ζYear 9 – End of Year RevisionDr Frost

PercentagesBe careful: Are you trying to find the new value or the old value? In the first case, you multiply, in the second case, you divide. Percentage change is based on the old value.

A jumper is bought in for £30 and marked up by 40%. What is it sold for?

Answer: 30 x 1.4 = £42

After one year the value of a care fell by 20% to £9600. What was its original value?

Answer: £12000

I put £15,000 into a savings account. It accrues 2.6% interest. What is in my account in one year’s time?

Answer: £15390

Lucy made 20% profit on the picture frame she sold at £35. What did she buy it in for?

Answer: £29.17

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Percentages

The interest rate for a savings account is 2.5% p.a. with compound interest. The principal is £1500. How much do I have in 10 years time?

Answer: £1500 x 1.02510 = £1920.13

My Bentley depreciates in value 10% each year. It is bought new for £150,000. How much is it worth in 5 years time?

Answer: £150,000 x 0.95 = £88573.50

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Compound Measures

A cat travels at 15km/s. It races around a 50km track. How much time did it take him?

Answer = 3.33s

The density of a hamster is 1.3kg/m3. Its volume is 0.03m3. What is the hamster’s mass?

Answer = 0.039kg

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GraphsMatch the graphs with the equations, and identify what type of equation it is.

1

2

3

4

5

6

7

8

9

10

11

y = -2x3 + x2 + 6xy = 4x

y = 2x - 3y = x2 + x – 2y = 5 – 2x2

y = 2x3

y = 5 – xy = x3 – 7x + 6y = -3x3

6 Cubic11 Exponential9 Straight Line1 Quadratic2 Quadratic

8 Reciprocal

5 Cubic10 Straight Line3 Cubic4 Cubic

7 Reciprocal

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Graphs

y = x3 – 2x2 - 5x + 6

x -3 -2 -1 0 1 2 3 4y -24 0 8 6 0 -4 0 18

When sketching, ensure you sketch a curvy line (i.e. don’t join up your points with lines), or you’ll lose a mark.

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Changing the SubjectChange the subject of the formula to the indicated letter.

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(b)

Changing the Subject

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Changing the SubjectThe following require you to factorise at some point.

Make a the subject of the formula:

n = _3a_a+1

a = n2-PnP-1? ?

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Simultaneous Equations

You can either use elimination or substitution.

3x + 2y = 105x – 2y = 14

3x + 2y = 44x + 3y = 7

x = 3, y = 0.5 x = -2, y = 5? ?

Probability

Question: Give there’s 5 red balls and 2 blue balls. What’s the probability that after removing two balls from the bag, we have a red ball and a blue ball?

R

B

R

B

R

B

57

27

46

26

56

16

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?Answer =

1021?

Probability

What’s the probability that when I roll 10 dice, I see the same number on every die? 𝒑 (𝒔𝒂𝒎𝒆𝒏𝒖𝒎𝒃𝒆𝒓 )= 𝟔

𝟔𝟏𝟎=𝟏𝟔𝟗

What’s the probability that when I roll 10 dice, the total of the dice is 10?

𝒑 (𝒔𝒖𝒎𝒊𝒔𝟏𝟎)= 𝟏𝟔𝟏𝟎?

Difficult: What’s the probability that when I roll 3 dice, I see exactly two sixes. 𝒑 (𝒕𝒘𝒐𝟔 𝒔 )=𝟏𝟓

𝟔𝟑 =𝟓𝟕𝟐?

Total outcomes

Matching outcomes

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6 29 3

Probability

If I have two dice, one numbered 1, 2, 3 and the other numbered 2, 3, 4, what’s the probability the sum is at least 5?

+ 2 3 4

1 3 4 5

2 4 5 6

3 5 6 7

Second DieFi

rst D

ie

p(sum ≥ 5) = =

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Sequences

Determine the formula for the following sequences.

5, 8, 11, 14, 17, ... 10, 8, 6, 4, 2, 0, -2, ...

3, 9, 17, 27, 39, ...

Un = n2 + 3n - 1

Un = 3n + 2 Un = 12 – 2n

1, 3, 6, 10, 15, ...

Un = 0.5n(n+1)

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Expanding brackets

Expand the following.

(x+1)(x-2) = x2 – x – 2(x-4)(x-8) = x2 – 12x + 32(x+1)(y+1) = xy + x + y + 1(x2+1)(y2-1) = x2y2 – x2 + y2 – 1(2x+1)(2x-1) = 4x2 – 1(x + 1)(x + y + 1) = x2 + xy + 2x + y + 1 x(y-x)-y(x-y) = y2 – x2

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This is known as the: difference of two squares

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Factorisation

Factorise the following

x2 + 7x + 12 = (x + 4)(x + 3)x2 + 2x – 3 = (x – 1)(x + 3)x2 – 10x + 24 = (x – 4)(x – 6)2x2 – 5x – 12 = (2x + 3)(x – 4)12x2 + 5x – 3 = (4x + 3)(3x – 1)x2 – 9 = (x + 3)(x – 3)4 – y2 = (2 + y)(2 – y)x3 – x = x(x + 1)(x – 1)16x2y2 – 9z4 = (4xy + 3z2)(4xy – 3z2)x4 + 2x2 – 143 = (x2 + 13)(x2 – 11)

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x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

Object

Enlarge the shape by a scale factor of 2 about the point (0,-2)

Enlargement

Image

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

Object

Enlarge the shape by a scale factor of -1 about the point (0,2)

Enlargement

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

Object

Enlarge the shape by a scale factor of -0.5 about the point (0,2)

Enlargement

Trigonometry

60 ° x

12

30 °

4

xx = 13.96 x = 3.46

65 °x

15x = 6.99

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2

3

θ

1

3

1 1θ

6

θ8

a

b c

d

θ

Trigonometry

θ = 33.69° θ = 70.53°

θ = 45°

θ = 48.59°? ??

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Trigonometry

What is the cosine of the angle between the internal diagonal of a cube and the bottom face of the cube?

Answer = √2√ 3?

Solving EquationsSolve the following equations for x.

x(2x + 1)(x – 2) = 0 x = 0 or -0.5 or 2x2 = 4 x = 2 or -2x2 = 3x x = 0 or 3x2 + 5x – 6 = 0 x = -6 or 1x3 = x x = -1, 0 or 1x2 + 32 = 12x x = 4 or 825x2 – 4 = 0 x = 2/5 or -2/5

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2x + 2

x3x - 2

Determine x

Answer:By Pythagoras, x2 + (2x+2)2 = (3x-2)2

Expanding and simplifying, we get 4x2 – 20x = 0Solving, x = 5 (we reject the 0 solution).

Solving Equations

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Similarity

5

3

10 x

x = 16?

Similarity

A square is inscribed in a right-angled triangle with length 4 and height 3. Find the width of the square.

3

4

Length of square = 127?

Loci

AB

3km4km

A Spotted Studdert Sheep is known to be within 3km of A and 4km of B. What region could the sheep be in?

Loci

AB

3km4km

Now the sheep is also known to be of equal distance from A and B. Where can it be?

Loci

AB

3km4km

Now the sheep is within 3km of A, but at least 4km away from B. Where could it be?

Loci

I’m equidistance from two lines AB and AC. Where could I be?

A

B

C

Loci

I’m the indicated distance away from the walls of a building. Where could I be?

Circular corners.

Straight corners.

Loci

My sheep is attached to a fixed point A on a square building, of 10m x 10m, by a piece of rope 20m in length. Both the sheep and rope are fire resistant. What region can he reach?

10m

20m A

Dimensional Analysis

(all variables are lengths)

Expression Length Area Volume None of these

2rh

πr + 4h

(r+h)2

3b

b3 + rh

πr2(h + r)

bhr_ (b+h)

Click your choice.

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Ratio

My fish tank has black and yellow fish in the ratio 3:1. A fish plague, Sanjotitus, wipes out a third of my fish. I then restock my fish tank with just black fish, so that I have the same number of fish as before. What’s the new ratio of black to yellow fish?

Answer = 5 : 1?

Method 1:Suppose a full tank has 12 fish. Then 9 fish are black and 3 yellow. The plague leaves 6 black fish and 2 white. Then if we fill up the rest of the tank with black fish, we have 10 black fish and 2 yellow. This ratio is 5:1.

Method 2: of the tank is black and is yellow. The plague leaves us with of the fish, so the tank is now full of black fish and of yellow fish. Now the of the tank wiped out is replenished with black fish, so that’s black fish (and still yellow fish). This ratio is 5:1.

Proportion

x 16 8 24y 10 5 15

Given that y is proportional to x, find the missing values.

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Inverse proportion

x 16 50 0.25L 5 2.83 40

Given that L is inversely proportional to √x, fill in the missing values in this table.

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Inequalities

Solve the following.

2x > x - 6

-x + 1 ≤ 6

x > - 6

x ≥ -5

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1 ≤ 2x + 3 < 9 -1 ≤ x < 3?

− 𝑥2 ≤1 𝑥≥−2?

Inequalities on a number line.

2 ≤ x < 4 x < -1 or x > 4

0 1 2 3 4 5

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-1 0 1 2 3 4

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Inequalities on a number line.

2 ≤ x < 5x < 3 or x > 4

0 1 2 3 4 5

Draw the range of x on the number line given that both of these inequalities hold.

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-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

-4 < y ≤ -2

-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

y ≤ x + 1 and x ≤ 6 and y > 2

Inequalities

When would

When all of x, y and z are negative, or one of x, y and z are negative.

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Arcs and Sectors

5

Sector area = 10.91

Arc length = 4.36 Area = 20

Radius = 4.122.1cm

Sector area = 4.04cm2

Arc length = 3.85cm

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50°

105°

135°

(Hint: Plug values into your formula and rearrange)

The shape PQR is a minor sector.The area of a sector is 100cm2.The length of the arc QR is 20cm.

a) Determine the length PQ.

Answer: 10cm

b) Determine the angle QPR

Answer: 114.6°

P

Q

R?

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Arcs and Sectors

Volume of a prism

10cm

4cm

6cm

8cm

Volume = 400cm3 ?

1m

5m

3m

5m

6mVolume = 17m2 x 6

= 102m3 ?

Surface Area

8m

4m

2m

Surface Area = 112m3?