A-LEVEL OCR 2024 MATHEMATICS A PAPER 1 MARK
SCHEME
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - Correct Answer--formula of an
arithmetic series
the sum of the terms of an arithmetic sequence - Correct Answer--what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - Correct Answer--nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - Correct Answer--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 - Correct Answer--formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - Correct Answer--divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - Correct Answer--convergent sequence
a / 1-r - Correct Answer--sum to infinity of a geometric series
,- Correct Answer--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 -
Correct Answer--recurrence relation of form Un+1 = f(Un)
it is decreasing - Correct Answer--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) - Correct Answer--if Un+k = Un for
all n ∈ ℕ, what is true of the sequence
(x+y)(x-y) - Correct Answer--x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - Correct Answer--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 - Correct Answer--using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - Correct Answer--completing the square to
find the turning point
< is dotted line
≤ is solid line - Correct Answer--using lines to represent < and ≤
,x=0 and y=0 - Correct Answer--where are the asymptotes of y = k/x
translation up by a units - Correct Answer--y = f(x) + a
translation left by a units - Correct Answer--y = f(x+a)
stretch vertically by scale factor a - Correct Answer--y = af(x)
stretch by scale factor 1/a horizontally - Correct Answer--y = f(ax)
reflection in x-axis - Correct Answer--y = -f(x)
reflection in y-axis - Correct Answer--y = f(-x)
m = (y2 - y1)/(x2 - x1) - Correct Answer--calculating the gradient with 2 points
y-y1=m(x-x1) - Correct Answer--another way to calculate equation of a line
y= -(1/m)x - Correct Answer--equation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - Correct Answer--distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - Correct Answer--equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - Correct Answer--equation of circle centre (a,b)
, centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - Correct Answer--centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - Correct Answer--a tangent to a circle is ...... to the radius of the circle at the
point of intersection
the centre of a circle - Correct Answer--the perpendicular bisector of a chord will go through.....
a right angle - Correct Answer--the angle in a semicircle is always
- Correct Answer--if ∠PRQ = 90° then R lies on the circle with diameter PQ
-find the equations of the perpendicular bisectors of 2 different chords
-find the coordinates of the intersection of the perpendicular bisectors - Correct Answer--find
the centre of a circle given any 3 points
if f(p) = 0 then (x-p) is a factor of f(x) - Correct Answer--factor theorem
starting from known facts or definitions then using logical steps to reach the desired conclusion
- Correct Answer--proof by deduction
breaking the statement into smaller cases and proving each case separately - Correct Answer--
proof by exhaustion
an example that does not work for the statement - Correct Answer--proof by counter-example
SCHEME
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - Correct Answer--formula of an
arithmetic series
the sum of the terms of an arithmetic sequence - Correct Answer--what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - Correct Answer--nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - Correct Answer--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 - Correct Answer--formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - Correct Answer--divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - Correct Answer--convergent sequence
a / 1-r - Correct Answer--sum to infinity of a geometric series
,- Correct Answer--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 -
Correct Answer--recurrence relation of form Un+1 = f(Un)
it is decreasing - Correct Answer--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) - Correct Answer--if Un+k = Un for
all n ∈ ℕ, what is true of the sequence
(x+y)(x-y) - Correct Answer--x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - Correct Answer--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 - Correct Answer--using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - Correct Answer--completing the square to
find the turning point
< is dotted line
≤ is solid line - Correct Answer--using lines to represent < and ≤
,x=0 and y=0 - Correct Answer--where are the asymptotes of y = k/x
translation up by a units - Correct Answer--y = f(x) + a
translation left by a units - Correct Answer--y = f(x+a)
stretch vertically by scale factor a - Correct Answer--y = af(x)
stretch by scale factor 1/a horizontally - Correct Answer--y = f(ax)
reflection in x-axis - Correct Answer--y = -f(x)
reflection in y-axis - Correct Answer--y = f(-x)
m = (y2 - y1)/(x2 - x1) - Correct Answer--calculating the gradient with 2 points
y-y1=m(x-x1) - Correct Answer--another way to calculate equation of a line
y= -(1/m)x - Correct Answer--equation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - Correct Answer--distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - Correct Answer--equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - Correct Answer--equation of circle centre (a,b)
, centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - Correct Answer--centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - Correct Answer--a tangent to a circle is ...... to the radius of the circle at the
point of intersection
the centre of a circle - Correct Answer--the perpendicular bisector of a chord will go through.....
a right angle - Correct Answer--the angle in a semicircle is always
- Correct Answer--if ∠PRQ = 90° then R lies on the circle with diameter PQ
-find the equations of the perpendicular bisectors of 2 different chords
-find the coordinates of the intersection of the perpendicular bisectors - Correct Answer--find
the centre of a circle given any 3 points
if f(p) = 0 then (x-p) is a factor of f(x) - Correct Answer--factor theorem
starting from known facts or definitions then using logical steps to reach the desired conclusion
- Correct Answer--proof by deduction
breaking the statement into smaller cases and proving each case separately - Correct Answer--
proof by exhaustion
an example that does not work for the statement - Correct Answer--proof by counter-example
