Module 10 - Self Study Handbook

Contents:

2. N10.1 - Growth and Decay 4. N10.2 - Number Work6. N10.2 - Rationalising the Denominator8. A10.1 - Finding Approximate Formulae 10. A10.2 - Algebraic Fractions 12. A10.3 - Solving Quadratics 14. A10.4 - Solving Simultaneous Equations - Quadratic and Linear16. A10.5 - Graph Transformations18. S10.1 - Surface Areas of 3-D Shapes20. S10.2 - Congruence and Shape Properties22. S10.3 - Non-Right Angle Triangles24. S10.4 - Trigonometrical Functions26. S10.5 - Vectors28. Self Assessment Sheet

The Data Handling Units have not been included. Two of the topics covered are 'Sampling' and 'Comparing Data', both of which are covered in your coursework. The final topic is 'Moving Averages' which you will have covered in Module 8

Useful websites:

Hamilton Maths: http://hamiltonmaths.wordpress.com/

Past papers:http://www.cheam.sutton.sch.uk/departments/mathematics.htm

BBC Bitesize: http://www.bbc.co.uk/schools/gcsebitesize/maths/

MathsNet: http://www.mathsnet.net/

Nrich: http://www.nrich.maths.org.uk/public/index.php

WisWeb: http://www.fi.uu.nl/wisweb/en/

10ticks: http://www.10ticks.co.uk/

Count On: http://www.counton.org/

Skoool: http://lgfl.skoool.co.uk/keystage4.aspx?id=314

Page 1

N10.1 - Growth and DecayPractice Questions

Key ideas:

Exponential functions are functions of the form y = abx. Since b0 = 1, the graphs of these functions always go through (0 , a)

To solve an exponential equation when the power is unknown, use the power key and trial and improvement

Exponential functions usually take the form:N = No rt

Where: N = Value at the required time, No = Initial Value, r = rate of change, t = time

Questions:

Solve the following questions:

1. The size, y, of a population of bacteria is growing according to the rule:y = 25 x 1.02t

where t minutes is the measured time

a. How many bacteria are there at time t = 0?

b. What will the population be 5 hours after starting to measure the time?

c. Find out how long the population took to double in size

2. In 2000, the value of Bharat's stamp collection was £85. Assume the value increases at 5% each year.

a. What will the value be in 2004?

b. Write as simply as possible, an expression for its value x years after 2000

c. Find by calculation how many years it will take for the value of the stamp collection to double

3. £5000 is invested at 3% compound interest per year

a. Calculate the value of the investment after 20 years

b. Find, to the nearest year, the time taken for the investment to be worth £12 000

4. A car cost £16 000 when new and depreciates in value by 16% each year.

a. State a formula for the value of the car after n years

b. Find, in years to 1 d.p., the age of the car when its value is £5000

5. A population of butterflies is declining at 8% per year. The population in August 2001 was 850

a. Find a formula for the population of butterflies t years after 2001

b. Find the population in August 2006

c. In August of which year is the population first below 400?

Page 2

N10.1 - Growth And Decay Worked Exam Questions

1. The population of Milford is growing at a rate of 15% per year. A newspaper reports that the population will double in 5 years. Use growth rate to decide whether this statement is correct. Explain your answer clearly.

[3]

2. The number of Lapwings is falling. In 2002 the population in one area was 8200 Lapwings. After t years the population of Lapwings, L, is given by the formula:

L = 8200 x 0.94t

a. What will be the population of Lapwings in 2007?

[2]b. In which year will the population be half that in 2002?

[2]

3. The value of a car decreases every year. The value, £V, of the car after t years is given by the formula:

V = 9500 x 0.8t

a. Find the original value of the car

[1]b. Another car that decreases in value at the same rate is worth £4096 after 3 years. What was the original value of the car?

[3]

4. In 1990 the population of animals in a colony was 640. The population, P, after t years is given by the equation:

P = 640 x 0.9t

a. By what percentage is the population changing each year?

[1]b. Work out an estimate of the population in 2015?

[2]Answers: 1.a. 25 bacteria b. 9506 bacteria (4 s.f.) c. 35 mins (2 s.f.)2.a. £103.32 (2 d.p.) b. V = 85 x 1.05x c. 14.2 years (1 d.p.)3.a. £9030.56 (2 d.p.) b. 30 years4.a. V = 16 000 x 0.84n b. 6.7 years (1 d.p.)5.a. P = 850 x 0.92t b. 560 butterflies (3 s.f.) c. 10 years

Page 3

N10.2 - Number WorkPractice Questions

Key Ideas: Any recurring decimal can be written as a fraction The rules of indices are:

→ an x am = am+n

→ an ÷ am = an-m

→ (an)m = anxm

→ a0 = 1→ a1 = a

→

→

You are expected to know the square numbers to 15 and the cubes of 1, 2, 3, 4, 5 and 10

The most common exam mistake is to try to add or subtract ax and ay which cannot be done

Questions:1. Write in index form:

a. b. the reciprocal of c.

Work out the following without using a calculator. Give the answers as a rational number.

2.a.

b.

c. 27-1 d

.e. 270

3.a.

b.

c. 160 d

.e.

4.a.

b.

c.

d.

5.a.

b.

c.

d.

6. Write as powers of 2 as simply as possible

a. 32 b. c.

d. 0.25 e. f.

7. Convert these recurring decimals to fractions or mixed numbers in their lowest terms:a. b. c. d. e.

Page 4

N10.2 - Number WorkWorked Exam Questions

1.a. Evaluate , give your answers as a fraction

[3]b. Express as a fraction in its simplest form

[3]2. Express as a fraction

[2]3.a. It is given that . Show that 99N = 57

[2]b. Hence express as a fraction in its lowest terms

[2]Answers:

1.a. b. c.

2.a. 3 b. 81 c. d. 3 e. 1

3.a. 4 b. ½ c. 1 d. 64 e. 128

4.a. 125 b. 6 c. 100 d.

5.a. 1000 b. c. d. 0

6.a. 25 b. 22 c. 23 d. 2-2 e. 23n f. 23n-8

7.a. b. c. d. e.

Page 5

N10.2 - Rationalising the DenominatorPractice Questions

Key Ideas: A rational number can be written as a fraction or ratio with both numerator and

denominator as integers An irrational number is a number which cannot be written as a ratio or fractions with

both numerator and denominator as integers. It is a number which, as a decimal, does not terminate or recur

To rationalise a fraction with an irrational denominator, multiply the numerator and the denominator by the surd that is in the denominator

Surds can be simplified using or

Questions:Simplify the following and state if the answer is rational or irrational:1. √12 2. √45 3. √8 x √2 4. √80 x √50 If x = 5 + 2√3 and y = 4 - 3√2 simplify:5. (3 + √5) + (4 - 3√5) 6. x√3 7. x2 8. y2

Simplify the following including rationalise the denominator:

9. 10. 11. 12.

13. 14. 15. 16.

17. Find an exact expression for the shaded areas between these two squares, simplifying as much as possible

19. Find the exact value of x, expressing your answer as simply as possible

18. Find, as simply as possible, an exact expression for the area of this circle

20. Find an expression for the total area of this shape, formed by a square and semicircle, simplifying as much as possible

Page 6

√15

√5

√7

√3

x

√12

√7

N10.2 - Rationalising the DenominatorWorked Exam Questions

1. Simplify

2. Simplify , give your answer in the form , where a, b and c are integers

3. Simplify, giving your answers in the form :

4. Simplify, giving your answers in the form :

Answers:1. 2√3 Irr 2. 3√5 Irr 3. 4 Rat 4. 20√10 Irr5. 7 - 2√5 6. 6 + 5√3 7. 37 + 20√3 8. 34 - 6√2

9. 10. 11. 12.

13. 14. 15. 16.

17. 10 18. 7π 19. √10 20.

Page 7

A10.1 - Finding Approximate FormulaePractice Questions

Key ideas: When writing general rules in algebraic form, you must:

Define all variables precisely, including the unitsMake sure that the correct rules of algebra are followed in writing the formula that fits the rule

If a relationship is thought to be of the form y = m f(x) + c, this can be tested by plotting f(x) against y.

m and c can be estimated using the line of best fit;m is the gradient and c is the y-intercept

Questions:1.

Jane buys some books for £3 each. She sells some of them for £5 each. To get rid of the remainder she sells them at £2 each.Find, and simplify, a formulae for her profit

2.

To find the volume of a pyramid, multiply the area of the base by the height and divide by 3.Write a formula for the volume of a pyramid with a base that is:a. Squareb. Rectangle

3.

Here is a sequence:2 , 5 , 10 , 17 , 26 , …

Find:a. the 7th termb. the nth termc. the 20th term

4.

Newton's Law of Gravitation states that the force of attraction between two bodies is inversely proportional to the square of the distance between them.Write this relationship algebraicallya. Using the symbol b. As a formula using the constant k

5.

State the variables you would need to plot to obtain a straight line graph of the following equations:a. s = 4t3 + 6b. A = πr2

c.

6.

By drawing a graph, or otherwise, calculate, calculate the values of a and b in the equation y = ax3 + b satisfied by the following values:

x 1 2 5 10y -37 -16 335 2960

7.

t 1 2 3 4 5s 1.0 1.25 1.44 1.6 1.74

a. How can you tell from the table that s and t are not linearly related?b. It is thought that s is proportional to √t. Draw an appropriate graph and obtain the equation connecting s and t. Explain, giving your reason, whether s is proportional to √t.

8.

Some rope is sold in different sizes. These sizes are the circumference, c millimetres, of the rope measured when it is unstretched. This table shows the price, p pence, per metre for the different sizes.

c 10 20 30 40 50p 24 36 56 84 120

a. Plot a graph of p against c2 and hence obtain the equation connecting p and c.b. What is the cost per metre of one of these ropes with circumference 70mm?

Page 8

A10.1 - Finding Approximate FormulaeWorked Exam Questions

1. This set of values of t and y are the results from an experiment. It is thought that

these data satisfy a relationship of the form .

a. Complete the table and plot y against . [2]

b. Draw a line of best fit onto your graph. Use this find to find an equation which

approximately connects y and . [3]

2. In an experiment, the values of s and r are recorded. It is believed that s and r are connected by an equation of the form s = ar3 + b. The values of s against r3 are plotted on the graph.Find approximate values for a and b.

[3]

3. In a mechanics experiment different masses were hung on the end of a spring. The total length, L centimetres, of the spring was measured for each different mass, M grams. The results are shown on the graph below.Use this graph to find a formula that approximately connects L and M.

[3]What is the length of the unstretched length of spring?

[1]

a.

b.

Answers: 1. N = total number of books, P = profit, n = number of books sold for £5. Formula: P = 3n - N

2. a. a = side length, h = height, V = Volume. Formula: . b. l = length, w = width, h = height, V = volume. Formula:

3.a. 37 b. n2 + 1 c. 401 4. d = distance, F = force of attraction. a. b.

5. a. s against t3 b. A against r2 c. P against 1/x 6. y = 3x - 40 , a = 3 , b = -407. a. The difference between the s values is not constant. b. s = 0.6 √t + 0.4 , yes proportional as points form a linear graph8.a. p = 0.04 c2 + 20 b. 216p or £2.16

Page 9

A10.2 - Algebraic FractionsPractice Questions

Key Ideas: When adding or subtracting fractions, put them over a common denominator When cancelling algebraic fractions, factorise if necessary. Only cancel factors When equations involve fractions, multiply through by the common denominator to

remove the fractions

Questions:

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

Page 10

A10.2 - Algebraic FractionsWorked Exam Questions

1. Simplify

[3]

2. Simplify

[3]

3.a. Express as a single fraction

[2]

b. A formula used in Optics is

Rearrange this formula to express v in terms of f and u

[2]

4. Prove that

[3]Answers:

1. 2. 3. x 4.

5. 6. 7. 3x1x16xx2

8.

9. 1x5x1517x3x2

10

.11. x = 2 12

. x = 3

13. x = 4 14. x = 5 15

.16. x = -4 , x = -3

17. x = 5 , x = -2 18. x = ¾ , x = 1 19

. x = ½ , x = 4 20. x = -½ , x = 5

Page 11

A10.3 - Solving QuadraticsPractice Questions

Key Ideas:To calculate the roots of a quadratic equation which will not factorise: Complete the square

→ Multiply if necessary so the coefficient of x2 is a perfect square→ Add a number to both sides so the left-hand side is a complete square→ Take the square root of both sides→ Solve the resulting linear equations

Or

Use the formula when ax2 + bx + c = 0,

Questions:

Solve these equations by completing the square1. x2 - 2x - 2 = 0 2. x2 + 6x - 4 = 03. x2 + 8x + 3 = 0 4. x2 - 10x + 6 = 05. x2 + 3x - 5 = 0 6. x2 - 5x + 2 = 07. 4x2 + 3x - 2 = 0 8. 2x2 + 12x + 3 = 09. 3x2 + 2x - 2 = 0 10. 5x2 - 6x - 4 = 0

Use the formula to solve these equations. Give the answers correct 2 d.p.11. x2 + 7x + 4 = 0 12. 2x2 - 3x - 4 = 0

13. 3x2 + 2x - 2 = 0 14. 5x2 - 13x + 7 = 0

15. 5x2 + 9x + 3 = 0 16. 3x2 + 4x - 2 = 0

17. 7x2 - 5x - 1 = 0 18. 3x2 + 2x - 7 = 0

19. A pen is constructed along an existing wall using 20 m of fencinga. If the width is x m, write an expression for the area enclosedb. Write down an equation and solve it to find the dimensions to give an area of 40 m2

c. By completing the square, find the maximum area possible with this amount of fencing

20. A rectangular lawn measuring 22 m by 15 m is surrounded by a path x m wide. Form a simplified expression for the total area of lawn and path. Write down an equation and solve it to find the width of path correct to the nearest cm if the area is 400m2

Page 12

xx

15 m

22 m

A10.3 - Solving QuadraticsWorked Exam Questions

1.a. Write x2 - 6x + 4 in the form (x + a)2 + b

[3]b. Find the minimum value of x2 - 6x + 4

[1]2. The lengths in this trapezium are in centimetres

a. Write down an expression for the area of this trapezium

[1]b. The area of the trapezium is 20 cm2, show that 3x2 + 5x - 40 = 0

[2]c. Solve 3x2 + 5x - 40 = 0. Hence find the height of the trapezium, correct to 1 decimal place

[3]

Answers: (all answers to 1d.p. unless stated otherwise)1. x = 2.7 , -0.7 2. x = 0.6 , -6.6 3. x = -0.4 , -7.64. x = 9.4 , 0.6 5. x = 1.2 , -4.2 6. x = 4.6 , 0.47. x = 0.4 , -1.2 8. x = -0.3 , -5.7 9. x = 0.5 , -1.2

10. x = 1.7 , -0.5 11. x = -0.6 , -6.4 12. x = 2.4 , -0.913. x = 0.5 , -1.2 14. x = 1.8 , 0.8 15. x = -0.4 , -1.416. x = 0.4 , -1.7 17. x = 0.9 , -0.2 18. x = 1.2 , -1.9

19.a. x(20 - 2x) b. x = 2.25 (2dp) c. Max Area = 50m2

20. Area = (2x + 22)(2x + 15) x = 0.9 m

Page 13

2x

x

x + 5

A10.4 - Solving Simultaneous Equations - Quadratic and LinearPractice Questions

Key ideas:

The equation of a circle centre (0 , 0) and radius r is x2 + y2 = r2

Simultaneous equations consisting of a linear equation and a quadratic or circle equation can be solved graphically. Draw the graph of each equation on the same set of axes and the points of intersection are the solutions

When solving, algebraically, simultaneous equations consisting of a linear equation and a quadratic or circle equation, use substitution. Use the linear equation to find one letter in terms of the other, then substitute this in the remaining equation

The quadratic formula to solve a quadratic of the form ax2 + bx + c = 0 is

Questions:

Solve the following simultaneous equations by method of substitution:(Give your answers correct to two decimal places, where appropriate)

1. y = 2x - 3 2. 4x - 2y = 3 3. 3x - y = 7and and and

7x - 4y = 10 x - y = 0 5x + 2y = 8

4. 2x + 3y = 7 5. y = 4 - 3x 6. y = 2x - 3and and and

5y = 11 - 3x y = x2 - 6x - 6 y = x2 - 4x + 5

7. y = x2 + x + 3 8. y = x2 + x - 2 9. y = x2 + 3and and and

2x + y = 1 x + 5y + 2 = 0 y = 3x + 7

10. y = x2 - 5x + 3 11. x2 + y2 = 49 12. x2 + y2 = 169and and and

7x + 2y = 11 y = 7 - x y = x + 7

13. x2 + y2 = 25 14. x2 + y2 = 100 15. x2 + y2 = 64and and and

x + y = 5 y = x + 2 y = 2x + 8

16. x2 + y2 = 4 17. x2 + y2 = 5 18. x2 + y2 = 36and and andy = x y = x + 2 y = 2x - 1

Page 14

A10.4 - Solving Simultaneous Equations - Quadratic and LinearWorked Exam Questions

1. The line y = x + 8 and the circle x2 + y2 = 100 meet at two points A and B.a. Show that the x-coordinates of A and B satisfy the equation x2 + 8x - 18 = 0

[3]b. Calculate the x-coordinates of A and B, give your answers to 1 decimal place

[2]2. Solve, algebraically, these simultaneous equations. Give your answers correct to 2 decimal places.

x2 + y2 = 4y = 2x + 1

[6]3. The straight line y = 5x - 2 intersects the parabola y = 3x2 + 3x - 10 at two points A and B. By solving two simultaneous equations, find the coordinates of A and b.

[6]Answers: (all correct to 2dp where appropriate)

1. x = 2 2. x = 1.5 3. x = 2 y = 1 y = 1.5 y = -1

4. x = 2 5. x = 5, y = -11 6. x = 4, y = 5y = 1 and x = -2, y = 10 and x = 2, y = 1

7. x = -1, y = 3 8. x = 0.8, y = -0.56 9. x = 4, y = 19and x = -2, y = 5 and x = -2, y = 0 and x = -1, y = 4

10. x = 2.5, y = -3.25 11. x = 7, y = 0 12. x = 5, y = 12and x = -1, y = 9 and x = 0, y = 7 and x = -12, y = -5

13. x = 5, y = 0 14. x = 6, y = 8 15. x = 0, y = 8and x = 0, y = 5 and x = -8, y = -6 and x = -6.4, y = -4.8

16. x = 1.41, y = 1.41 17. x = 0.22, y = 2.22 18. x = 3.1, y = 5.15and x = -1.41, y = -1.41 and x = -2.22, y = -0.22 and x = -2.3, y = -5.55

Page 15

A10.5 - Graph TransformationsPractice Questions

Key Ideas: The graph of y = f(x) + a is the graph of y = f(x) translated by

The graph of y = f(x - a) is the graph of y = f(x) translated by

The graph of y = k f(x) is a one way stretch of the graph of y = f(x) parallel to the y-axis with scale factor k

The graph of y = f(kx) is a one way stretch of the graph of y = f(x) parallel to the x-

axis with scale factor

The graph of y = -f(x) is a reflection in the x-axis of the graph of y = f(x) The graph of y = f(-x) is a reflection in the y-axis of the graph of y = f(x)

Questions:

1. State the equation of the graph of y = x2 - 1, after:

a. reflection in the y axis b. reflection in the x axis

2. State the equation of the graph of y = 4x + 1, after:

a. a one way stretch parallel to the y axis with scale factor 4b. a one way stretch parallel to the x-axis with scale factor 0.5

3. Describe transformations to map y = h(x) onto:

a. y = h(x) - 2 b. y = 3h(x)c. y = h(0.5x) d. y = 4h(2x)

4. State the equation of the graph of y = x2, after it has been translated by

5. Write down the equations ( in terms of f(x) ), of the following graphs, given the first graph is f(x)

y = f(x) a. b.

6. Sketch on the same diagram the graphs of:

a. y = x2 b. y = (x + 2)2 c. y = (x + 2)2 - 3d. What transformation maps a. onto c.?

Page 16

A10.5 - Graph TransformationsWorked Exam Questions

1. This diagram shows the graph of y = f(x)

The two graphs below are transformations of y = f(x). Choose the correct equation from the list below for each graph.

y = f(x + 2) y = f y = f(x - 2) y = ½ f(x)

y = f(2x) y = f(x) - 2 y = 2f(x) y = f(x) + 2

a. b.

[2] [2]

[3]2. Sketch the following graphs on the axes below. In each case the graph of y = x 2 is given to help you.a. y = x2 + 2 b. y = (x - 2)2

[1] [1]

c. y = d. y = ½ x2

[1] [1]

Answers: (all answers to 1d.p. unless stated otherwise)1.a. y = (-x)2 - 1 b. y = -(x2 - 1) 2.a. y = 16x + 4 b. y = 8x + 1

3.a.Translation

b. One way stretch, parallel to y-axis, s.f. 3 c. One way stretch, parallel

to x-axis, s.f. 3 d.One way stretch,

parallel to y-axis, s.f. 4 and one way stretch parallel to x-axis, s.f.

½ 4. y = (x - 3)2 + 4 5.a. y = 2 f(x) b. y = f(2x)

6.a. b.Translation

Page 17

S10.1 - Surface Areas of 3-D ShapesPractice Questions

Key Ideas: Volume of pyramid = ⅓ length x width x height

Volume of sphere =

Surface area of sphere =

Volume of cone =

Curved surface area of cone = πrl(h = perpendicular height, l = slant height)

Questions:

1. Calculate the curved surface area of these cylinders:a. radius = 2.7cm, height = 3.4cm b. radius = 7.2cm, height = 0.157m

2. A cylinder 6.3cm long has a curved surface area of 170cm2. Calculate the radius of the cylinder

3. Calculate the surface area of a sphere of radius 4.7 cm

4. Calculate the total surface area of a solid cone with base radius 7.1cm and slant height 9.7cm

5. A sphere has a surface area of 157.6cm2. Calculate its radius

6. Calculate the base area of a cone with slant height 8.2cm and curved surface area 126cm2

7. Calculate the total surface area of a solid hemisphere of radius 5.2cm

8. A cone has slant height 7cm and curved surface area 84cm2. Calculate the total surface area of the cone

9. A wastepaper bin is a cylinder with no lid. It is 50cm high and the diameter of its base 30cm. The outside of the bin is painted white; the inside is painted black. Calculate the area that is painted black

10. All the sloping edges of this square-based pyramid are 8cm long.Calculate the volume of the pyramid

Page 18

6cm6cm

8cm8cm

S10.1 - Surface Areas of 3-D ShapesWorked Exam Questions

1. The base of a right pyramid is a rectangle measuring 18 cm by 10cm. The pyramid has a vertical height 12 cm.Calculate the total surface area of the pyramid

[5]2. This cone is designed to hold chips. Calculate the curved surface area of this cone

[4]3. This metal rubbish bin is the frustum of a hollow cone. It is open at the top and closed at the bottom.Calculate the total surface area of the outside of the bin

[4]4. A pencil case is made by joining together a cone and cylinder. The radius of the cone and the radius of the cylinder are both 4cm. The height of the cone is 5cm and the height of the cylinder is 18cm.Calculate the total surface area of the outside of the pencil case.

[5]Answers:

1. a. 18.36π cm2

b. 226.08π cm2 2. 4.29 cm (2dp)

3. 88.36π cm2 4. 56.58π cm2

5. 3.54 cm (2dp) 6. 4.89 cm (2dp)7. 81.12π cm2 8. 129.84 cm (2dp)9. 1725π cm2 10. 81.39 cm3 (2dp)

Page 19

S10.2 - Congruence and Shape PropertiesPractice Questions

Key ideas:

Two triangles are congruent if any of these four sets of conditions is satisfied:→ Two sides and the included angle of one

triangle are equal to the two sides and the included angle of the other triangle (side, angle, side or SAS)

→ The three sides of one triangle are equal to the corresponding three sides of the other triangle (side, side, side or SSS)

→ Two angles and the side 'linking' them (the included side) in one triangle are equal to the corresponding two angles and side in the other triangle (angle, side, angle or ASA)

→ Each triangle is right-angled and the hypotenuse and one side of the triangle are equal to the hypotenuse and the corresponding side in the other triangle (Right Angle, Hypotenuse, Side or RHS)

Questions: (For the word questions, a sketch may prove useful)

1. Which of the triangles (i) (ii) (iii) (iv) (v) (vi) are congruent to any of triangles A, B or C? Explain your answer in each case.

2. Sketch an equilateral triangle. Join the midpoints of each side to make a second triangle. Prove that this triangle is equilateral.

3. Prove that the diagonals of a rectangle are equal in length.

Page 20

S10.2 - Congruence and Shape PropertiesWorked Exam Questions

1. ABCD is a parallelogram. E is the point of intersection of the diagonals.Prove that triangles ABE and CDE are congruent.Remember to give a reason for each step of your proof.

[3]2. James has used a ruler and compassed to construct the bisector of angle AOB.

By proving two triangles congruent show that angle AOC = angle BOC.

[3]

Answers: 1. Congruent Pairs: A and (iv) - SAS, B and (iii) - ASA and C and (i) - RHS

2. 3.

Page 21

A B

CD

E

S10.3 - Non-Right Angle TrianglesPractice Questions

Key Ideas: The sine rule: or

Use the sine rule when you know:a. two angles and one side b. two sides and the non-included angle

The cosine rule: a2 = b2 + c2 -2bc cos A or

Use the cosine rule when you know:a. all the sides b. two sides and the included angle To calculate the area of a triangle when you don't know the height, use:

Area = ½ ab sin C

Questions:

1. Find A , B and b

2. Find b , C and c

3. Find length BC

4. Find S

5. Find the areas of the following triangles:a.

b.

6. Town B is 45 km due East of town A. Town C is on a bearing 057o from town A and a bearing of 341o from town B. Find how far town C is from towns A and B

7. A parallelogram has sides of length 5.1 cm and 2.5 cm. Adjacent sides are separated by an angle of 70o. Find the length of each of the diagonals of the parallelogram

8. The area of a triangle PQR is 273cm3. Given that PQ = 12.8 cm and the angle PQR is 107o, find QR

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C

B

A

a

b

c

C

B

A

72 m71o

100 m

CB

A52o

81o

10 mm

CB

A59.5o

8.7 cm 9.2 cm

S

T

R

5.57 cm2.81 cm

4.32 cm

62o

3.8 m

3.1 m

81o

7 m

6.5 m67o

B

NN C

A45 km

S10.3 - Non-Right Angle TrianglesWorked Exam Questions

1. Planning permission has been granted to build houses on the triangular field shown in the diagram. The housing density permitted is 1 house for each 200 m2.Calculate how many houses may be built on this field

[5]2. This diagram shows the cross-section of a roof. Calculate the size of angle x

[3]3. Two coastguard stations, P and Q, are 32 kilometres apart. Q is due south of P. A boat, B, is on a bearing of 145o from P and 040o from Q.Calculate the distance QB

[3]4. The bearing of a ferry (F) from a lighthouse (L) is 063o. The bearing of a yacht (Y) from the ferry is 158o. The yacht is 18.5km from the ferry and 21 km from the lighthouse.Calculate the bearing of the yacht from the lighthouse

[3]

Answers:

1.A = 42.9o (1dp)B = 66.1o (1dp)

b = 96.7 m (1dp)2.

b = 12.5 mm (1dp)C = 47o

c = 9.3 mm (1dp)3. 8.9 cm (1dp) 4. S = 95.3o (1dp)

5. a. 5.2 m2 (1dp)b. 12.1 m2 (1dp) 6. a = 22.3 km (1dp)

b = 43.9 km (1dp)7. 4.9 cm (1dp) 8. b = 44.6 cm (1dp)

Page 23

S10.4 - Trigonometrical FunctionsPractice Questions

Key Ideas: The shapes and main features of trigonometrical graphs:

Your calculator and the symmetry of these graphs will help you to find solutions of trigonometrical equations

The 'length' of a repeating pattern is called the period. For a wave, the amount it varies from the mean is called the amplitude

The graphs of y = a sin bx and y = a cos bx each have amplitude a and period

Questions:

1. State the amplitude of:a. y = 5 cos x b. y = 2 sin 3xc. y = 4 sin ⅓ x d. y = 2 cos 0.5x

2. Sketch the graph of y = sin xo for values of x from 0 to 540o. Use your graph and calculator to find four solutions of sin x = 0.8. Give your answers to 1dp

3. One value of x for which cos xo = -0.3 is 107o, to the nearest integer. Without using your calculator, use the symmetry of the graph y = cos xo to find two other solutions of this equation between 0 and 540o

4. Give two other angles (between -180o and 360o) which have a tan value equal to:a. tan 40o b. tan -80o c. tan 30o d. tan 135o

5. Find all the solutions of cos 3x = -1 between 0o and 360o

6. Find all the solutions of 2 sin xo = 1 for 0 < x ≤ 360o

Page 24

S10.4 - Trigonometrical FunctionsWorked Exam Questions

1.a. Sketch the graph of y = sin x for 0o ≤ x ≤ 360o

[1]

b. Given that sin 18o = 0.31, find the values of x between 0o and 360o, for which sin x = -0.31 [2]

198o 342o

2.a. Sketch the curve of y = sin 2x for 0o ≤ x ≤ 360o.

[3]b. When x = 15o, sin 2x = 0.5. Use your graph to find another value of x which is a solution sin 2x = 0.5

75o

195o

155o

[1]3. Which of these equations is the equation of the curve shown below?

y = 4 sin ½x y = ¼ sin 2x y = 4 cos 2xy = 4 sin 2x y = 4 cos ½x y = ½ sin ½x

[3]Answers: (all answers to 1d.p. unless stated otherwise)

1.a. 5 b. 2 c. 4 d. 22. 53.1o 126.9o

413.1o 486.9o

3. 253o

377o

4.a. -140o 220o b. 100o 280o

c. -150o 210o d. -45o 315o

5. 60o 180o 300o 6. 30o 150o

Page 25

S10.5 - VectorsPractice Questions

Key Ideas:

A vector has magnitude (length) and direction but can start at any point To add or subtract column vectors, add or subtract the two components separately To multiply a column vector by a number, multiply each component by that number The resultant of two vectors is the third side of the triangle formed by those vectors If , then ABC is a straight line and BC is n times AB in length If , then AB and CD are parallel and CD is n times AB in length

Questions:

1. Write down the column vectors for , , , and .

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7

2. Find where the point is mapped by the vector.

a. (1 , 2) by b. (-3 , 2) by

3. Given that p = , q =

, work out:a. 2p b. p + qc. q - p d. 2p + qe. 3q - 2p

4. ABCD is a parallelogram. E, F, G, H are the midpoints of the sides.

= a and = b.

a. Write down the vectors , , , , and in terms of a and/or

b.

b. What does this show about EF and HG?

5.

ABCD is a kite. E is the point where the diagonals cross, BE = ED and CE = 3 x AE in length. = a and = b.

a. Work out the vectors , , , and in terms of a and/or b.

b. Explain why the vectors show that BC is not parallel to AD

Page 26

AC

D

B A B

C

E

DF

G

H

A

B

C

D

E

S10.5 - VectorsWorked Exam Questions

1. , , , Q is the midpoint of AN.

Express as column vectors: a.i. ii. iii.

[1] , [1] , [1]b. What is the relationship between MN and PQ?

[2]2. and . A is the midpoint of OC. X is the midpoint of AB.Find , in terms of a and b, in its simplest form.

[4]

3. In the diagram, , , , .a. Work out in terms of a and b. i. ii. iii.

[1] , [1] , [2]b. State two facts about the relationship between AB and CD.

[2]Answers:1. , , , , 3.a. b. c.

d. e. 2.a. (4 , 4) b. (-8 , 0) 4.a. a , b , a + b , b , a , b + a5.a. b - a , ½(b - a) , ½(a + b) , , -2a - b

5.b. Not parallel as has a and b components, whilst only has b components

Page 27

a

b

O

A

X

B

C

AC

BDO

Module 10 - Self Assessment Sheet

TL Topic Comment ExamTL

NUMBER

N10.1 - Use calculators to explore exponential growth and

decay.

N10.2 - Convert a recurring decimal to a fraction; simplify

expressions involving powers or surds including rationalising a

denominator.

ALGEBRA

A10.1 - Find formulae that approximately connect data;

express them in symbolic form.

A10.2 - Manipulate algebraic expressions including fractions.

A10.3 - Solve quadratic equations by completing the

square and using the quadratic formula.

A10.4 - Solve exactly, by elimination of an unknown, two simultaneous equations in two

unknowns, one of which is linear, the other equation

quadratic in one unknown or of the form x2 + y2 = r2 .

A10.5 - Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a), y = af(x), for linear, quadratic, sine and cosine functions f(x).

SHAPE

S10.1 - Solve problems involving surface areas of

pyramids, cylinders, cones and spheres, segments of circles

and frustums of cones.

S10.2 - Understand and use SSS, SAS, ASA and RHS

condition to prove the congruence of triangles; verify standard ruler and compass

constructions; use congruence to show that translations, reflections and rotations

preserve length and angle.

S10.3 - Calculate the area of a triangle using ½absinC; use the sine and cosine rules to solve

2-D and 3-D problems

S10.4 - Draw, sketch and describe the graphs of

trigonometric functions for angles of any size, including

transformations involving scalings in either or both the x

and y directions.S10.5 - Understand and use

vector notation; calculate, and represent graphically the sum

of two vectors, the difference of two vectors and a scalar

multiple of a vector; calculate the resultant of two vectors;

understand and use the commutative and associative properties of vector addition;

solve simple geometrical problems in 2-D using vector

methods.

DATA

D10.1 - Compare data sets (including grouped discrete and

continuous data); draw conclusions.

D10.2 - Identify seasonality and trends in time series, from

tables or diagrams; interpret graphs modelling real

situations.

D10.3 - Select a representative sample from a population using random and stratified sampling;

criticize sampling methods.