A-level Maths (OCR) Prep Exam With
Solved Solutions.
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - Answer formula of an arithmetic series
the sum of the terms of an arithmetic sequence - Answer what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - Answer nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - 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 - Answer formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - Answer divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - Answer convergent sequence
a / 1-r - Answer sum to infinity of a geometric series
- 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 - Answer
recurrence relation of form Un+1 = f(Un)
it is decreasing - 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) - Answer if Un+k = Un for all n ∈ ℕ, what
is true of the sequence
(x+y)(x-y) - Answer x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - 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 - Answer using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - Answer completing the square to find the turning
point
< is dotted line
≤ is solid line - Answer using lines to represent < and ≤
x=0 and y=0 - Answer where are the asymptotes of y = k/x
translation up by a units - Answer y = f(x) + a
, translation left by a units - Answer y = f(x+a)
stretch vertically by scale factor a - Answer y = af(x)
stretch by scale factor 1/a horizontally - Answer y = f(ax)
reflection in x-axis - Answer y = -f(x)
reflection in y-axis - Answer y = f(-x)
m = (y2 - y1)/(x2 - x1) - Answer calculating the gradient with 2 points
y-y1=m(x-x1) - Answer another way to calculate equation of a line
y= -(1/m)x - Answer equation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - Answer distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - Answer equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - Answer equation of circle centre (a,b)
centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - Answer centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - Answer a tangent to a circle is ...... to the radius of the circle at the point of intersection
the centre of a circle - Answer the perpendicular bisector of a chord will go through.....
Solved Solutions.
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - Answer formula of an arithmetic series
the sum of the terms of an arithmetic sequence - Answer what is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - Answer nth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - 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 - Answer formula of first n terms of a geometric sequence
the sum of the values tend towards infinity - Answer divergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - Answer convergent sequence
a / 1-r - Answer sum to infinity of a geometric series
- 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 - Answer
recurrence relation of form Un+1 = f(Un)
it is decreasing - 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) - Answer if Un+k = Un for all n ∈ ℕ, what
is true of the sequence
(x+y)(x-y) - Answer x^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - 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 - Answer using the discriminant to find number of roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - Answer completing the square to find the turning
point
< is dotted line
≤ is solid line - Answer using lines to represent < and ≤
x=0 and y=0 - Answer where are the asymptotes of y = k/x
translation up by a units - Answer y = f(x) + a
, translation left by a units - Answer y = f(x+a)
stretch vertically by scale factor a - Answer y = af(x)
stretch by scale factor 1/a horizontally - Answer y = f(ax)
reflection in x-axis - Answer y = -f(x)
reflection in y-axis - Answer y = f(-x)
m = (y2 - y1)/(x2 - x1) - Answer calculating the gradient with 2 points
y-y1=m(x-x1) - Answer another way to calculate equation of a line
y= -(1/m)x - Answer equation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - Answer distance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - Answer equation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - Answer equation of circle centre (a,b)
centre: (-f,-g)
radius: sqrt (f^2 + g^2 -c) - Answer centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - Answer a tangent to a circle is ...... to the radius of the circle at the point of intersection
the centre of a circle - Answer the perpendicular bisector of a chord will go through.....
