firrhill high school mathematics department level...

Firrhill High School

Mathematics Department

Level 5

Home Exercise 1 - Basic Calculations

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Int 2 Unit 1

1. Round these numbers to 2 significant figuresa) 409000 (b) 837500000 (c) 562d) 0.00000009 (e) 0.000305 (f) 0.25

2. Write the numbers in Question 1 in Scientific Notation(Standard Form):

3. Write these numbers in full:a) 4.27 x 105 (b) 3.55 x 107 (c) 2.01 x 102

d) 4.38 x 10-8 (e) 7.27 x 10-5 (f) 4 x 10-3

4. a) A spaceship travels at a speed of 2.57 x 104 miles per hour.How far is it to the moon if it takes the spaceship 72hours to reach it?

b) A textbook has a thickness of 1.35 x 10-2 metres. The book contains 240pages. Calculate the thickness of one page, be careful!

5. Calculate the compound interest on:a) £1500 at a rate of 6% p.a. for 3 yrs. (b) £12000 at a rate of 4.5% p.a. for 5 yrs.

6. Miss Cairns is a secondary school teacher. She earns £28 500 per annum.Her union agree a 3-year pay deal which will see her get an annual rise of 2.5%.How much will Miss Cairns earn in 3 years time?

7. A painting valued at £68 000 in 2005 has appreciated at a steady rate of 12% per annumfor each of the last 6 years. What was the value of the painting in 2011?

8. A van rental company purchases vans costing £18 000 each. The value of a vandepreciates by 25% in its first year and then by 12% in the second year.

a) What is the van worth after the 2 years?

b) A van is replaced when its value falls below half its original price.After how many years will the company replace a van if it continues to fall by 12% p.a.?

9. The cost of a new laptop has fallen by 16% since 2009. A laptop currently costs £441.How much did a laptop cost in 2009?

d) (e) (f)

2. Calculate the volume of these composite shapes:

a) (b) (c)

3. A waste paper bin is in the shape of a large cone with asmaller cone removed.

The large cone has radius 14cm and height 22cm.The small cone has radius 8cm.

Calculate the volume of the bin.

4. The diagram shows a cylinder with a cone cut from it.The cone and the cylinder both have radius 24cmand height 40cm. Calculate the volume of the solidonce the cone has been removed.

Home Exercise 2 - Volumes


Int 2 Unit 1

1. Calculate the volume of each shape below:

a) (b) (c)

2. Write down the gradient and y-intercept of each line below:

a) y = 3x - 5 (b) y = -2x + 4 (c) 2y = 8x - 6

d) x + y = 7 (e) 3x + 2y = 12 (f) 5y - 15x + 20 = 0

3. Calculate the gradient of the line connecting these points:

a) (4 , 5) & (8 , 17) (b) (-2 , 3) & (4 , 12) (c) (-1 , 0) & (-4 , -6)

4. Find these equations:

a) m = 2, y-int = 5 (b) m = -3, y-int = 2 (c) m = ½, (0 , -6)

d) (0 , 5) & (6 , 17) (e) (-2 , 3) & (0 , 12) (f) (1 , 0) & (0, -6)

5. Find the equation of each of these lines:

a) (b) (c)

6. Write down the gradient of: (i) a VERTICAL line (ii) a HORIZONTAL line

Home Exercise 3 - Straight Lines

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Int 2 Unit 1

1. Sketch the following:

a) y = 2x + 3 (b) x - y = 4 (c) 3x + 2y = 6

1. Expand the following:

a) 5(2x + 1) (b) 7(3x + 3) (c) 9(2x + 1)

d) x(4x + 5) (e) m(2x + 4) (f) x2(2x + 9)

g) (x + 3)(x + 7) (h) (p – 1)(p + 6) (i) (u – 5)(u – 6)

j) (2m – 3)(5m + 6) (k) (3w – 2)2 (l) (5t – 3)(2t – 7)

m) (x + 2)(x2 + 3x – 1) (n) (2p – 3)(3p2 – 3p + 2) (p) (3u – 4)(5u2 – 3u – 1)

2. Factorise the following:

a) 6x - xy (b) 3p + 6pq (c) x2 - 6x

d) 2n - 4n2 (e) x2 - 16 (f) 2y - 10xy

g) 100 – p2 (h) 4m2 - 49n2 (i) 36 - 9n2

j) x2 + 7x + 12 (k) a2 - a - 20 (l) c2 - 11c + 30

m) y2 - 3y - 10 (n) u2 + 5u - 14 (p) w2 - 10w + 24

q) 2a2 - a - 15 (r) 3p2 + 13p + 12 (s) 5u2 + 4u - 12

t) 4x2 - 8x + 3 (u) 4x2 + x - 3 (v) 6x2 + 11x - 2

w) 12x2 - 40x + 32 (x) 30p2 - 5p - 10 (y) 8n2 - 28n + 12

Home Exercise 4 - Algebra 1


Int 2 Unit 1

x

2. In each diagram below calculate the angle at the centre:

a) (b)

3. In each circle, centre O, below, calculate x:

a) (b) (c)

4. The diagram shows a prism whose cross-section is thearea between two sectors. One sector has radiusOA = 12cm and the other has radius OC = 15cm.Find the shaded area and hence the volume of this prism.

15cm

Home Exercise 5 - Circles


Int 2 Unit 1

1. For each diagram below, calculate the: a) arc length (b) sector area.

i) (ii) (iii)

2. A ladder 5.2 metres long is resting againstthe top of a wall, as shown.

Calculate h, the height of the wall.

3. The length of this football pitch is 105 mand the breadth of the pitch is 60m.

Calculate the distance, shown by a dottedline, from one corner flag to the oppositecorner flag on this pitch.

4. The diagram opposite shows a rhombus ABCD.

Given the information in the diagram,calculate the length of AB.

5. A ship sails 18km due east from the harbour andthen 8km due south before getting into difficulty.

A lifeboat leaves the same harbour and sails directlyto the sinking ship. How far does the lifeboat sail?

Home Exercise 6 - Pythagoras


Int 2 Unit 2

1. Calculate x in each of the following:

a) (b) (c)

1. Calculate the length of the sides marked x, y and z.

a) (b) (c)

2. Calculate the length of the angles marked a, b and c.

a) (b) (c)

3. a) The diagram shows a plane comingin to land. Calculate the size of theangle the plane will make with theground as it lands.

b) A long handled brush is leaning against a wall.The brush is 1.8 metres long.It makes an angle of 72° with the ground.Calculate how far up the wall the top of thebrush handle reaches.

c) Triangle ABC is isosceles.Triangle ADC is right angled.AC and BC are both 30 centimetres long.AB is 20 centimetres long.Calculate the size of the angle marked x °.

Home Exercise 7 - Basic Trigonometry


Int 2 Unit2

The diagram shows a triangular shaped garden.A jogger runs from one corner of the garden toanother, as shown. Calculate:

a) how far he ran. b) the garden‛s Area.

3. Three towns Applegrove, Bananafield andCarrotsville form a triangle on a mapTheir distances are shown in this diagram.

Find the size of the angle ABC.

4. Another three towns Portland, Queensland andRidgeland form a triangle on a map.

Calculate the size of angle RPQ.

5. Two wires holding a flagpole are fastened atthe same point on the flagpole and two pointson the ground 12 metres apart.The longer wire is frayed and has to be replaced.

What length of wire must be bought.

A

C

B

Home Exercise 8 - Trig 1


Int 2 Unit 2

1. Calculate:

a) the Area

b) the length of side PR.

2.


Int 2 Unit 2

1. a) Sketch this pair of equations on the same diagram: y = 2x + 2

y = 6 - 2x

b) Write down the point of intersection.

2. Solve:

a) a + 2b = 7 (b) 2a + 3c = 9 (c) 4x + 2y = -103a - 2b = 13 3a + c = 10 3x - 5y = -1

3. 6 mugs of coffee and 3 double chocolate muffins cost a total of £18

4 mugs of coffee and 5 double chocolate muffins cost a total of £16.50

How much will a mug of coffee cost, and how much will a double chocolate muffin cost?

4. The senior pupils are organising an end of year dance.Tickets for the dance cost £3 for students and £6 for teachers.Altogether the school takes in £270 from ticket sales.

a) Let x represent the number of students attending the dance and y representthe number of teachers. Write down an equation involving x and y.

b) Given that 70 people attend the dance, write down another equation in x and y.

c) How many students attended the dance?

d) What is the ratio of students to teachers in its simplest form?

Home Exercise 9 - Simultaneous Equ.‛s

1. The percentage marks of 24 pupils in a test are shown below.

1 0 3 92 1 4 5 7 93 2 3 3 5 6 7 3 4 represents 34%4 0 0 1 1 2 2 2 4 8 9 n = 24

a) Show this information in a boxplot.

b) Find the semi-interquartile range of these marks.

2. The marks of 18 pupils in a test were

26 35 35 42 19 35 26 35 44 40 24 45 34 56 59 11 55 34

a) Find the median, lower and upper quartiles of these marks.

b) Draw a boxplot to illustrate these marks.

c) Another class of 18 pupils sat the same test. A boxplot of their marks is shown below.Which class did better overall?

3. The grades in a higher Maths exam obtainedby a group of 72 pupils are shown.

Calculate the angles that are needed to draw a Pie Chart.

Do NOT draw the Pie chart!

4. A survey of 2500 cars was carried out into their place of origin.The results are shown opposite.

a) List the countries in order starting with the highest.

b) How many of the cars surveyed came from the UK?

c) How many cars came from Japan?

Home Exercise 10 - Charts & Graphs


Int 2 Unit 2

1. Calculate the mean and standard deviation of:

a) 14 15 18 20 23 18 (b) 41 45 34 45 46 47 50

2. a) The number of pupils in 7 S3 classes are: 25 24 28 22 24 30 22

Calculate the mean and standard deviation of the class sizes.

b) In the same school the mean and standard deviation of the number of pupils in7 S4 classes are 22 and 4.4 respectively.

Make two comparisons between the class sizes in third year and in fourth year.

3. For the word MATHEMATICS, what is the probability that a letter chosen at randomwill be a: a) vowel (b) T (c) X?

4. The colour of t-shirts worn by 40different S2 pupils on a non-uniformday was recorded as follows:a) Copy and complete the table.

b) Construct a 5-figure summaryfrom this table.

c) Calculate the Semi-IQR.

d) Draw a boxplot to illustrate the info.

5. The scattergraph below shows the Maths &Physics marks of a group of pupils in an S4examination. A line of best fit has beendrawn on the diagram.

a) Find the equation of the line ofbest fit.

b) What would a pupil who scored 72 inPhysics get in Maths using your equation?

Colour Frequency Cumulative Freq.

Blue 4

Green 3

White 5

Red 8

Yellow 2

Other

Home Exercise 11- Statistics


Int 2 Unit 2

2. Simplify these fractions:

a) (2x10

4)6

4)(3x8)(3x (b) (5

23m)2 (2

3x)5)(2(3m3x) (c)

15x5x7x2x2

2

3. Change the subject of each formulae below to m:

a) R = 3m - 2 (b) P = 5 + 7m2 (c)2

85mT

4. Simplify:

a) 24 (b) 96 (c)3672

d) 20 125 45 (e) 3 8 242 72 (f) 7 63 28

5. Rationalise the denominators for the following:

a)2 2

3 (b)6

63 (c)3

73

6. Simplify, expressing your answers with positive powers:

a) x2x53

x(b) (m 4 )3 (c) 3p 53

42p

d) 2x6

24

3x5x (e) find 3

5q , when q = 8 (f) find x

13 , when m = 27

Home Exercise 12 - Algebra 2

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Int 2 Unit 3

1. Express the following as a single fraction and simplify:

a)65

43

32 (b)

65

43

32 (c) a x 3

5a2

bab+ - × ÷ ÷

+ --

--

--

-

- - -

-

- -

-

+

++ +

++

+

=

×

× ×

2. Write down the equation in the form y = (x - a)2 + b, or y = b - (x - a)2 for these graphs:

a) (b)

3. For question 2, use the DIAGRAMS TO WRITE DOWN the solutions to f(x) = 0.

4. Solve by factorisation:

a) 2x2 - 4x = 0 (b) x2 + 5x + 6 = 0 (c) 2x2 - 5x - 3 = 0

5. Solve using the quadratic formula:

a) x2 + 2x - 4 = 0 (b) 2x2 + 3x - 4 = 0 (c) 3x2 - 6x + 2 = 0

6. For the following find the axis of symmetry, turning point & whether it‛s a max or min:

a) y = (x - 4)2 - 10 (b) y = 16 - (x + 6)2 (c) y = (x + 2)2 - 6

-4 8

3 9

Home Exercise 13 - Quadratics


Int 2 Unit 3

1. Write down the equation in the form y = ax2 for these graphs:

a) (b)


Int 2 Unit 3

1. For the following Quadratics in the forms, y = (x - a)2 + b or b - (x - a)2, find the

(i) turning point and its nature (ii) y-intercept by making x = 0:

a) y = (x - 3)2 + 2 (b) y = (x + 5)2 - 4 (c) y = 12 - (x + 6)2

d) y = 5 - (x - 2)2 (e) y = (x - 12)2 - 8 (f) y = 5 - (x - 9)2

2. The diagram below shows the graph of y = (x + a)2 + b.

a) Find the values of a and b.

b) State the equation of the axis of symmetry.

3. The parabola in the diagram below is y = (x – 4)2 – 25.

a) State the coordinates of the turning point

b) Find the coordinates of C.

c) A is the point (-1 , 0). State the coordinates of B.

4. Each quadratic function below has an equation in the form y = ax2.Write down the equation of each function.

a) (b) (c)

5. Solve the following correct to 2 d.p.

a) x2 + 3x + 2 = 0 (b) x2 + 5x + 2 = 0 (c) 2x2 + 3x - 4 = 0

d) 3x2 - 6x + 2 = 0 (e) 3 + 4t - 4t2 = 0 (f) 2x2 + 3x - 4 = 0

Home Exercise 14 - Quadratic Function 2

c) (d)

3. Solve these equations for 0° ≤ x° ≤ 360°

a) 2sinx – 1 = 0 (b) 4cosx – 3 = 0 (c) 5tanx – 12 = 0

d) 3sinx + 6 = 7 (e) 4tanx – 3 = 10 (f) 6cosx – 2 = 3

4. a) Write down the equation of thegraph opposite in the form y = asin x.

b) The line y = 2 meets thisgraph at the points P and Q.Find the coordinates of P and Q.

Home Exercise 15 - Trig 2


Int 2 Unit 3

1. Sketch the following graphs for 0° ≤ x° ≤ 360°

a) y = 3sin2x (b) y = cos2x + 3 (c) y = 8sin3x - 2

2. Write down the equations for each of these graphs:

a) (b)

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