pure

Question 1

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

Answer 1

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

Question 2

when is binomial theorem valid

Answer 2

when x lies between -1 and 1

Question 3

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

Answer 3

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

Question 4

the chain rule

Answer 4

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

Question 5

integrate

Answer 5

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

Question 6

product rule

Answer 6

use when u and v are both functions of x

u x dv/dx+ v x du/dx

Question 7

quotient rule

Answer 7

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

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

Question 8

remainder theorem/ find the remainder

Answer 8

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

Question 9

the factor theorem

Answer 9

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

Question 10

the discriminant

Answer 10

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

Question 11

stationary points

Answer 11

where dy/dx=0

Question 12

classifying stationary points

Answer 12

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

Question 13

f(x) moved a upwards

Answer 13

f(x)+a

Question 14

F(x)-a

Answer 14

moved a downwards

Question 15

f(x) moved a leftwards

Answer 15

f(x+a)

Question 16

F(x-a)

Answer 16

a rightwards

Question 17

stretch by scale factor a parallel to the y axis

Answer 17

af(x)

Question 18

f(ax)

Answer 18

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

Question 19

reflection on x axis

Answer 19

-f(x)

Question 20

f(-x)

Answer 20

reflection on y axis

Question 21

y=e^x reflected on y=x

Answer 21

y=loge^x

Question 22

e^x differentiated

Answer 22

e^x

Question 23

trapezium rule

Answer 23

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

Question 24

area under speed time graph

Answer 24

distance

Question 25

gradient of distance time graph

Answer 25

speed

Question 26

gradient of speed time graph

Answer 26

acceleration

Question 27

area under acceleration time graph

Answer 27

change in velocity

Question 28

prove k^3-k is divisible by 6

Answer 28

=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

Question 29

prove root 2 is irrational

Answer 29

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

Question 30

prove there are an infinite number of primes

Answer 30

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

Question 31

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

Answer 31

so a+3b=1/5 | a and b are integers so a+3b must = integer- contradiction exists

Question 32

position to term rule of an arithmetic sequence

Answer 32

un=u1+(n-1)d

Question 33

position to term rule of a geometric sequence

Answer 33

un=ar^n-1

Question 34

term to term rule of an arithmetic sequence

Answer 34

un=un-1+d

Question 35

term to term rule of a geometric sequence

Answer 35

un+1=run

Question 36

sum of arithmetic sequence

Answer 36

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

Question 37

partial sum of geometric sequence

Answer 37

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

Question 38

sum to infinity of geometric sequence

Answer 38

=a/1-r

Question 39

if a geometric sequence converges, what is its limit

Answer 39

1-a/r

Question 40

cosec*

Answer 40

1/sin*

Question 41

sec*

Answer 41

1/cos*

Question 42

cot*

Answer 42

cos*/sin* or 1/tan*

Question 43

tan*

Answer 43

sin*/cos*

Question 44

sin^2*+cos^2*

Answer 44

=1

Question 45

sin(A+B)

Answer 45

=sinAcosB+cosAsinB

Question 46

cos(A+B)

Answer 46

=cosAcosB-sinAsinB

Question 47

tan(A+B)

Answer 47

=tanA+tanB/1-tanAtanB

Question 48

derivative of sinx

Answer 48

cosx

Question 49

derivative of cosx

Answer 49

-sinx

Question 50

derivative of tanx

Answer 50

sec^2x

Question 51

cosine rule

Answer 51

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

Question 52

to find a bearing

Answer 52

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

Question 53

base^power=number- logarithm

Answer 53

logbase^number=power

Question 54

arc length in °

Answer 54

x/360 x 2πr

Question 55

area of sector in °

Answer 55

x/360 x πr^2

Question 56

arc length in radians

Answer 56

Xr

Question 57

area of sector in radians

Answer 57

(Xr^2)/2

Question 58

what's a mapping

Answer 58

a rule that connects input to output

Question 59

what's a domain

Answer 59

set of all possible inputs

Question 60

codomain

Answer 60

set of all numbers from which you can choose your outputs

Question 61

range

Answer 61

all outputs from the domain

Question 62

well defined mapping

Answer 62

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

Question 63

function

Answer 63

well-defined map

Question 64

how do you work out the graph of the inverse function

Answer 64

graph of function reflected on y=x

Question 65

for an inverse to be a well defined map

Answer 65

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

Question 66

volume of prism

Answer 66

area of base x h | divide by 2 if triangular prism

Question 67

volume of cylinder

Answer 67

πr^2h

Question 68

SA of cylinder

Answer 68

2πrh+2πr^2

Question 69

volume of a sphere

Answer 69

4/3∏r^3

Question 70

volume of a cone

Answer 70

1/3∏r^2h

Question 71

volume of a pyramid

Answer 71

1/3 w x l x h

Question 72

SA of a sphere

Answer 72

4∏r^2

Question 73

SA of a cone

Answer 73

∏r^2+∏rl where l is slant height