Web viewArea of a Rhombus = h 2 a+b where a and b are the length of the top and bottom

Leytonstone School

Edexcel GCSE MathematicsHigher Paper

Flash Cards

© Mr A 2016

Card One – Formula to memorise

Area of circle ¿ π r2

Circumference of circle ¿2π r

Area of a rectangle ¿ l×w

Area of a triangle ¿ 12 bh

Speed ¿ distancetime

Density ¿ massvolume

Area of a Rhombus ¿ h2

(a+b )

where a and b are the length of the top and bottom

lengthof arc= θ360

×2πr

Areaof sector= θ360

×π r2

Pythagoras’s theorem: a2=b2+c2,where a is the longest side of a right angle triangle

sin θ=opphyp

cosθ= adjhyp

tanθ=oppadj

Card Two – Calculations

1. How do you work out the reciprocal of a fraction?

Turn it upside down e.g. the reciprocal of 34 is 43

2. How do you work out the reciprocal of a whole number?

Turn it into a whole number and then turn it upside down e.g. to work out the reciprocal of 3, turn it into 31 and then turn it upside down 13

3. What happens when you times a number by its reciprocal?

The answer is always 1. E.g. 3× 13=1

4 Fill in this table:

+ ×∨÷ + = +

+ ×∨÷ - = -

- ×∨÷ + = -

+ ×∨÷ + = +

5. Fill in this table

xa× xb = xa+ b

xa÷ xb = xa−b

xab

= xab

x−a = 1xa

x0 = 1

xmn = n√ xm

x−mn = 1

n√xm

Leytonstone School, Edexcel GCSE Maths, Flash Cards © Mr A 2016

Card Three – Percentages

1. How do you write a percentage as a fraction?

Divide your number by 100 e.g. 80% = 80100 , you should

always simplify where possible so 80100=810

= 45

2. How do you write a fraction as a percentage?

Turn the fraction into a decimal and then times by

100. e.g. 320=0.15=15 %

3. How do you work out the original amount after a percentage increase?

Original Amount= New Amount

1+ Percentage increase100

e.g. Mr T earns 5 shillings a week after a 20% increase, what did he earn originally.

Original Amount= 5

1+ 20100

Original Amount= 51+0.2

Original Amount= 51.2

Original Amount=4

3. How do you work out the original amount after a percentage decrease?

Original Amount= New Amount

1−Percentage decrease100


Card Four – Ratio and Proportion

1. What is the difference between a ratio and a fraction?

A fraction compares a portion to a total e.g.23

compares a portion of two to a total of three. A ratio compares a portion to another portion 2:1, compares a portion of 2 to a portion of 1.

2. Split 48 in the ratio 5:7. What are the three steps?

i) Add the total number of parts e.g. 5+7 = 12ii) Divide the total by the total number of parts

by the total., e.g. 48 divided by 12 = 4iii) Times the value of one part by the number of

parts allocated to each group, e.g. 5 times 4 = 20 and 7 times 4 = 28. So 48 split in the ratio 5:7 = 20:28

3. What does ‘y∝x’ mean?

y is proportional to x, as x increases y increases by the same proportion. When doing calculations we replace ‘y∝ x’ with y=kx, where k is the constant of proportionality.

4. What does ‘y∝ 1x ’ mean?

y is inversely proportional to x, as x increases y decreases by the same proportion. When doing

calculations we replace ‘y∝ 1x ’ with ‘y= kx ’ where k is

the constant of proportionality.

5. what does ‘y∝x2’mean?

y is proportional to the square of x

6. what does ‘y∝1x2 ’ mean

y is inversely proportion to the square of x


Card Five – Standard Form and Surds

1. What does standard form mean?

It means a number in the form x×10a, it is often used in science etc. to manipulate large and small numbers.

2. What sort of function is this ax?

Exponential, as x increases the function increases at a greater and greater rate, hence exponentially.

3. What would axrepresent if a > 1

Exponential growth

4. What would axrepresent if a < 1

Exponential decay

5. Fill in the rest of this table

√a×√b=√ab

√a÷√b=√ ab

6 Simplify the following √2×√8

√2×√8=√16=4

6. What is wrong with this surd?

1√2

We are not allowed to have a surd as a denominator. To rationalise the denominator times both the top and the bottom by √2 , e.g

1√2

X √2√2

=√22


Card Six – Bounds

1. What is a bound?

When we round a measurement to a degree of accuracy, for example 2m to the nearest m. The real distance could be somewhere between the smallest number that would round up to the required measure of accuracy or the largest number that would round down. These are called the bounds. The bounds of 2m to the nearest m are 1.5 m and 2.5 m.

2. How do we use bounds in calculations? Complete this table:

Calculation

UB of a+b UB of a+UBof b

LBof a+b LBof a+LBof b

UBof a−b UB of a−LBof b

LBof a−b LBof a−UBof b

UB of a×b UB of a×UBof b

LBof a×b LBof a×LBof b

UB of a÷b UB of a÷ LB of b

LBof a÷b LBof a÷UBof b


Card Seven – Mathematical Vocabulary

1. What does coefficient mean?

Is the amount a number or pi is times by. E.g. 3a, the coefficient of a is 3, or in 5x2, the coefficient of x2 is 5.

2. What is an expression?

An expression is a combination of letters, numbers, and mathematical signs, it does not equal anything. E.g. ‘5x – 3’ is an expression.

3. What is an equation?

An equation is an expression which is equal to something, often another expression. E.g. 5x -3 = 2x + 6

4. What is an identity?

An identity is an equation which is true for all values of the unknown e.g. 3x + 4x = 7x, no matter what the value of x both sides are equal.

5. What is a formula?

A formula is an equation with real life application e.g. acceleration= changeof speed

time is a formula


Card Eight – Linear (straight line) graphs

1. In the graph y = mx + c, what do m and c stand for?

m stands for the gradient (how steep the line is) and c stands for the y-intercept.

2. What does perpendicular mean?

Perpendicular means at right angles, for example the y-axis is perpendicular to the x-axis since they make right angles with each other.

3. How do you work out gradient?

Gradient=Change∈ yChange∈x

or in other words

Gradient=y2− y1x2−x1

4. What is the gradient of the line that passes through (0,-2) and (2,-6)

Gradient=−6−−22−0

Gradient=−42

Gradient=−2

5. What is the gradient of the line with gradient m?−1m

6. What is the formula for a line perpendicular to the line y = 2x + 3 and passes through (1, 5)

We know the gradient of our perpendicular line is

going to be −12 , so it is going to take the form

y=−12

x+c. To work out c, we substitute the values of

the point we know it passes through 5=−1

2+c ,therefore c=5.5 ,

so the formulais y=−12

x+5.5.


Card 9 – Simultaneous Equations Part 1

1. What are the two ways you can solve simultaneous equations?

Graphically and algebraically

2. Describe how you would solve an equation graphically.

Make y the subject of both formulas then plot both lines and the point where they intercept is the solution.

3. What needs to be the same to solve simultaneous equations algebraically.

The coefficient of either x or y

4. If the coefficient of x or y are identical do you add or subtract the equations?

Subtract

5. if the coefficient of x or y are identical except for their sign do you add or subtract equations?

Add

Card 10 – Quadratic Equations – Part 1

1. What is a quadratic equation?

Any equation where x2 is the highest power. e.g. x2 + 7x – 8 = is a quadratic equation.

2. What does a quadratic equation HAVE to equal before you can attempt to solve it?

Zero

3. Describe the steps of factorising.

i. List all the factor pairs of the constant (or ‘c’)

ii. then we add them e.g. in the equation x2 + 7x – 8, the constant is – 8 and the factor pairs are:

-8 and 1, their sum is -7

-4 and 2, their sum is -2

-2 and 4, their sum is 2

-1 and 8, their sum is 7

iii. Then we check whether the sum of any of the factor pairs is equal to the coefficient of x, in this case the sum of -1 and 8 is equal to 7.


iv. take those factor pairs and put them in brackets in this form (x-1)(x+8)

4. Complete the square of this equation

x2+6 x−5=0

x2+6 x=5

coefficient of x is 6, to complete the square we need to add half of the coefficient of x and then square it, half of 6 is 3, the square of 3 is 9. We need to add 9 to both sides

x2+6 x+9=5+9

x2+6 x+9=14

Then we can factorise x2+6 x+9, which gives us(x+3)(x+3)=14

(x+3)2=14

Square root both sides

x+3=±√14x=−3±√14x=−3+√14∨−3−√14


Card 11 - Inequalities

1. Which of these means x is greater than 4 and which one means x is less than 4

x < 4, x > 4

x > 4 means x is greater than 4

x < 4 means x is less than 4

2. What is an inequality?

An inequality is an equation that expresses whether an unknown is greater or less than an amount. e.g.

x>3

is an equality since it tells us that x is any number bigger than 3.

3. What does and mean on a number line?

means a number is less than or greater than the number it is above, but not equal to.

means a number is less than or greater than the number it is above and it can also be equal to it.

4. What should you avoid doing when solving an inequality?

Times or dividing by a negative number

5. What happens if you do have to do that?

The greater than or less than sign reverses.


Card 12 – Trial and Improvement

1. The answer on this Trial and Improvement is right, but this answer wouldn’t get full marks. Why?

The square root of 29 lies between 5 and 6, find the square root of 29 by trial and improvement to 1.d.p.

x x2 Too Big/Too small

Bounds

5<x<6

5.5 30.25 Too big 5<x<5.5

5.3 28.09 Too small 5.3<x<5.5

5.4 29.16 Too big 5.4<x<5.5

Since 5.4 is closer, the answer must be 5.4 to one d.p

Although this answer is correct, it would not get full marks as the candidate has not PROVEN that the real answer will always round to 5.4. An extra line is required:

5.35 28.6225 Too small 5.35<x<5.4

Since all the values between 5.35 and 5.4 round to 5.4, you have now PROVEN that it is closer to 5.4.


Card 13 – Quadratic graphs

1. Is this a ‘happy’ or a ‘sad’ graph, y = x2 + 2x – 8

Happy, as the coefficient of x2 is positive (1), the graph look like this:

2. Is this a ‘happy’ or ‘sad’ graph, y =-x2-2x+8

Sad, as the coefficient of x2 is negative (-1), the graph looks like a sad face:

3. What is the u or n shaped graph called?

Parabola

4. How do you solve a quadratic equation graphically?

The points where the parabola (curved line) crosses the x axis are the solutions.


Card 14 – Circular graphs and advanced simultaneous equations

1. What does this graph look like, x2 + y2 = 9

A circle with radius 3 and centre 0,0, since 3 is the square root of 9

2. What does this graph look like, x2 + y2 = r2?

A circle with radius r, since r is the square root of r2

3. What are the two ways of solving simultaneous equations where one of the equations is quadratic?

Graphically and Algebraically

4. How would you solve an advanced simultaneous equation graphically?

Draw both lines, the points of intersection are the solution

5. Describe step by step how you would solve the following simultaneous equation:

x2 + y2 = 10x + y = 2

i) Make y the subject of the linear equation, y = 2 – x

ii) Substitute your equation of y into the quadratic

x2+(2−x)2=10

iii) expand the brackets x2+ (2−x ) (2−x )=10

x2+4−4 x+x2=10

iv) simplify and make it equal 0

2 x2−4 x−6=0

v) if you can, divide through by the coefficient of x2

x2−2 x−3=0

vi) then solve as a regular quadratic( x−3 ) ( x+1 )=0 , x=3∨−1

vii) substitute your values of x into the original linear equation to get your values for y

y=−1whenx=3 , y=3when x=−1

Card 15 – Graphs


What type of graphs are these:

Cubic Exponential e.g. 2x

Linear Reciprocal e.g. 1/x

Quadratic


Card 16 – Transformations

1. What are the four transformations?

Translation, Rotation, Enlargement and Reflection. (TRER)

2. What is translation and how is it described?

Translation is when is shape is moved without changing size or direction. It is described in terms of a

vector [ xy ], where x is the movement horizontally and

y is the movement vertically

3. How is a reflection described? How many pieces of information do you need?

You need one piece of information, the line which the shape is reflected in. e.g. Shape A is reflected in the line y = x

4. How is a rotation described? How many pieces of information do you need?

You need 3 pieces of information. The centre of rotation, the size of the turn and its direction (clockwise or anti clockwise) e.g. Shape A has been rotated 90o clockwise around point (2,1)

5. How would you describe an enlargement? How many pieces of information do you need?

You need 2 pieces of information. The centre of enlargement and scale factor. e.g. Shape A has been enlarged by scale factor 4 with a centre of enlargement at (2,1). Remember a scale factor can be negative as well as positive.


Card 17 – Similarity and Congruence

1. What do we mean when we describe a shape as congruent to another one?

The angles and lengths of both shapes are identical.

2. How do we prove two triangles are congruent?

We must show that either

1. Two sides and one angle are identical in both triangles.

2. Two angles and one side are identical in both triangles

3. All three sides are identical in both triangles.

3. What do we mean when we say two shapes are similar?

Two shapes are similar if one is an enlargement of the other.

4. How do we prove two triangles are similar?

We must show that either:

1. All corresponding angles are equal

2. All corresponding sides are in the same proportion i.e. they’ve all been increased by the same scale factor

3. Two corresponding sides are in the same proportion and at least one angle is identical between both triangles.


Card 18 – Regular Polygons

1. What do we mean when we say a polygon is regular? e.g. a regular hexagon.

All sides and angles are equal.

2. Draw in an exterior angle on this shape

3. How do we work out the interior and exterior angles of a regular polygon?

exterior angle= 360number of sides

interior angle=180−exterior angle


Card 19 – Bearings and Constructions

1. Where are bearings ALWAYS measured from?

North

2. What is wrong with this bearing, 60o?

It only has two digits, bearings ALWAYS have three digits, it should have read 060o

3. What does bisector mean?

Bisector is maths term for cutting exactly in half. So if you bisect an angle of 40o, you cut it into two equal angles of 20o

4. What is a perpendicular bisector?

It is a line that cuts another line in half and is at right angles to the midpoint of the original line.

5. What is an angle bisector?

An angle bisector cuts an angle exactly in half.

Watch Mathswatch clips 128 and 129 for demonstrations of how to draw perpendicular and angle bisectors.

Card 20 – Plans, Nets and Elevations

1. What is a plan?

A plan is a ‘birds-eye’ view of a shape. It is the view from the top. It is always a 2d drawing.

2. What is an elevation?

An elevation is a view from a particular point of view. A front elevation meets the view of a shape from the front. It is always a 2d drawing.

3. What is a net?

A net is 2d shape that can be folded together to make a 3d shape. The net of a cube looks like this:


Card 21 – Pythagoras’s Theorem

1. Describe Pythagoras’s Theorem

Pythagoras’s theorem states that in a right angle triangle the square of the longest side is equal to the sum of the squares of the remaining two sides.

2. How do you work out the distance between two co-ordinates in 2d?

Distance=√(x2−x1 )2+( y2− y1 )2

So thedistance between points (6,3 )∧(10,6 )would be

Distance=√(10−6 )2+(6−3 )2Distance=√42+32Distance=√16+9Distance=√25Distance=5

3. How do you work out the distance between two co-ordinates in 3d?

Distance=√(x2−x1 )2+( y2− y1 )2+(z2−z1)2


Card 22 – Trigonometry Part 1 – SOHCAHTOA!

1. What is the Hypotenuse?

On a right angled triangle, the hypotenuse is the longest side which is always opposite the right angle.

2. What is the Opposite?

On a right angled triangle, the opposite is the side opposite the angle we know or the angle we’re trying to find out.

3. What is the Adjacent?

On a right angled triangle, the Adjacent is the side next to the angle we know or we’re trying to find out.

4. What order should you label your triangle?

Hypotenuse, Opposite and Adjacent

5. What does Sin equal?

sin θ=opphyp

6. What does Cos equal?

cosθ= adjhyp

7. What does Tan equal?

tanθ=oppadj

8. What type of triangle does SOHCAHTOA only work on?

Right angled triangles.


Card 23 – Advanced Trigonometry – Sin and Cos rule

1. What is wrong with the way this triangle has been labelled?

Lower case letters must be opposite upper case angles.

2. What is the Sin Rule (both forms)?a

sin (A )= bsin (B )

= csin (C )

sin(A)a

=sin(B)

b=sin(C )

c

3. What is the Cos Rule (with a2 as subject and Cos A as subject)

a2=b2+c2−2bcCos(A)

cos ( A )=b2+c2−a2

2bc

4. The Sin rule is easier to use than the Cos rule, what are the two circumstances when we have to use the Cos rule?

i. When we are trying to find an angle and the only information we have is 3 lengths e.g.

ii. When we have two sides and an angle, but the angle we know is NOT opposite the other sides. e.g.

5. What is the useful to do when answering advanced trig questions?

Ignore the labels in the exam and label your own triangle.


Card 24 – Advanced Trig - Area rule

1. What is the area rule?

Areaof triangle=12abSin(C)2. Why can’t you use the area

rule on this triangle? What would you need to do first?

We can only use the area rule, when we know the length of two sides and the angle between them.

We can’t use the area rule because we don’t know the angle between the two sides between sides we do know (8cm and 6cm). To find that other angle we would need to use the Sin rule to find the angle opposite the 6cm. With that we can work out the remaining angle by taking the two other angles away from 180.

Card 25 – Trigometric Graphs

1. What are the values of x to complete a whole cycle of a trig graph?

From 0 to 360, and then the graph repeats itself.

2. What are the maximum and minimum values of y, when y = sin(x) or y = cos (x)

1 and -1

3. Sketch y = sin (x)


4. Sketch y = cos (x)

5. Sketch y = tan (x)

6. What is the line called when the graph approaches infinity?

An asymptote

Card 26 – Circle Theorems Part 1

1. What is the angle between a tangent to a circle and the radius?

90 degrees

2. What shape do two tangents make when they meet?

An isosceles triangle


3. What is the angle within a semicircle?

90 degrees

4. What is the relationship between an angle subtendered at the centre and the circumference?

Angle at Centre is twice angle at circumference


Card 27 – Circle Theorems Part 2

5. What is special about angles which subtender the same segment?

They are the same size

6. What is special about the angles of a quadrilateral within a circle?

Opposite angles add up to 180 degrees, e.g. B+D = 180 and A+C = 180

7. What is special about the angle between a chord and a tangent, and the angle that subtends that chord at the circumference (if that makes no sense, look at the diagram; what is the relationship between the two green angles?)

They are the same size


Card 28 – Shapes in 2D

1. What is the formula for the area of a circle?

Area=π r2

2. What is the formula for a circumference?Circumference=2πr

3. What is an arc, and how do you work out its length?

An arc is a proportion of a circle’s circumference, the formula to work it out is

lengthof arc= θ360

×2πr

4. What is a sector and how do you work out its area?

A sector is a proportion of a circle, the formula to work out its area is

Areaof sector= θ360

×π r2

5. What is a segment and how do you work it out its area?

The part shaded green is a segment, to work out its area the formula is

Areaof segment= π r2θ360

−12r2 sin (θ)

or to put it another way that is the area of the sector take away the area of the triangle (using the area rule) to leave the segment.


Card 29 – Shapes in 3D

1. What is a prism?

A prism is a 3D shape with the same cross-section all along its length

2. How do we work out the volume of a prism?

Area of cross section times its length

3. How do you work out the surface area of a cylinder?

2πrh+2 π r2

To put that another way, the area of the curved part plus the base and the top

4. How do you work out the volume of a cylinder?

π r2h

5. How do we work out the volume of a frustrum?

A frustrum is made from a large cone, with a smaller cone chopped off the top, so work out the volume of the big cone and then take away the volume of a smaller cone

6. How would we describe the triangles used to make up a frustrum?

Similar, which means the angles are identical and the sides are in the same proportion. Or in other words, the triangles are enlargements of each other.

7. What do I need to do to this cone to work out the volume?

Since the formula requires the height of the cone, we need to use Pythagoras’s theorem to calculate the height. In this case

height=√l2−r2=√25−9=√16=4


Card 30 – Dimensions

1. How many dimensions does an area have?

2

2. How many dimensions does a volume area?

3

3. To represent a volume or an area, all components of a formula must be to the same power. Why can’t the following expression represent a volume? a,b, and c represent lengths.

a3+b3+c2

While a and b are raised to the third dimension, c is only raised to the second. For a formula to represent a volume all the components must be raised to the 3 dimension, therefore the expression cannot represent a volume


Card 31 – Vector

1. What is a vector?

A vector is mathematical term for a movement with a specified length and direction.

2. What are the 3 ways of writing vectors?

a or [ xy ] or AB→

3. What is a position vector?

A position vector is a vector which defines the location of given point. Position vectors are nearly always from the point O.

4. How do you know if two vectors are parallel?

if vector a is parallel to vector b then vector a = kb, where k is a number. For example the vector a+b is parallel to 2(a+b), since you can multiply the first vector by 2 to get the second. However a+b is not parallel to 3(a-b), since there is no number you can times a+b by that equals 3(a-b)

5. How do you know a vector is parallel to the x-axis?

The y component of the vector is = 0, e.g. the vector

[40 ] is parallel to the x-axis.

6. How do you know a vector is parallel to the y-axis?

The x component of the vector is = 0, e.g. the vector

[ 0−2] is parallel to the y-axis.


Card 32 – Function Notation and transformation of graphs, Part 1

1. y = x2 + 3, is a translation of y=x2, remember that translation is a maths word for movement. Describe the translation.

y= x2 + 3 is translation of y = x2 by vector [03]2. y = (x+3)2 is a translation of y=x2, describe the translation.

y= (x+3)2 is translation of y = x2 by vector [−30 ]3. y =(x+a)2 +b is a translation of y=x2, describe the translation.

y= (x+a)2 +b is translation of y = x2 by vector [−ab ]

4. Use completing the square to find the minimum value (vertex) of this equation, x2+6 x−5

x2+6 x−5=0

x2+6 x=5x2+6 x+9=5+9( x+3 )2=14 ( x+3 )2−14=0

By completing the square we have put the equation is in the form y= (x+a)2 +b. We can now see it is a

translation of y=x2 with vector [ −3−14] The vertex of y=x2

is at (0,0) so if we translate the vertex as well, the vertex of our new graph is at (-3,-14)

5. What does f stand for in f(x) = sin (x)?

f stands for the function (mathematical action) to sin an input to produce an output, it does not represent a number.


Card 33 – Function Notation and transformation of graphs, Part 2

6. If f(x) = x2, what does f(3) and f(Mr. T) equal?

f(3) = 9, f(Mr.T) = (Mr. T)2

7. Describe the transformation of f(x+a)+b from f(x)

y= f(x+a) +b is translation of y = f(x) by vector [−ab ]

8. Is the graph of y=2sin(x) a stretched or squashed version of y = sin(x). Has it been stretched or squashed vertically or horizontally?

It has been stretched vertically by a scale factor of 2

9. Is the graph of y=12sin(x) a stretched or squashed

version of y = sin(x). Has it been stretched or squashed vertically or horizontally?

It has been squashed vertically by a scale factor of 2

10. Is the graph of y = sin(2x) a stretched or squashed version of y = sin(x). Has it been stretched or squashed vertically or horizontally?

It has been squashed horizontally by a scale factor of 2

11. Is the graph of y = sin(12x) a stretched or squashed

version of y = sin(x). Has it been stretched or squashed vertically or horizontally?

It has been stretched horizontally by a scale factor of 2.

12. Describe the transformation of y=-f(x) from y = f(x)

y=-f(x) is a reflection of y = f(x) in the x –axis.

13. Describe the transformation of y=f(-x) from y = f(x)

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

14. Fill in the blanks in this sentence:“If the change is outside the brackets of a function e.g. y = f(x) + a, 2f(x) or –f(x) then the transformation is _______

If the change is inside the brackets of a function e.g. y = f(x+a), f(2x) or f(-x), then the transformation is _______”

First one is vertical, second one is horizontal.



Card 34 – Loci

1. What does Locus mean?

Locus is a set of points that obey a given rule. Or in English, it means if you pick a point and it fits the rule then it is part of the locus. If that doesn’t make any sense, the following examples should help.

2. How do you work out the locus of the points equidistant (of equal distance) between two points, A and B.

The Locus, in this case is the perpendicular bisector of the line AB

3. What would the locus of the points equidistant from the following lines look like and how would you draw it?

It would look like this and you would bisect the angle to draw it:

4. What does the locus of the points r distance from point O and from the line AB look like:


A circle centred around point O, with radius r and two parallel lines joined with two semicircles of radius r.


Documents

Web viewArea of a Rhombus = h 2 a+b where a and b are the length of the top and bottom