AS Linea:
PURE:
1. If you multiply the gradients of two lines and this product is equal to -1, then both lines are
perpendicular.
2. Lines of a triangle are not collinear – they don’t have the same gradient
3. When using b2-4ac, sub in the lower value and upper value into the original equation to see if
it’s true. This will tell you if the symbols “<” and “>” should have an equal sign
4. When using b2-4ac, remember if there is a negative before the ‘c’ term to include that as well
5. To prove a statement of x and y is true, do working out until you reach a known fact. Then
cross this out and work back to the original equation given in the question to prove it’s true.
Don’t start from the statement, remember.
6. When proving (and you do the backwards and forwards shit), remember to show when the
division/multiplication happened and say it’s because the denominator or multiplier was
positive – applies only for proving with inequalities.
7. Proof by:
Exhaustion: Every single scenario
Deduction: Using a known fact
Counter-example: Using example values to prove/disprove
8. If a straight line and curve only have 1 solution (intersect once), the line is a tangent to the
circle
9. Set notation with an ‘∅’ is an empty set. No values. E.g., {x:x<0} = ∅ means no values for x<0
because x is never less than 0
10. Union symbol in set notation basically means ‘or’ and intersection means ‘and’
11. When giving a range of values for x in set notation, remember to do it like: {x E IR:x<-1/3} for
example
12. If you use the discriminant to prove that a quadratic equation>0, then state the value of the
discriminant, state if its U or n shaped and is above/below the x-axis, how many roots it has
and then ‘therefore equation>0’
13. If asked to state, giving a reason, if a statement is always true, sometimes or never true then
you have to use examples to prove your point
14. Triple roots in cubic curves causes it to flatten out at the x-axis where the triple root is
located
15. Direct proportion graphs go through the origin and are straight lines. They can be written as
y = kx, where k is the constant (of proportionality).
16. Circumcircle is when a circle passes through all the vertices of a triangle. Remember, each
side of this triangle is a chord and is a perpendicular bisector to the circle’s radius.
17. A set of values where it’s ‘above’ the graph (i.e., when the quadratic curve>0), it is union
(‘or’)
18. The minimum value of a quadratic curve is the lowest y value. This is always equal to f(x), if
asked in the question
19. To find the rth entry in the nth row of Pascal’s triangle, it’s given by n-1Cr-1. E.g., 6th entry of 10th
row of Pascal’s triangle is 9C5
20. Remember, when asked to work out a minimum/maximum value of something, use the next
value from the inequalities ‘<’ or ‘>’. Vice versa for maximum.
, 21. If asked to find any real values of x in a log sum, remember to state which values of log are
undefined and therefore what values of x are the right answers. E.g., log 4(2-(4+4√ 2)) is
undefined, therefore x = 4-4√ 2
22. If its sinxcosx = 7sinx for example, sinx can be factored out and set to a value to find sinx =
that value. Don’t ever divide both sides of a trigonometric equation by a factor. Rearrange to
both sides, factorise and solve always
23. If asked to prove from first principles, in the final step, mention: ‘as h->0, f’(x) = [ans]’
24. For a graph of P=abx, a is the initial something and b is the ‘appreciation rate of [what
changes in the graph e.g., population per year]’ or ‘the rate of increase of […] per [unit of …]’
depending on the context
25. If you have to sketch the graph of something complex like y = (1/2) x-3, separate it into 2 parts
and set the part with x equal to f(x) and simply the remaining equation. You should get a
transformation sign e.g., left with y = 8f(x). Sketch this.
26. After differentiation, if asked to work out the value of k (or any constant in front of the
differentiated equation), say that it’s a decreasing function and interpret this in context by
saying the decrease is exponential (if k is negative and vice versa if k is positive).
27. Log can never have base 1.
28. log(x) can only be defined for positive values of x. No solutions for negative values of x
29. y = ln(x) is a reflection of the graph y = e x in the line y = x
30. In log modelling with exponentials, to estimate the values of 2 constants given in the
question, log the original equation on both sides. This is because:
1. Work out the gradient of
the line of best fit or
something and that’s one
of the constant values
you have to work out
2. Read the y-intercept of
the graph and that’s the
other constant you have
to find
31. If y = abx, the value of b represents a percentage decrease. E.g., if b = 0.84, there is a
percentage decrease of 16% per [value of x]
32. If asked to find out a range of values, check the question to make sure its above 0 etc.
33. To prove a line is parallel to the x or y-axis, show how 2 coordinates on the line have the
same y or x coordinate respectively
34. When calculating the bearing of something and you’re given 2 points that the thing was at,
Draw an x-y axis and plot the first point location. Then draw an N upwards. Plot the second
point location. Work out the angle between the line connecting both point locations and the
N arrow upwards. This is the bearing it’s travelling on
35. Remember to check specific values given in the question at the start and remember to
include this relevant information in your explanations/working out
36. ‘per’ something always means gradient – remember to include this if relevant
37. If asked whether a statement is always true, sometimes true, or never true, give examples of
the scenarios & remember to write conclusions at the end
PURE:
1. If you multiply the gradients of two lines and this product is equal to -1, then both lines are
perpendicular.
2. Lines of a triangle are not collinear – they don’t have the same gradient
3. When using b2-4ac, sub in the lower value and upper value into the original equation to see if
it’s true. This will tell you if the symbols “<” and “>” should have an equal sign
4. When using b2-4ac, remember if there is a negative before the ‘c’ term to include that as well
5. To prove a statement of x and y is true, do working out until you reach a known fact. Then
cross this out and work back to the original equation given in the question to prove it’s true.
Don’t start from the statement, remember.
6. When proving (and you do the backwards and forwards shit), remember to show when the
division/multiplication happened and say it’s because the denominator or multiplier was
positive – applies only for proving with inequalities.
7. Proof by:
Exhaustion: Every single scenario
Deduction: Using a known fact
Counter-example: Using example values to prove/disprove
8. If a straight line and curve only have 1 solution (intersect once), the line is a tangent to the
circle
9. Set notation with an ‘∅’ is an empty set. No values. E.g., {x:x<0} = ∅ means no values for x<0
because x is never less than 0
10. Union symbol in set notation basically means ‘or’ and intersection means ‘and’
11. When giving a range of values for x in set notation, remember to do it like: {x E IR:x<-1/3} for
example
12. If you use the discriminant to prove that a quadratic equation>0, then state the value of the
discriminant, state if its U or n shaped and is above/below the x-axis, how many roots it has
and then ‘therefore equation>0’
13. If asked to state, giving a reason, if a statement is always true, sometimes or never true then
you have to use examples to prove your point
14. Triple roots in cubic curves causes it to flatten out at the x-axis where the triple root is
located
15. Direct proportion graphs go through the origin and are straight lines. They can be written as
y = kx, where k is the constant (of proportionality).
16. Circumcircle is when a circle passes through all the vertices of a triangle. Remember, each
side of this triangle is a chord and is a perpendicular bisector to the circle’s radius.
17. A set of values where it’s ‘above’ the graph (i.e., when the quadratic curve>0), it is union
(‘or’)
18. The minimum value of a quadratic curve is the lowest y value. This is always equal to f(x), if
asked in the question
19. To find the rth entry in the nth row of Pascal’s triangle, it’s given by n-1Cr-1. E.g., 6th entry of 10th
row of Pascal’s triangle is 9C5
20. Remember, when asked to work out a minimum/maximum value of something, use the next
value from the inequalities ‘<’ or ‘>’. Vice versa for maximum.
, 21. If asked to find any real values of x in a log sum, remember to state which values of log are
undefined and therefore what values of x are the right answers. E.g., log 4(2-(4+4√ 2)) is
undefined, therefore x = 4-4√ 2
22. If its sinxcosx = 7sinx for example, sinx can be factored out and set to a value to find sinx =
that value. Don’t ever divide both sides of a trigonometric equation by a factor. Rearrange to
both sides, factorise and solve always
23. If asked to prove from first principles, in the final step, mention: ‘as h->0, f’(x) = [ans]’
24. For a graph of P=abx, a is the initial something and b is the ‘appreciation rate of [what
changes in the graph e.g., population per year]’ or ‘the rate of increase of […] per [unit of …]’
depending on the context
25. If you have to sketch the graph of something complex like y = (1/2) x-3, separate it into 2 parts
and set the part with x equal to f(x) and simply the remaining equation. You should get a
transformation sign e.g., left with y = 8f(x). Sketch this.
26. After differentiation, if asked to work out the value of k (or any constant in front of the
differentiated equation), say that it’s a decreasing function and interpret this in context by
saying the decrease is exponential (if k is negative and vice versa if k is positive).
27. Log can never have base 1.
28. log(x) can only be defined for positive values of x. No solutions for negative values of x
29. y = ln(x) is a reflection of the graph y = e x in the line y = x
30. In log modelling with exponentials, to estimate the values of 2 constants given in the
question, log the original equation on both sides. This is because:
1. Work out the gradient of
the line of best fit or
something and that’s one
of the constant values
you have to work out
2. Read the y-intercept of
the graph and that’s the
other constant you have
to find
31. If y = abx, the value of b represents a percentage decrease. E.g., if b = 0.84, there is a
percentage decrease of 16% per [value of x]
32. If asked to find out a range of values, check the question to make sure its above 0 etc.
33. To prove a line is parallel to the x or y-axis, show how 2 coordinates on the line have the
same y or x coordinate respectively
34. When calculating the bearing of something and you’re given 2 points that the thing was at,
Draw an x-y axis and plot the first point location. Then draw an N upwards. Plot the second
point location. Work out the angle between the line connecting both point locations and the
N arrow upwards. This is the bearing it’s travelling on
35. Remember to check specific values given in the question at the start and remember to
include this relevant information in your explanations/working out
36. ‘per’ something always means gradient – remember to include this if relevant
37. If asked whether a statement is always true, sometimes true, or never true, give examples of
the scenarios & remember to write conclusions at the end
