OCR a-level maths Questions & Answers
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - ANSWERSformula of an
arithmetic series
the sum of the terms of an arithmetic sequence - ANSWERSwhat is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - ANSWERSnth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - ANSWERSnth 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 - ANSWERSformula of first n terms of a geometric sequence
the sum of the values tend towards infinity - ANSWERSdivergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - ANSWERSconvergent sequence
a / 1-r - ANSWERSsum to infinity of a geometric series
- ANSWERSseries 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 - ANSWERSrecurrence relation of form Un+1 = f(Un)
it is decreasing - ANSWERSif 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) - ANSWERSif Un+k = Un
for all n ∈ ℕ, what is true of the sequence
(x+y)(x-y) - ANSWERSx^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - ANSWERSrationalising 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 - ANSWERSusing the discriminant to find number of
roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - ANSWERScompleting the square
to find the turning point
< is dotted line
≤ is solid line - ANSWERSusing lines to represent < and ≤
x=0 and y=0 - ANSWERSwhere are the asymptotes of y = k/x
translation up by a units - ANSWERSy = f(x) + a
translation left by a units - ANSWERSy = f(x+a)
stretch vertically by scale factor a - ANSWERSy = af(x)
stretch by scale factor 1/a horizontally - ANSWERSy = f(ax)
reflection in x-axis - ANSWERSy = -f(x)
reflection in y-axis - ANSWERSy = f(-x)
m = (y2 - y1)/(x2 - x1) - ANSWERScalculating the gradient with 2 points
y-y1=m(x-x1) - ANSWERSanother way to calculate equation of a line
y= -(1/m)x - ANSWERSequation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - ANSWERSdistance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - ANSWERSequation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - ANSWERSequation of circle centre (a,b)
centre: (-f,-g)
, radius: sqrt (f^2 + g^2 -c) - ANSWERScentre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - ANSWERSa tangent to a circle is ...... to the radius of the circle at the
point of intersection
the centre of a circle - ANSWERSthe perpendicular bisector of a chord will go
through.....
a right angle - ANSWERSthe angle in a semicircle is always
- ANSWERSif ∠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 - ANSWERSfind
the centre of a circle given any 3 points
if f(p) = 0 then (x-p) is a factor of f(x) - ANSWERSfactor theorem
starting from known facts or definitions then using logical steps to reach the desired
conclusion - ANSWERSproof by deduction
breaking the statement into smaller cases and proving each case separately -
ANSWERSproof by exhaustion
an example that does not work for the statement - ANSWERSproof by counter-example
(n+1)th row - ANSWERSwhich row of pascal's triangle gives the coefficients of the
expansion of (a+b)^n
n * (n-1) * (n-2) * ... *3 * 2 * 1 - ANSWERSn!
n!/r!(n-r)! - ANSWERSnCr
a^n + nC1*a^n-1*b + nC2*a^n-2*b^2 + ... + nCr*a^n-r*b^r + ... + b^n -
ANSWERSbinomial expansion of (a+b)^n with nCr
- ANSWERSif x is small the first few terms in a binomial expansion can be used to find
an approximate value for a complicated expression
a^2 = b^2 + c^2 - 2bcCosA - ANSWERScosine rule
a/sinA = b/sinB = c/sinC - ANSWERSSine rule
sinθ = sin(180-θ) - ANSWERSambiguous case of the sine rule θ
first quadrant: A sinθ , cosθ and tanθ are all positive
- Sn = n/2 (2a + (n-1)d)
- Sn = n/2 (a + l) where a is the first term and l is the last term - ANSWERSformula of an
arithmetic series
the sum of the terms of an arithmetic sequence - ANSWERSwhat is an arithmetic series
- Un = a + (n-1)d
- a = the first term
- d = the common difference - ANSWERSnth term of an arithmetic sequence
- Un = ar^(n-1)
- a = first term
- r = common ratio - ANSWERSnth 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 - ANSWERSformula of first n terms of a geometric sequence
the sum of the values tend towards infinity - ANSWERSdivergent sequence
- the sum of the values tend towards a specific number
- it is only convergent if |r|<1 - ANSWERSconvergent sequence
a / 1-r - ANSWERSsum to infinity of a geometric series
- ANSWERSseries 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 - ANSWERSrecurrence relation of form Un+1 = f(Un)
it is decreasing - ANSWERSif 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) - ANSWERSif Un+k = Un
for all n ∈ ℕ, what is true of the sequence
(x+y)(x-y) - ANSWERSx^2-y^2
* (a-sqrt(b) / a-sqrt(b)) - ANSWERSrationalising 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 - ANSWERSusing the discriminant to find number of
roots
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q) - ANSWERScompleting the square
to find the turning point
< is dotted line
≤ is solid line - ANSWERSusing lines to represent < and ≤
x=0 and y=0 - ANSWERSwhere are the asymptotes of y = k/x
translation up by a units - ANSWERSy = f(x) + a
translation left by a units - ANSWERSy = f(x+a)
stretch vertically by scale factor a - ANSWERSy = af(x)
stretch by scale factor 1/a horizontally - ANSWERSy = f(ax)
reflection in x-axis - ANSWERSy = -f(x)
reflection in y-axis - ANSWERSy = f(-x)
m = (y2 - y1)/(x2 - x1) - ANSWERScalculating the gradient with 2 points
y-y1=m(x-x1) - ANSWERSanother way to calculate equation of a line
y= -(1/m)x - ANSWERSequation of line perpendicular to y = mx
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 ) - ANSWERSdistance between (x1,y1) and (x2,y2)
x^2 + y^2 = r^2 - ANSWERSequation of circle centre (0,0)
(x-a)^2 + (y-b)^2 = r^2 - ANSWERSequation of circle centre (a,b)
centre: (-f,-g)
, radius: sqrt (f^2 + g^2 -c) - ANSWERScentre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
perpendicular - ANSWERSa tangent to a circle is ...... to the radius of the circle at the
point of intersection
the centre of a circle - ANSWERSthe perpendicular bisector of a chord will go
through.....
a right angle - ANSWERSthe angle in a semicircle is always
- ANSWERSif ∠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 - ANSWERSfind
the centre of a circle given any 3 points
if f(p) = 0 then (x-p) is a factor of f(x) - ANSWERSfactor theorem
starting from known facts or definitions then using logical steps to reach the desired
conclusion - ANSWERSproof by deduction
breaking the statement into smaller cases and proving each case separately -
ANSWERSproof by exhaustion
an example that does not work for the statement - ANSWERSproof by counter-example
(n+1)th row - ANSWERSwhich row of pascal's triangle gives the coefficients of the
expansion of (a+b)^n
n * (n-1) * (n-2) * ... *3 * 2 * 1 - ANSWERSn!
n!/r!(n-r)! - ANSWERSnCr
a^n + nC1*a^n-1*b + nC2*a^n-2*b^2 + ... + nCr*a^n-r*b^r + ... + b^n -
ANSWERSbinomial expansion of (a+b)^n with nCr
- ANSWERSif x is small the first few terms in a binomial expansion can be used to find
an approximate value for a complicated expression
a^2 = b^2 + c^2 - 2bcCosA - ANSWERScosine rule
a/sinA = b/sinB = c/sinC - ANSWERSSine rule
sinθ = sin(180-θ) - ANSWERSambiguous case of the sine rule θ
first quadrant: A sinθ , cosθ and tanθ are all positive
