1 9

AC and BD share a midpoint and therefore they bisect each other. This means that _{ABCD is a parallelogram (diagonals bisect). 9} ₍₃₎

4.3 a) Using sides, simply prove that adjacent sides are not equal. (ABCD is a ||gm) AB =

₍

₃

₁

₎

2

₍

₁

₄

₎

2 ₌

₂₉

₉

AD =

₍

₃

₍

₃

₎₎

2

₍

₁

₍

₃

₎₎

2 ₌

₄₀

₉

Adjacent sides are not equal and therefore parallelogram ABCD is not a _rhombus.9 (3)

b) Diagonals of a rhombus bisect at 90q. Using gradients: mAC = 2 1 ) 5 ( 3 = 3 8 9 mBD = ) 3 ( 4 ) 3 ( 1 = 7 4 9

mBD u mAC z –1, so diagonals are not perpendicular. Parallelogram ABCD is

therefore not a rhombus. 9

(3)

(30)

4.4 mAD = ) 3 ( 1 ) 3 ( 3 = 2 6 = 3 9 mDC = 2 3 ) 5 ( 3 = 5 2 9

Since mDC u mAD z –1, there is no right angle between AD and DC. 9

Since ABCD does not have four right angles, it cannot be a rectangle. 9 $OWHUQDWHSURRIVKRZWKDW$&%' (4)

5. Triangle 1

The longest side in a right-angled triangle is always the hypotenuse. 9 In this triangle, the hypotenuse is not the longest side, which is impossible. 9 If we try to solve for x using Pythagoras, we will not be able to find a solution because the triangle does not make sense.

Triangle 2

In this triangle, the sum of the angles is not 180q 9 (29q + 63q + 90q = 182q). This triangle also does not make sense. 9 If we try to use trig ratios to solve for x, we will get a slightly different answer depending on which angle we use. 9 This is because a right-angled triangle can not have a 29q angle and a 63q angle – these angles would belong to different triangles, hence the two different answers.

(5) 6.1

Blue shirt, white stripes;; blue shorts Blue shirt, white stripes;; white shorts White shirt, blue stripes;; blue shorts White shirt, blue stripes, white shorts 99

(2) 6.2 4 2 = 2 1 999 (3)

(31)

7.1 99 P(B) = 0,3 9 (3) 7.2 P(A or B) = 0,4 + 0,2 + 0,3 9 = 0,9 9 (or, use 1 – 0,1 = 0,9) (2) [TOTAL: 60 marks]

(32)

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