Wrenn Academy Year 11 Mathematics Revision Session Paper 1 Wednesday 25 th May 2016

Number Topics

Estimate the value of: x 7.85 Answer Round each number to 1 significant figure! 80 2 x = 5

Estimation and Place Value Use approximations to estimate the value of – 316 x You must show your working First make sensible approximations 300 x Then work out 1200 ÷ ÷ 2 = 6000

Using the information that 14 x 23 = 322 Write down the value of.. (i) 1.4 x 2.3 (ii) 322 ÷ 2.3 Answer This question has both values with one decimal point movement 1.4 x 23 = x 2.3 = 3.22 Re-arrange the original calculation first! 322 ÷ 23 = ÷ 2.3 = 140 Notice that the decimal point has moved the other way when you divide!

Using the information that 58 x 117 = 6786 Write down the value of.. (i) 0.58 x 117,000 (ii) ÷ 5.8 Answer 5.8 x 117 = x 117,000 = 67, ÷ 58 = ÷ 58 = ÷ 5.8 = 11.7 Notice which way the decimal point moves! 0.58 x 117 = 67.86

Write each of these numbers correct to 1 significant figure.. a) 26,366 b) c) 45,071 d) Answer 30, , All answers have only one digit other than 0!

Find the Highest Common Factor (HCF) of 54 and 135 Prime factors of 54 = 2x27 = 2x3x9 = 2x3x3x3 of 135 = 3x45 = 3x3x15 = 3x3x3x5 Common factors are 3x3x3 Highest Common Factor = 27 Find the Least Common Multiple (LCM) of 28 and 42 Multiples of 28: 28; 56; 84; 112; 140; ………… Multiples of 42: 42; 84; 126; ……… Least Common Multiple = 84 Factors, Multiples, HCF and LCM

The cost of a CD player is £200 + vat. The rate of vat is 17.5%. What is the cost of the CD player? Answer VAT is easy to work out by hand if you do the following.. 10% of 200 = 20 5% of 200 = % of 200 = = £235

Put these in order of size, starting with the smallest; % Answer First turn these values into decimals Now sort them starting with the smallest 4 10 =

Work out: 1212 x 4545 Give your answer in its simplest form Answer When multiplying fractions, just multiply the top two numbers then the bottom two numbers = = Now simplify it!

Work out: Answer When adding fractions you have to make the denominators (bottom numbers) the same! = 1818

Work out: 2323 ÷ 4 Answer To divide fractions, turn the second fraction upside down (invert) and then multiply the fractions 4 1 x ÷ = 2 12 You can leave your answer like this or simplify it if asked!

Work out: 2323 ÷ Answer First convert the mixed numbers to top heavy (improper or vulgar) fractions ÷ Now deal with it the same way as question 3! 5 9 x 25 27

Mrs Jones inherits £ She divides the £ between her three children Laura, Mark and Nancy in the ratio 7:8:9 How much does Laura receive? Laura, Mark and Nancy receive shares (24 shares) between them One share is worth £ ÷ 24 = £500 Laura receives 7 shares Laura receives £500 x 7 = £3500 Ratio and Proportion

If 3 coffees cost £4.17, what would 7 coffees cost? Answer 4.17 ÷ 3 = £1.39 £1.39 x 7 = £9.73 One coffee!

These two triangles are similar. Find the lengths of x and y y 30 cm 20 cm 12 cm x 5 cm Answer Similar means that one triangle is an enlargement of the other And..x = 20 ÷ 2.5 = 8cm 30 ÷ 12 = 2.5 scale factor So..y = 5 x 2.5 = 12.5cm

Standard Form Here are six numbers written in standard form. 2·6 x ·75 x ·84 x 100 8·2 x ·5 x ·9 x x 10 5 = x 10 6 = x 100 = x = x = x = Write down the largest number 1.75 x 10 6 Write down the smallest number 8.2 x 10 -3

a) Write 2.7 x 10 3 as an ordinary number b) Write 3.12 x as an ordinary number Answer

a) Write 38,500,000 in standard form b) Write in standard form Answer 3.85 x x Standard form must start with values between 1 and 10!

Algebra Topics

Find the value of 3x + 4y When x = 6 and y = -3 This is a SUBSTITUTION question If x = 6, 3x = 3 lots of 6 And if y = -3, 4y = 4 lots of -3 So … 3x + 4y = (3 x 6) + (4 x -3) = Adding a negative number means you take it away So the answer is 6 Substitution

Find the value of 5p + 2q when p = 4 and q = -7 Find the value of u 2 – v 2 when u = 5 and v = 3 5p + 2q will be (5 x 4) + (2 x -7) This is (20) + (-14) Adding a negative number is the same as taking it away So 20 – 14 = 6 u 2 – v 2 will be 5 2 – 3 2 This is 25 – 9 So the answer is 16

Simplify 5x + 3y – 2y + 4y Simplify means collect together like terms, So that’s all the xs and all the ys It can be easier to do the adding first So … 5x + 3y + 4y – 2y So that gives … 5x + 7y – 2y = 5x +5y Simplifying Expressions

Simplify the expression… 5p 2 Answer + 3q– p 2 + 2q 4p 2 + 5q

Expand and simplify 4(m + 3) + 3(2m – 5) First multiply out the brackets 4m m – 15 Then simplify 10m - 3 Expanding and Simplifying

7z + 2 = 9 – 3z 7z + 3z = 9 – 2 10z = 7 z = 7 ÷ 10 z = 0∙7 Sort out the 7z and 3z and the numbers first Watch out here with signs Solving Equations

Solve the equation….. 4(3n + 7) = 16 Answer Brackets first! 12n+ 28= n = -12 n = -1 It is ok to have a negative answer!

Solve the equation… 4(x + 2) = 6x + 4 Answer Brackets first, then group the terms! 4x+ 86x+ 4 = x 4 = 2x x = 2

6r + 2 = 8 6r = 8 – 2 6r = 6 r = 1 Solve these equations 7s + 2 = 5s + 3 7s – 5s = 3 – 2 2s = 1 s = ½ Be careful here! And here! 12- y = – y = 5 x 3 12 – y = – 15 = y -3 = y (which is the same as y = -3) And especially here! And then, of course, there’s here!

Expand and simplify (4x - 3)(x + 5) Solve 4(z - 1) = 2(z + 3) Multiply out the brackets first.. 4z - 4 = 2z + 6 Then solve as before – careful with the signs!!! 4z – 2z = z = 10 z = 5 This is harder. Multiply everything in the first bracket by everything in the second bracket 4x x – 3x – 15 Simplify 4x x - 15

Solve this simultaneous equation x + 3y = 13 Answer Multiply the first equation by 3 to make the x values the same 3x + 2y = 4 ( x 3)3x + 9y = 39 Now subtract (there are no x’s left then!) 7y = 35 y = 5 Now substitute y=5 into the first equation x + 3(5) = 13 x = -2

Give some values of x and y that make this equation true x + 3y = 7 3x - 6y = 6 These equations are simultaneous. What does this mean? How can I change the first equation to make the x or y Co-efficients the same? ( x 2) 2x + 6y = 14 What should you do now? Hint: if I added them, what would happen? + 5x=20 So…. x = 4…how do we find y? + 0y What does this word mean?? Ooh look! Two 6’s

Expand 4x(x 2 + 5) Expand means multiply out the brackets That means mutliply both terms inside the bracket by 4x So the answer is 4x x Expanding Brackets

Expand these brackets………. a) 7(x + 3) b) x(x + 3) c) 2y(3y – 5) Answer Remember to multiply out both terms! 7x + 21 x 2 + 3x 6y 2 – 10y

Expand and simplify…… Expressions 2 2(4a + 2) – 3(2a – 4) Answer Multiply out the brackets first! + 12 Notice that this sign has changed to a +, because of the – outside of the brackets 2a a+ 4– 6a

Simplify…… i) 3a 2 b x 7ab 3 ii) (x + 2) 2 (x + 2) Answer 21a 3 b 4 (x + 2) Notice how the powers have changed! ii) (x + 2) (x + 2)

Expand these brackets………. a) (x + 1)(x + 3) b) (x – 6)(x + 2) c) (x – 4)(x + 7) Answer You should get 4 terms when you multiply out the brackets! x 2 + 3x + 1x + 3 x 2 + 4x + 3 x 2 + 2x - 6x - 12 x 2 - 4x - 12 x 2 + 7x - 4x - 28 x 2 + 3x - 28

Factorise 2x + 6 This means take out the factors that are common to both terms. So that would be 2, giving an answer of 2(x + 3) Factorise x 2 + 6x Here the x is common to both terms, giving an answer of x(x + 6) Factorising

Factorise….. 12x + 4 Answer Find the common factor(s) in all terms. In this case it is 4! 44 (3x + 1) Notice that this is 1 and not 0! 12x + 4

Factorise …. 6x + 18y Answer Find the common factor(s) in all terms. In this case it is 6! 66(x + 3y) 6x + 18y

Factorise…… 8xy + 12x Answer Find the common factor(s) in all terms. In this case it is 4x! 4x (2y + 3) Why is this just 3? 8xy + 12x

Factorise….. 6x 2 – 3xy Answer Find the common factor(s) in all terms. In this case it is 3x! 3x (2x – y) 6x 2 – 3xy

Factorise………. a) x 2 + 2x b) y 2 – 6y c) 8x 2 – 20xy Answer Find the common factors in all the terms x (x + 2) y (y – 6) 4x (2x – 5y)

Factorise………. a) x 2 + 3x + 2 b) x 2 + 7x + 12 c) x 2 + 2x - 15 d) x 2 - 2x - 35 Answer This means separate into two brackets! (x + 1)(x + 2) (x + 3)(x + 4) (x + 3)(x - 5) (x + 5)(x - 7) The number values in each set of brackets must multiply together to make the 3 rd term in the quadratic, and must add together to make the number of x’s in the 2 nd term!

Complete this table of values for y = 2x - 1 To get the y value, you double the x value and then take away 1 So when x = 0, double 0 is still 0, take away 1 = -1 When x = 2, double 2 = 4, take away 1 = 3 A useful check that you are right, is to notice that this equation gives a straight line, so there will be a pattern in the y numbers. There should be the same difference between all of them. Here the difference is 2 3 Plotting Straight Line Graphs / Curved Graphs

On the grid, draw and label the lines y = -4 and y = 2x + 1 The point y = -4 is here. So the line y = -4 is the line where all the y coordinates are -4 y = -4 To find the line y = 2x + 1 Substitute some values for x to get y If x = 0, 2x + 1 = 1, y = 1 If x = 1, 2x + 1 = 3, y = 3 If x = 2, 2x + 1 = 5, y = 5 And so on Join the dots! x x x Write down the point where the lines y = -4 and y = 2x + 1 cross Y = 2 x + 1 (-2∙5, -4)

Complete the table of values for y = x Then, on the grid, draw the graph of y = x x y1-3-2 When you see x 2 you expect to see a curve. You also expect to see a symmetrical number pattern in the table. Work out the positive values first When y = 2, y 2 will be 4, take away 3 will be 1 When y = 3, y 2 will be 9, take away 3 will be 6 When y = -1, y 2 will be 1, (NOT -1!!) take away 3 will be x x x x x x Plot the points and join with a curve

Write down the inequality shown by the following diagram Solve this in the same way as you would an equation 3x + 8 < 29 So 3x < 29 – 8 3x < 21 x < 21 ÷ 3, so x < 7 This shows that x is less than 2. The empty circle means 2 is not included So x < 2 Representing and Solving Inequalities

Solve the inequality… 2x + 7 > 1 Answer Solve it like an equation 2x + 7 > x > -6 x > -3

Write down all the values of m if -2 < m ≤ 4 and m is an integer…… Answer m can be any of these numbers, but not -2

Write down all the possible values of p when -3 ≤ p < 2 and p is an integer…. Answer p can be any of these numbers, but not 2

x This diagram shows an inequality We can write the inequality using symbols

How? x is somewhere between the spots A filled in spot means that x includes that number An empty spot means that x DOESN’T include that number

So….. In words….. x is more than or equals -2 and is less than 5 In symbols….. x ≥ -2 and x < 5 Which is written as -2 ≤ x ‹ 5

Shape and Space Topics

Transformations Describe fully the single transformation which takes shape A to shape B The diagram shows two identical shapes, A and B Imagine a hook attached at one end to (0,0) and the other end to a corner of shape A If this was rotated half a turn anticlockwise, shape A would move to position B So the single transformation taking shape A to shape B is a rotation of 180º anticlockwise about (0,0)

AREA OF COMPOUND SHAPES -A shape has dimensions as shown Calculate the shaded area First, split this shape into smaller shapes that we recognise A C B Area of triangle A b x h 2 3 x 4 2 = 6 cm 2 Area of triangle B b x h 2 2 x 4 2 = 4 cm 2 Area of rectangle C l x w = 10 x 1 = 10 cm 2 Total area = = 20 cm 2

The area of the cross section of the triangular prism is 20cm 2. The length is 12cm Work out the volume of the prism 12cm 20cm 2 Answer Volume = cross sectional area x length Volume = 20 x 12 Volume = 240cm 3 SURFACE AREA AND VOLUME

What do the external angles total? What is the value of x? x x x x x Answer The external angles of a polygon always add up to..360 o 360 ÷ 5 = 72 0

The exterior angle of a regular polygon is 45 o. How many sides has the polygon? Answer 45 o 45 o 45 o 45 o 45 o 45 o 45 o 45 o The polygon has 360 ÷ 45 = 8 sides

Find the value of x 107 o x 34 o 48 o 47 o 63 o Answer x = 360 x = 61 o

Find the size of the interior angle of this regular octagon Answer Find the exterior angle first! 360 ÷ 8 = 45 o 45 o So the interior angle must be?? 180 – 45 = 135 o 135 o

Angle Facts The diagram shows a regular octagon Calculate the size of the exterior angle of the regular octagon, marked y on the diagram. We know this is a REGULAR octagon, so all the exterior angles are equal The total of all the exterior angles is 360º (a full turn) So angle y will be 360 ÷ 8 Angle y = 45º

Work out the size of angles x and y. Give reasons for your answers 125 o x y x = Corresponding or F angles 125 o 55 o y = 55 0 Angles on a straight line = Answer

Work out the size of angles x and y. Give reasons for your answers 62 o x y 40 o x = 118 o Notice the inside angles are complementary 118 o 62 o y = 62 o Alternate or Z angles Answer

Find the angle marked g. Give a reason for your answers 28 o g 95 o Work out this angle first (angles on a straight line) 57 o g = 57 0 Alternate or Z angles Answer

Find angles x and y. Give reasons for your answers 68 o x y 120 o x is 68 o Z angles y = 52 0 Angles in a triangle 68 o 60 o 52 o Answer 60 0 Angles on a straight line

Handling Data Topics

Pie Charts The holiday destinations of 30 students were recorded Draw a clearly labelled pie chart to represent this information First, work out the sizes of the angles that will represent each destination There are 360° in a circle and there are 30 students, so each student will be represented by 12° (360 ÷ 30) Italy France will be 6 x 12 = 72° Spain will be 9 x 12 = 108° Italy will be 3 x 12 = 36° Greece will be 7 x 12 = 84° America will be 5 x 12 = 60° France Spain Greece America

STEM and LEAF -The stem and leaf diagram shows the ages, in years, of 15 members of a badminton club (a)What is the median age of the members The median is the middle value, when the values are in order The middle value of 15 will be the eighth value This is 42 (b) What is the range of their ages The range is the difference between the youngest and the oldest member. So that would be 62 – 27 = 35 years

PROBABILITY - Tom and Sam take turns to throw a dart at a target The probability that Tom hits the target is 0·3 The probability that Sam hits the target is 0·2 Complete the tree diagram 0·7 0·8 0·2 0·8

Probability The diagram shows a spinner. When the arrow is spun, the probability of scoring 2 is 0∙3. The arrow is spun twice and the scores are added. (a) Complete the tree diagram (b) What is the probability that the total score is 4? ∙3 0∙3 0∙7 0∙7 0∙7 To get a 4, you need to score 2 on both spins. There is only one route for this The probability of a total of 4 is found by multiplying 0∙3 by 0∙3 So this is 0∙09 The probabilities must add up to one

Complete this probability tree for throwing a fair red dice and a fair blue dice Red Dice Blue Dice Six Not Six 1616 Answer

CUMULATIVE FREQUENCY CURVES - 60 students took a test. The graph shows information about their marks Cumulative Frequency Marks a) What was the median mark? b) What was the lowest mark? c) Estimate how many students scored 12 or less marks d) Estimate the interquartile range Answer Question = 4.5

BOX PLOTS - This box plot shows information about 40 students’ test marks Which of these statements are true? a) The top mark was 54 b) The range was 35 c) The median was 42 d) The interquartile range was 12 e) ½ the students scored less than 38 Answer True False - 53 False - 30 False - 38