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a level maths > pure > Flashcards

pure Flashcards

1
Q

how to find coefficient eg of 4 when (3-x)^8

A

8C4 a(in this case 3)^4 b(in this case -x)^4

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2
Q

when is binomial theorem valid

A

when x lies between -1 and 1

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3
Q

what to do if you have a surd as a denominator in a fraction

A

rationalise the denominator by using a complete the square form eg times ur fraction by 1-root2/1-root2

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4
Q

the chain rule

A

define g(x) as thing u do first, f(x) as thing you do second and then f’(g(x))g’(x)

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5
Q

integrate

A

Add 1 to the power and divide by the power +1 and then substitute in values to find what you ahem to add(c) to get correct equation

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6
Q

product rule

A

use when u and v are both functions of x

u x dv/dx+ v x du/dx

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7
Q

quotient rule

A

use for a fraction with functions of x as top and bottom

u= numerator, v= denominator
dy/dx=(vu’-uv’)/v^2

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8
Q

remainder theorem/ find the remainder

A

to find the remainder when f(x) is divided by (x-b) do f(b)/ substitute b into the equation and see what the answer is

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9
Q

the factor theorem

A

the factors must be factors of the +c at the end of the equation

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10
Q

the discriminant

A

b^2-4ac- if =less than 0 there are no solutions, if =0 there is 1, if =more than 0 there are 2

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11
Q

stationary points

A

where dy/dx=0

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12
Q

classifying stationary points

A
  1. make a table and sub in the x values into dy/dx- if +, 0, - its a max point, if -, 0, + its a minimum point
  2. differentiate again- less than zero= max point, if =0 use the grid method, more than 0= minimum point
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13
Q

f(x) moved a upwards

A

f(x)+a

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14
Q

F(x)-a

A

moved a downwards

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15
Q

f(x) moved a leftwards

A

f(x+a)

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16
Q

F(x-a)

A

a rightwards

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17
Q

stretch by scale factor a parallel to the y axis

A

af(x)

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18
Q

f(ax)

A

f(x) stretched by scale factor 1/a parallel to the x axis

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19
Q

reflection on x axis

A

-f(x)

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20
Q

f(-x)

A

reflection on y axis

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21
Q

y=e^x reflected on y=x

A

y=loge^x

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22
Q

e^x differentiated

A

e^x

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23
Q

trapezium rule

A

1/2h(y1+last y coordinate)+2(all rest added together)

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24
Q

area under speed time graph

A

distance

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25
Q

gradient of distance time graph

A

speed

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26
Q

gradient of speed time graph

A

acceleration

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27
Q

area under acceleration time graph

A

change in velocity

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28
Q

prove k^3-k is divisible by 6

A

=k(k^2-1)
=k(k-1)(k+1)
=(k-1)k(k+1) 3 consecutive integers
at least one is even so will have a multiple of 2 and as there are 3 consecutive numbers one must be a multiple of 3, thus it must be divisible by both 3 and 2 and so 6

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29
Q

prove root 2 is irrational

A

assume it is rational and therefore can be represented as a/b where a and b are integers and a/b is in its simplest form
so 2=a^2/b^2
2b^2=2a^2- a must be even as its square is divisible by 2 so a=2k
hence b^2=k^2 so b must also be even and therefore a/b is not in its simplest form as both can be divided by 2

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30
Q

prove there are an infinite number of primes

A

assume there are a finite number of primes and make p the product of all prime numbers +1 (so p. is larger than any prime)

  • if p is prime we have found another prime exists
  • if p is not prime then p has a prime factor which must be a factor of 1
  • this is not possible as they are all whole numbers, hence there is a contradiction
31
Q

prove there exists no integers A,b for which 5a+15b= 1

A

so a+3b=1/5

a and b are integers so a+3b must = integer- contradiction exists

32
Q

position to term rule of an arithmetic sequence

A

un=u1+(n-1)d

33
Q

position to term rule of a geometric sequence

A

un=ar^n-1

34
Q

term to term rule of an arithmetic sequence

A

un=un-1+d

35
Q

term to term rule of a geometric sequence

A

un+1=run

36
Q

sum of arithmetic sequence

A
  1. add 1st and last term in the sequence
  2. divide that sum by 2
  3. multiply by the total number of terms in the sequence
37
Q

partial sum of geometric sequence

A

sn=a(1-r^n)/1-r

38
Q

sum to infinity of geometric sequence

A

=a/1-r

39
Q

if a geometric sequence converges, what is its limit

A

1-a/r

40
Q

cosec*

A

1/sin*

41
Q

sec*

A

1/cos*

42
Q

cot*

A

cos/sin or 1/tan*

43
Q

tan*

A

sin/cos

44
Q

sin^2+cos^2

A

=1

45
Q

sin(A+B)

A

=sinAcosB+cosAsinB

46
Q

cos(A+B)

A

=cosAcosB-sinAsinB

47
Q

tan(A+B)

A

=tanA+tanB/1-tanAtanB

48
Q

derivative of sinx

A

cosx

49
Q

derivative of cosx

A

-sinx

50
Q

derivative of tanx

A

sec^2x

51
Q

cosine rule

A
  1. cosA=(b^2+c^2-a^2)/2bc

2. a^2=b^2+c^2-2bcCosA

52
Q

to find a bearing

A
  1. draw diagram with North/South line
  2. join the points you want to find the distance between and use the cosine rule to do so
  3. work out the angle using the sine rule
  4. bearing should be the angle between north and the point so probably won’t just be internal angle
53
Q

base^power=number- logarithm

A

logbase^number=power

54
Q

arc length in °

A

x/360 x 2πr

55
Q

area of sector in °

A

x/360 x πr^2

56
Q

arc length in radians

A

Xr

57
Q

area of sector in radians

A

(Xr^2)/2

58
Q

what’s a mapping

A

a rule that connects input to output

59
Q

what’s a domain

A

set of all possible inputs

60
Q

codomain

A

set of all numbers from which you can choose your outputs

61
Q

range

A

all outputs from the domain

62
Q

well defined mapping

A

everything in the domain has an output and only 1 output in the codomain

63
Q

function

A

well-defined map

64
Q

how do you work out the graph of the inverse function

A

graph of function reflected on y=x

65
Q

for an inverse to be a well defined map

A

f(x) must be 1-1 and range=codomain

66
Q

volume of prism

A

area of base x h

divide by 2 if triangular prism

67
Q

volume of cylinder

A

πr^2h

68
Q

SA of cylinder

A

2πrh+2πr^2

69
Q

volume of a sphere

A

4/3∏r^3

70
Q

volume of a cone

A

1/3∏r^2h

71
Q

volume of a pyramid

A

1/3 w x l x h

72
Q

SA of a sphere

A

4∏r^2

73
Q

SA of a cone

A

∏r^2+∏rl where l is slant height

a level maths (3 decks)

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