OCR A-Level Math (PURE MATH). All Exam Questions and Correct
Answers | Graded A+ | 2025/2026 Update|
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - formula of an arithmetic
series
the sum of the terms of an arithmetic sequence - what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - nth term of a geometric sequence
- Sn = a(1-r^n) / 1-r
- Sn = a(r^n - 1) / r-1
where r does not equal 1 - formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - convergent sequence
a / 1-r - sum to infinity of a geometric series
,- series can be shown using sigma notation
- defines each term of a sequence as a function of the previous term
- to find the members of the sequence substitute in n=1, n=2 ... using the previous terms given -
recurrence relation of form Un+1 = f(Un)
it is decreasing - if Un+1 < Un for all n ∈ ℕ, what is true of the sequence
- it is periodic
- means that the terms repeat in a cycle
- k = the order of the sequence (how often the terms repeat) - if Un+k = Un for all n ∈ ℕ,
what is true of the sequence
(x+y)(x-y) - x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - rationalising the denominator of e.g. 1/sqrt(b)+a
b^2 - 4ac > 0 has 2 distant real roots
B^2 -4ac = 0 has on real repeated root
b^2 - 4ac < 0 has no real roots - using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - completing the square to find the
turning point
< is dotted line
≤ is solid line - using lines to represent < and ≤
x=0 and y=0 - where are the asymptotes of y = k/x
, translation up by a units - y = f(x) + a
translation left by a units - y = f(x+a)
stretch vertically by scale factor a - y = af(x)
stretch by scale factor 1/a horizontally - y = f(ax)
reflection in x-axis - y = -f(x)
reflection in y-axis - y = f(-x)
m = (y2 - y1)/(x2 - x1) - calculating the gradient with 2 points
y-y1=m(x-x1) - another way to calculate equation of a line
y= -(1/m)x - equation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - equation of circle centre (a,b)
centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
Answers | Graded A+ | 2025/2026 Update|
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - formula of an arithmetic
series
the sum of the terms of an arithmetic sequence - what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - nth term of a geometric sequence
- Sn = a(1-r^n) / 1-r
- Sn = a(r^n - 1) / r-1
where r does not equal 1 - formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - convergent sequence
a / 1-r - sum to infinity of a geometric series
,- series can be shown using sigma notation
- defines each term of a sequence as a function of the previous term
- to find the members of the sequence substitute in n=1, n=2 ... using the previous terms given -
recurrence relation of form Un+1 = f(Un)
it is decreasing - if Un+1 < Un for all n ∈ ℕ, what is true of the sequence
- it is periodic
- means that the terms repeat in a cycle
- k = the order of the sequence (how often the terms repeat) - if Un+k = Un for all n ∈ ℕ,
what is true of the sequence
(x+y)(x-y) - x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - rationalising the denominator of e.g. 1/sqrt(b)+a
b^2 - 4ac > 0 has 2 distant real roots
B^2 -4ac = 0 has on real repeated root
b^2 - 4ac < 0 has no real roots - using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - completing the square to find the
turning point
< is dotted line
≤ is solid line - using lines to represent < and ≤
x=0 and y=0 - where are the asymptotes of y = k/x
, translation up by a units - y = f(x) + a
translation left by a units - y = f(x+a)
stretch vertically by scale factor a - y = af(x)
stretch by scale factor 1/a horizontally - y = f(ax)
reflection in x-axis - y = -f(x)
reflection in y-axis - y = f(-x)
m = (y2 - y1)/(x2 - x1) - calculating the gradient with 2 points
y-y1=m(x-x1) - another way to calculate equation of a line
y= -(1/m)x - equation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - equation of circle centre (a,b)
centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
