OCR a-level maths
variance (msmsm)
(Σ(x-x̄)^2) / n = (Sxx)/n
mean
(Σx) / n
- the weight acts straight downwards - the reaction force is at a right angle to the plane
- direction of forces for a particle on an inclined plane
-menu -7 -3: inverse normal -put area equivalent to 0.6 -enter standard deviation and mean -calculator will work out b
-if P(X>b) = 0.4 find b using a calculator
dy/dx
1 / (dx/dy)
180/π
1 radian in degrees
Cosec^2(x)
1+cot^2(x)=
Sec^2(x)
1+tan^2(x)=
A/(x-2) + B/(x+6) + C/(x+3)
7 /(x-2)(x+6)(x+3) = ?
True
A function must have a distinct output value for every input value in the domain. True or False?
-swap x and y -rearrange
Finding inverse functions
Newton's Third Law
For every action there is an equal and opposite reaction
One-to-one
What type of function can have an inverse function?
R = sqrt( a^2 + b^2)
Where Rcos(α)= a and Rsin(α) = b What is R?
- P(X<Q1) = 0.25 -inverse normal function on calculator can be used to show that Q1 =89.88 - mean is 100 - 100-89.88 = 10.12 -Q1 is 10.12 from the mean -Q3 will also be 10.12 from the mean as the normal distribution is symmetrical -so IQR = 2*10.12 = 20.24
X~N(100,15^2) how would you calculate the IQR?
y= -(1/m)x
equation of line perpendicular to y = mx
-integrate the function -substitute the values of a point on a curve or the value of the function at a given point into the integrated function -solve the equation to find c
how to find the constant of integration, c
add the vectors
how to find the resultant force of 2 or more forces
join the middle of the top of each bar
how to form a frequency polygon
P(A) * P(B)
if A and B are independent what is P(A∩B)?
kloga(x)
loga(x^k) = ?
|a| = sqrt (x^2 + y^2)
magnitude of a (vector)
distance
magnitude of displacement vector
W
mg = ?
|bx|<1 or |x|< 1/|b|
the expansion of (1+bx)^n, where n is negative or a fraction is valid for
the hypothesis that you assume to be correct
the null hypothesis, H0
Sec(x) + c
∫sec(x)tan(x) dx =
tan(x) + c
∫sec^2(x) dx
-cos(x) + c
∫sin(x) dx =
mean of data in frequency table
(Σxf)/(Σf)
-menu -7 -3: inverse normal -put area equivalent to 0.75 -enter standard deviation and mean -calculator will work out a
-if P(X<a) = 0.75 find a using a calculator
-68% -95% -99.7% -100%
-what percentage of data is within: -1 standard deviation of the mean -2 standard deviations -3 standard deviations -5 standard deviations
P(A∩B)/P(B)
P(A|B)
P(A) + P(B) - P(A∩B)
P(A∪B) = ?
Where f"(x) = 0 and the curve changes from being convex to concave (or vice versa)
Point of inflection
acos(x) ± bsin(x)
Rcos(x ∓ α) can be expressed as what?
asin(x) ± bcos(x)
Rsin(x ± α) can be expressed as what?
a/sinA = b/sinB = c/sinC
Sine rule
class width
The difference between the upper and lower class boundaries
If f''(x) ≤ 0 for every x value of that interval
Where is f(x) concave on a given interval? ≤ ≥
If f''(x) ≥ 0 for every x value of that interval
Where is f(x) convex on a given interval? ≤ ≥
a^(kx)kln(a)
Y = a^(kx) what is dy/dx?
partial fractions
a single fraction with 2 distinct linear factors in the denominator can be split into two separate fractions with linear denominators called...
perpendicular
a tangent to a circle is ...... to the radius of the circle at the point of intersection
sinθ = sin(180-θ)
ambiguous case of the sine rule θ
Newton's First Law
an object at rest will stay at rest and that an object moving with constant velocity will continue to move with constant velocity unless unbalanced force acts on the object
g = 9.8
an object moving vertically in a straight line can be modelled as a particle with a constant downward acceleration of ?
y-y1=m(x-x1)
another way to calculate equation of a line
distance travelled
area between velocity-time graph and horizontal axis
frequency
area of bar / k = ?
qualitative
associated with non-numerical observations
Quantitative
associated with numerical observations
a^n + nC1*a^n-1*b + nC2*a^n-2*b^2 + ... + nCr*a^n-r*b^r + ... + b^n
binomial expansion of (a+b)^n with nCr
data which has pairs of values for two variables
bivariate data
-menu -7 -2: normal CD -making lower bound extremely negative so it is basically negative infinity -make upper bound 109 -enter mean and standard deviation -calculator will give the probability
calculating P(X<109) from normal distribution using calculator
-equivalent to 1-P(80<X<106) -this can be found with the calculator
calculating P(X<80 or X>106)
m = (y2 - y1)/(x2 - x1)
calculating the gradient with 2 points
continuous variable
can take any value in a given range
discrete variable
can take only specific values in a given range
observes and measures every member of a population
census
centre: (-f,-g) radius: sqrt (f^2 + g^2 -c)
centre and radius of x^2 + y^2 + 2fx + 2gy + c = 0
dy/dx = (dy/du)*(du/dx)
chain rule
-choose u by seeing what will be canceled out when swapping dx for du -e.g. if expression has sin in it -u = cosx -du/dx = -sinx -dx = -du/sinx (sin will be canceled)
choosing u for integration by substitution
maximum and minimum values in a class
class boundaries
removing anomalies
cleaning the data
Z = (X - μ) / σ
coding for the standard normal distribution μ σ
if f(x) = a(x+p)^2 + q, then the turning point is (-p,q)
completing the square to find the turning point
θ must be in radians
condition for A = 0.5r^2θ
θ must be in radians
condition for l = rθ
- the sum of the values tend towards a specific number - it is only convergent if |r|<1
convergent sequence
the nature of the linear relationship between two variables
correlation
cosAcosB-sinAsinB
cos(A+B)
Cos^2(A)-sin^2(A) = 2cos^2(A) -1 = 1-2sin^2(A)
cos2A
a^2 = b^2 + c^2 - 2bcCosA
cosine rule
the region of the probability distribution which, if the test statistic falls within it, would cause you to reject the null hypothesis
critical region
the first value to fall inside of the critical region
critical value
f'(y) dy/dx
d/dx (f(y)) = ?
x(dy/dx) + y
d/dx (xy) = ?
ny^(n-1) dy/dx
d/dx (y^n) = ?
Sqrt ((x2 - x1)^2 + (y2 - y1)^2 )
distance between (x1,y1) and (x2,y2)
the sum of the values tend towards infinity
divergent sequence
∫1/g(y) dy = ∫f(x)dx
dy/dx = f(x)g(x) then ∫
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)
(1+x)^0.5 * (1-x)^-0.5
express sqrt (1+x / 1-x) so that it can be expanded
1+nx+ (n(n-1)x^2)/2! + (n(n-1)(n-2)x^3)/3! +...+ (n(n-1)...(n-r+1)x^r)/r!+...
expression for binomial expansion of (1+x)^n
lim h->0 (f(x+h)-f(x))/h
f'(x)=?
anx^n-1
f(x) = ax^n f'(x)=?
loga(g(x))
f(x) = g(x) loga(f(x)) = ?
local minimum
f(x) has a stationary point at x=a and f''(a) > 0, is the point a local minimum or maximum
if f(p) = 0 then (x-p) is a factor of f(x)
factor theorem
f(g(x))
fg(x) = ?
e.g. in third quadrant -sinθ = sin(180+θ)
find sin, cos , tan of any positive or negative angle using the corresponding acute angle made with the x-axis, θ
-find the equations of the perpendicular bisectors of 2 different chords -find the coordinates of the intersection of the perpendicular bisectors
find the centre of a circle given any 3 points
-divide n by 4 -if it is a whole number the LQ is halfway between this data point and the one above -if it is not a whole number round up
finding lower quartile for discrete data
- s = Ut + 0.5at^2 - y = Usinαt -0.5gt^2 - t = x / Ucosα = x/u Secα - y = Usinα(x/cosα) - 0.5g(x/U secα)^2 - y = xtanα - gx^2 / 2U^2 Sec^2 U - y = xtanα - g/2U^2 x^2 sec^2α - y = xtanα - g/2u^2 x^2(tan^2α + 1) - y = xtanα - gx^2 ((1 + tan^2α) / 2U^2)
finding the equation of trajectory for a particle
where f'(x) = 0
finding the stationary point
-multiply n by 0.75 -if this is a whole number the UQ is halfway between this data point and the one above -if it is not a whole number round up
finding upper quartile for discrete data
y = xtanα - (gx^2)(1+tan^2α / 2U^2)
for a particle which is projected from a point on a horizontal plane with an initial velocity U and an angle α above the horizontal, and that moves freely under gravity: what is the equation of trajectory?
U^2Sin2α / g
for a particle which is projected from a point on a horizontal plane with an initial velocity U and an angle α above the horizontal, and that moves freely under gravity: what is the range on horizontal plane?
2Usinα / g
for a particle which is projected from a point on a horizontal plane with an initial velocity U and an angle α above the horizontal, and that moves freely under gravity: what is the time of flight?
Usinα / g
for a particle which is projected from a point on a horizontal plane with an initial velocity U and an angle α above the horizontal, and that moves freely under gravity: what is the time to reach greatest height?
P(A) * P(B)
for independent events what is (P A and B)
P(A) + P(B)
for mutually exclusive events, what is P(A or B)
X̄ - μ / (σ/sqrt(n)) and is normally distributed with Z~N(0,1)
for the sample mean of a normally distributed random variable X̄~N (μ , σ^2/n)? Z = ?
mean = 0 standard deviation = 1 X~N(0,1^2)
for the standard normal what is the mean and what is the standard deviation
- 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
- 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
- always resists the motion - Fmax is the maximum value of friction - Fmax = µR where µ is the coefficient of friction and R is the reaction force
friction in mechanics µ
gradient of a curve at a given point
gradient of the tangent to the curve
acceleration
gradient of velocity-time graph
frequency density
height of a bar on a histogram
y = a +bx
how is the regression line of y on x written
X̄~N (μ , σ^2/n)
how is the sample mean normally distributed for a random sample of size n (X̄ μ σ)
μ = np σ = sqrt(np(1-p)) and then apply a continuity correction
how to approximate the binomial distribution into the normal distribution? μσ
-menu -6:statistics -2: y=a+bx -enter the variables -OPTN -4: regression calc
how to calculate PMCC on a calculator
-menu -6: statistics - 2: y = a+bx - enter data -optn -4: regression calc
how to calculate r on a calculator
1. state null hypothesis (e.g. H0: ρ=0) 2. state alternative hypothesis (e.g. H1:ρ≠0 for 2 tailed tests or H1:ρ<0 if r<0 or H1:ρ>0 if r>0) 3. use sample size to find critical value from table (the significance level should be halved if it is a 2-tailed test) 4. conclude with context e.g. if observed outcome > critical value then H1 if not then H0
how to do hypothesis testing for correlation (ρ≠)
-differentiate the u to get an expression for dx -substitute this into the expression -substitute the rest of the expression so that xs are replaced by us -integrate this expression -the us can be converted back to xs
how to integrate by substitution
try ln|f(x)| and differentiate to check then adjust the constant
how to integrate ∫k(f'(x))/(f(x))dx
try (f(x))^(n+1) and differentiate to check then adjust any constant
how to integrate ∫kf'(x)(f(x))^n dx
-sketch y = ax+b -then reflect the section of the graph below the x-axis in the x-axis
how to sketch y = |ax+b|
one-tailed tests
hypothesis test with alternative hypotheses in the form H1:p< ... and H1:p>...
two-tailed tests
hypothesis tests with an alternative hypothesis in the form H1:p≠...
P(A) (as A is not affected by B)
if A and B are independent what is P(A|B)?
0 (won't both happen)
if A and B are mutually exclusive what is P(A∩B) ?
P(A) + P(B)
if A and B are mutually exclusive what is P(A∪B)?
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
μ=200 σ=4
if X~N(200,16) what is μ? what is σ?
- horizontal component is UcosA - vertical component is UsinA
if a particle is projected with an initial velocity of U, at an angle of A above the horizontal what are the horizontal and vertical components of the initial velocity?
P(X=r) = ((nCr)p^r)*(1-p)^(n-r)
if a random variable X has the binomial distribution B(n,p) the its probability mass function will be : ?
-dP = kP -∫(1/P)dP = ∫kdt -ln|P| = kt + c -logeP = kt + c - P = e^(kt+c) - P = Ae^(kt) where A = e^c
if dP/dt = KP find an expression for P (∫)
mean of x minus a all divided by b standard deviation of original data divided by b
if data is coded using y = (x-a)/b what is the mean of the coded data? what is the standard deviation?
(1/(x+1))x^(n+1)+c
if dy/dx = x^n y=?
(k/(x+1))x^(n+1)+c
if f'(x) = kx^n f(x)=
e^x
if f(x) = e^x what is f'(x)
a^x
if loga(n) = x what is n?
v = ds/dt
if s is expressed as a function of t, then velocity can be expressed as?
a = dv/dt = (d^2)s/d(t^2)
if v is expressed as a function of t, then velocity can be expressed as ?
ke^(kx)
if y = e^(kx) what is dy/dx ?
f'(x) + g'(x)
if y = f(x) + g(x) what is dy/dx
logk + xlogb
if y = kb^x for constants k and b then logy = ?
logy = logk + xlogb
if y = kb^x what is this in logs?
loga + nlogx
if y =ax^n for constants a and n then logy = ?
logy = loga + nlogx
if y=ax^n what is this in logs?
u = Ax (often)
integrating Sin(Ax) by substitution
velocity
integrating acceleration gives
Displacement
integrating velocity gives
difference between 2 given percentiles
interpercentile range
difference between upper quartile and lower quartile
interquartile range
- when friction is at its maximum - so a body is about to move or moving at a constant speed
limiting equilibrium
0
loga(1) = ?
loga(x^-1) = -loga(x)
loga(1/x) = ?
1
loga(a) = ?
loga(xy)
loga(x) + loga(y) = ?
loga(x/y)
loga(x)-loga(y) = ?
speed
magnitude of velocity vector
the middle value when the data values are put in order
median
integration by parts to get xlnx - x + c
method of integrating lnx
average of the class boundaries
midpoint
value or class that occurs most often
mode or modal class
1.dimensions of object are negligible so mass is concentrated at a single point 2.all dimensions but one are negligible so mass is concentrated along line, it has no thickness and is rigid 3. mass of the object is treated and tension the same at both ends of a light string 4. does not stretch under load so acceleration is the same in objects connected by the string 5.assume that there is no friction between the surface and any object on it 6.pulley has no mass and tension is the same on either side of the pulley
modelling assumptions of: 1.particles 2.rod 3.light object 4.inextensible string 5.smooth surface 6.smooth and light pulley
n * (n-1) * (n-2) * ... *3 * 2 * 1
n!
n!/r!(n-r)!
nCr
- Un = ar^(n-1) - a = first term - r = common ratio
nth term of a geometric sequence
- Un = a + (n-1)d - a = the first term - d = the common difference
nth term of an arithmetic sequence
consists of taking the sample from people who are available at the time the study is carried out and who fit the criteria you are looking for
opportunity sampling
the whole set of items that are of interest
population
fully describes the probability of any outcome in the sample space
probability distribution
P(A|B)
probability of A given B
- if f(x) = uv - f'(x)= uv' + u'v
product rule
an example that does not work for the statement
proof by counter-example
starting from known facts or definitions then using logical steps to reach the desired conclusion
proof by deduction
breaking the statement into smaller cases and proving each case separately
proof by exhaustion
first quadrant: A sinθ , cosθ and tanθ are all positive second quadrant: S only sinθ is positive third quadrant: T only tanθ is positive fourth quadrant: C only cosθ is positive
quadrants of CAST diagram θ
An interviewer or researcher selects a sample that reflects the characteristics of the whole population
quota sampling
-if f(x)= u/v -f'(x) = (vu' - uv') / v^2
quotient rule
difference between largest and smallest values in a data set
range
* (a-sqrt(b) / a-sqrt(b))
rationalising the denominator of e.g. 1/sqrt(b)+a
- 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)
-sum of all moments acting on a body
resultant moment
selection of observations taken from a subset of the population which is used to find out information about the population as a whole
sample
sampling units individually named or numbered to form a list
sampling frame
individual units of a population
sampling units
magnitude only
scalar quantity
A sample of size n selected from the population in such a way that each possible sample of size n has an equal chance of being selected.
simple random sample
sinAcosB + cosAsinB
sin(A+B)
2sinAcosA
sin2A
1
sin^2θ + cos^2θ ≡ ?
tanθ
sinθ / cosθ = ?
-sketch the graph of y = f(x) where x is greater than or equal to 0 -reflect this in the y-axis
sketching y = f(|x|)
-sketch y =f(x) -reflect any parts where f(x)<0 in the x-axis -delete the parts below the x-axis
sketching y = |f(x)|
θ θ 1-(θ^2)/2
small angle approximation of: sinθ tanθ cosθ
square root of the variance
standard deviation σ
The population is divided into mutually exclusive strata and a random sample is taken from each
stratified sampling
a / 1-r
sum to infinity of a geometric series
πrl (where l is length of slanted edge)
surface area of a cone
4πr^2
surface area of a sphere
1. u + at 2. 0.5(u+v)t 3. u^2 + 2as 4.ut + 0.5at^2
suvat equations: 1. v=? 2.s=? 3.v^2=? 4.s=?
the required elements are chosen at regular intervals from an ordered list
systematic sampling
(tanA+tanB)/(1-tanAtanB)
tan(A+B)
2tanA / 1-tan^2A
tan2A
Probability of incorrectly rejecting the null hypothesis
the actual significance level of a hypothesis test
tells us about the parameter if your assumption is shown to be wrong
the alternative hypothesis, H1
a right angle
the angle in a semicircle is always
|(b/a)x|<1 or |x|<|a/b|
the expansion of (a+bx)^n, where n is negative or a fraction is valid for
Newton's Second Law (F=ma)
the force needed to accelerate a particle is equal to the product of the mass of the particle and the acceleration produced
s = vt
the horizontal motion of a projectile is modelled as having constant velocity so what is displacement?
the centre of a circle
the perpendicular bisector of a chord will go through.....
- acceleration - due to gravity (a=g)
the vertical motion of a projectile is modelled as having constant what?
force * perpendicular distance (so may have to use trig to find the perpendicular distance)
turning moment = ?
a / |a|
unit vector in the direction of a
i and j
unit vectors along x- and y-axes are denoted by
-using the first 3 terms it comes to 543/256 -using the expression it comes to sqrt(9/2) -sqrt(9/2) = 3sqrt(2)/2 -this can be rearranged to find that sqrt(2) is approximately equal to 181/128
use the first 3 terms in the expansion of (4+5x)^0.5 to find an approximation of sqrt(2) with x=0.1
< is dotted line ≤ is solid line
using lines to represent < and ≤
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
has both magnitude and direction
vector quantity
Gradient
velocity on a displacement time graph
(πr^2h)/3
volume of a cone
(4πr^3)/3
volume of sphere
the sum of the terms of an arithmetic sequence
what is an arithmetic series
0.5h*(y0+2(y1+y2...+yn-1)+yn) where h = (b-a)/n and yi = f(a+ih)
what is the area between a and b of curve y roughly equal to?
1
what is the area under a continuous probability distribution equal to?
A' (not A- the complement of A)
what is the notation for this? (∩∪')
A∩B (A and B- the intersection of A and B)
what is the notation for this? (∩∪')
A∪B (A or B- the union of A and B)
what is the notation for this? (∩∪')
describes the linear relationship between 2 variables. it is a value between -1 and 1 (how close data is to a straight line)
what is the product moment correlation coefficient?
the domain of f(x)
what is the range of f-1(x)
- velocity is equal to zero - (significant in problem solving as the parabola of motion of the projectile is symmetrical and this point is at the top of the curve)
what is true of a projectile when it reaches its greatest height
-the resultant force in any direction and the resultant moment about any point will both be 0
what is true of a rigid body in equilibrium?
any vector parallel to a
what is λa
one-tailed test
what kind of test should you use these for: -H0: ρ=0 -H1: ρ>0 or ρ<0
two-tailed
what kind of test should you use this for: -H0: ρ=0 -H1: ρ≠0
-a measure of location -a measure of spread
what to comment on when comparing data sets
resolve it to find the component of the force that acts in the direction of motion (this can be done using trigonometry)
what to do if a force is applied at an angle to the direction of motion?
zero
when a rigid body is on the point of tilting about a pivot, the reaction at any other support (or tension in any other wire or string) is what?
also negative
when calculating the PMCC, if the correlation is negative then r is...
-when n is large and p is close to 0.5
when can you approximate the binomial distribution into the normal distribution?
classes
when data is presented in a group frequency table, the specific data values are not shown.
- if it is at rest and the resultant force acting on the particle is zero
when is a particle or rigid body in static equilibrium?
x=0 and y=0
where are the asymptotes of y = k/x
y=ax^n
which equation has a linear graph of log(y) against log(x) with a gradient n and vertical intercept a
y=ab^x
which equation has a linear graph of log(y) against x with a gradient log(b) and vertical intercept log(a)
(n+1)th row
which row of pascal's triangle gives the coefficients of the expansion of (a+b)^n
(x+y)(x-y)
x^2-y^2
reflection in x-axis
y = -f(x)
stretch vertically by scale factor a
y = af(x)
-Kcosec(kx)cot(kx)
y = cosec(kx) what is dy/dx?
kcosec^2(kx)
y = cot(kx) what is dy/dx?
reflection in y-axis
y = f(-x)
stretch by scale factor 1/a horizontally
y = f(ax)
translation up by a units
y = f(x) + a
translation left by a units
y = f(x+a)
y = x
y = ln(x) is a reflection of y = e^x in which line
1/x
y = ln(x) what is dy/dx
ksec(kx)tan(kx)
y = sec(kx) what is dy/dx?
kcos(kx)
y = sin(kx) what is dy/dx
ksec^2(kx)
y = tan(kx) what is dy/dx?
-ksin(kx)
y=cos(kx) what is dy/dx
y=x
y=f(x) and y=f-1(x) are reflections of each other in which line?
-there are a fixed number of trials, n -there are two possible outcomes (success or failure) -there is a fixed probability of success, p -the trials are independent of each other
you can model X with a binomial distribution, B(n,p) if:
5g -14 = 35N 35 * 1/7 = 5 5>3 so the box stays in equilibrium
µ = 1/7 a box weighs 5kg and a force of 3N is exerted on it there are 2 reaction forces and one is 14N will the box remain at rest or accelerate?
Fmax = µR
µ Fmax equation
1
ΣP(X=x) = ?
180
π radians in degrees
ln|x| + c
∫ 1/x dx =
ln|f(x)|
∫(f'(x))/(f(x)) dx =?
Sin(x) + c
∫cos(x) dx =
-cosec(x) + c
∫cosec(x)cot(x) dx =
-cot(x) + c
∫cosec^2(x) dx =
e^x + c
∫e^x dx =
(1/a)f(ax + b) +c
∫f'(ax + b) dx =
(x^(n+1)) / (n+1) + c
∫x^n dx =
