AS GCE (H635) A GCE (H645) Further Mathematics B (MEI

*3410625432*

Further Mathematics B (MEI)© OCR 2020 [603/1364/X]DC (ST) 203569 CST323

AS and A Level Further M

athematics B

(MEI)

INSTRUCTIONS• Do not send this Booklet for marking. Keep it in the centre or recycle it.

INFORMATION• This document has 16 pages.

AS Level Further Mathematics B (MEI) (H635)A Level Further Mathematics B (MEI) (H645)

Formulae Booklet

Oxford Cambridge and RSA

2

Further Mathematics B (MEI)© OCR 2020

Contents

A level MathematicsCore PureMechanicsFurther Pure with TechnologyExtra PureNumerical MethodsStatisticsStatistical tables

A level Mathematics

Arithmetic series

( ) { ( ) }S n a l n a n d2 1n 21

21= + = + -

Geometric series

( )S r

a r11

n

n

=--

forS ra r1 11=-3

Binomial series

( ) ( ),C C Ca b a a b a b a b b n Nn n n n n n nr

n r r n1

12

2 2 f f !+ = + + + + + +- - -

where ! ( ) !!C C

nr r n r

nnr n r= = =

-

J

LKKN

POO

( ) ,!( )

!( ) ( )

x nx x nn n

x rn n n r

x1 1 121 1 1

Rn r2 ff

f 1 !+ = + +-

+ +- - +

+ ^ h

Differentiation

( )f x ( )f xltan kx seck kx2

sec x sec tanx xcot x cosec x2-

cosec x cosec cotx x-

Quotient Rule , dd d

ddd

y vu

xy

v

v xu u x

v

2= =-

Differentiation from first principles

( )( ) ( )

f limf f

x hx h x

h 0=

+ -

"l

3

Further Mathematics B (MEI) Turn over© OCR 2020

Integration

( )( )

( )ff d ln fxxx x c= +

ly

( ) ( ) ( )f f d fx x x n x c11n n 1

=+

++l ^ ^h hy

Integration by parts dd d d

d du vx x uv v x

u x= -y y

Small Angle Approximations

, ,sin cos tan1 21 2. . .i i i i i i- where i is measured in radians

Trigonometric identities

( )sin sin cos cos sinA B A B A B! !=

( )cos cos cos sin sinA B A B A B! "=

( ) ( ( ) )tan tan tantan tanA B A BA B A B k1 2

1!

"!

! ! r= +

Numerical methods

Trapezium rule: {( ) ( )}dy x h y y y y y2a

bn n2

10 1 2 1f. + + + + +

-y , where h nb a

=-

The Newton-Raphson iteration for solving ( ) : ( )( )

f ff

x x x xx

0 n nn

n1= = -+ l

Probability

( ) ( ) ( ) ( )P P P PA B A B A B, += + -

( ) ( ) ( ) ( ) ( )P P P P PA B A B A B A B+ = = or ( ) ( )( )

P PP

A B BA B+

=

Sample Variance

s n S11

xx2 =

- where ( )S x x x n

xx nxxx i i

ii

2 22

2= - = - = - 2^ h/ /

//

Standard deviation, variances =

4


The Binomial Distribution

If ( , )BX n p+ then ( ) CP X r p qnrr n r= = - where q p1= -

Mean of X is np

Hypothesis test for the mean of a Normal distribution

If ( , )NX 2+ vn then ,NXn

2+ n

vJ

LKK

N

POO and

/( , )N

nX

0 1+v

n-

Percentage points of the normal distribution

z

%p21 %p2

1

p 10 5 2 1

z 1.645 1.960 2.326 2.576

Kinematics

Motion in a straight line Motion in two and three dimensions

v u at= + v u a t= +

s ut at21 2= + s u at t2

1 2= +

( )s u v t21= + ( )s u v t2

1= +

v u as22 2= + -

s vt at21 2= - s v at t2

1 2= -

5


Core Pure

Complex Numbers

De Moivre’s theorem:{ ( )} ( )cos isin cos isinr r n nn ni i i i+ = +

Roots of unity:

The roots of z 1n = are given by exp iznk2r

=J

LKK

N

POO for , , , ,k n0 1 2 1f= -

Vectors and 3-D geometry

Cartesian equation of a plane isn x n y n z d 01 2 3+ + + =

Cartesian equation of a line in 3-D is

dx a

dy a

dz a

1

1

2

2

3

3-=-

=-

Vector product a ba b a ba b a ba b a b

2 3 3 2

3 1 1 3

1 2 2 1

# =

-

-

-

J

L

KKKK

N

P

OOOO

sina bijk

a b na b a ba b a ba b a b

aaa

bbb

2 3 3 2

3 1 1 3

1 2 2 1

1

2

3

1

2

3

# i=

-

-

-

= = t

J

L

KKKK

N

P

OOOO

where , ,a b nt , in that order, form a right-handed triple.

Distance between skew lines is . ( )d d

d da a1 2

1 21 2

#

#- where a1 is the position vector of a point on the first line and

d1 is parallel to the first line, similarly for the second line.

Distance between point (x1, y1) and line ax by c 0+ + = is a b

ax by c2 2

1 1

+

+ +

Distance between point (x1, y1, z1) and plane n x n y n z d 01 2 3+ + + = is n n n

n x n y n z d

12

22

32

1 1 2 1 3 1

+ +

+ + +

Hyperbolic functions

cosh sinhx x 12 2- =

[ ( )]arsinh lnx x x 12= + +

[ ( ],)arcosh lnx x x x1 12 $= + -

,artanh lnxxx x2

111 1 11 1=-

+-

J

LKK

N

POO

6


Calculus

( )f x ( )f xlarcsinx

x11

2-arccosx

x11

2-

-arctanx

x112+

( )f x ( )f dx xy

a x12 2-

arcsinax x a1J

LKK ^N

POO h

a x1

2 2+ arctana ax1 J

LKKN

POO

a x12 2+

( )arsinh or lnax x x a2 2+ +J

LKKN

POO

x a12 2-

( ) ( )arcosh or lnax x x a x a2 2 2+ -J

LKKN

POO

The mean value of f(x) on the interval [a,b] is ( )f db a x x1a

b

-y

Area of sector enclosed by polar curve is dr21 2 iy

Series

( ) ( )r n n n61 1 2 1

r

n2

1= + +

=

/ ( )r n n41 1

r

n3

1

2 2= +=

/

( ) ( ) ( ) !( )

!( )

f f ff f

x x x r x0 0 20 0( )r

r2 f f= + + + + +lm

( ) ! !e exp x x xrx1 2

xr2

f f= = + + + + + for all x

( ) ( ) ( )ln x x x xrx x1 2 3 1 1 1rr2 3

1f f 1 #+ = - + - + - + -+

! ! ( ) ( ) !sin x x x xrx

3 5 1 2 1r

r3 5 2 1f f= - + - + -

++

+

for all x

! ! ( ) ( ) !cos x x x xr1 2 4 1 2

rr2 4 2

f f= - + - + - + for all x

( ) !( )

!( ) ( )

,x nxn n

x rn n n r

x x n1 1 21 1 1

1 Rn r2 ff

f 1 !+ = + +-

+ +- - +

+ ^ h

7


Mechanics

Motion in a circle

For motion in a circle,

tangential velocity is v ri= o

radial acceleration is orrv r2

2io towards the centre

tangential acceleration is rip

Further Pure with Technology

Numerical solution of differential equations

For ( , ):dd

fxy

x y=

Euler’s method: ( , )fx x h y y h x yn n n n n n1 1= + = ++ +

Modified Euler method (A Runge-Kutta method of order 2):

( , )fk h x yn n1 =

( , )fk h x h y kn n2 1= + +

, ( )x x h y y k k21

n n n n1 1 1 2= + = + ++ +

Runge-Kutta method of order 4:

( , )fk h x yn n1 =

( , )fk h x h yk

2 2n n21= + +

( , )fk h x h yk

2 2n n32= + +

( , )fk h x h y kn n4 3= + +

( )k ky y k k61 2 2n n1 1 2 3 4= + + + +

+

Gradient of tangent to a polar curve

For a curve ( ),f dd

dd cos sindd sin cos

r xy

r r

r ri

ii i

ii i

= =+

-

8


Extra Pure

Multivariable calculus

g g

g

g

g

gradx

y

z

d

2

2

2

2

2

2

= =

J

L

KKKKKKK

N

P

OOOOOOO

. If g(x, y, z) can be written as , )(fz x y= then g

f

fgradx

y1

22

22=

-

J

L

KKKKK

N

P

OOOOO

9


Numerical methods

Solution of equations

The Newton-Raphson iteration for solving ( ) : ( )( )

f ff

x x x xx

0 n nn

n1= = -+ l

For the iteration ( )gx xn n1 =+ the relaxed iteration is ( ) ( )gx x x1n n n1 m m= - +

+.

Numerical integration

To estimate ( )f dx xa

by :

The midpoint rule:

( )M h y y y yn n n21

23

23

21f= + + + +

- - where h n

b a=-

The trapezium rule:

{( ) ( )}T h y y y y y21 2n n n0 1 2 1f= + + + + +

- where h n

b a=-

Simpson’s rule

{( ) ( ) ( )}S h y y y y y y y y31 4 2n n n n2 0 2 1 3 2 1 2 4 2 2f f= + + + + + + + + +

- -

where h nb a2=-

These are related as follows:

( )T M T21

n n n2 = +

( ) ( )S M T T T31 2 3

1 4n n n n n2 2= + = -

Interpolation

Newton’s forward difference interpolation formula:

( ) ( )( )

( )!

( ) ( )( )f f f fx x h

x xx

hx x x x

x20

00 2

0 1 20 fT T= +

-+- -

+

Lagrange’s polynomial:

( ) ( ) ( )P L fx x xn r r=/ where ( )L x x xx x

rr i

i

i ri

n

0=

-

-

!=

%

10


Statistics

Discrete distributions

X is a random variable taking values xi in a discrete distribution with ( )P X x pi i= =

Expectation: ( )E X x pi in = =/Variance: ( ) ( )Var X x p x pi i i i

2 2 2 2v n n= = - = -//

Probability E(X) Var(X)

Uniform distribution over 1, 2, …, n ( )P X r n1

= =n 12+ ( )n12

1 12 -

Geometric distribution ( )P X r q pr 1= = -

q p1= -

p1

pp12-

Poisson distribution( ) !eP X r r

rm= = m-

Correlation and regression

For a sample of n pairs of observations ( , )x yi i

, ,S x nx

S y ny

S x y nx y

xx ii

yy ii

xy i ii i2

2

2

2

= - = - = -a ak k/

//

/ ///

product moment correlation coefficient: rS S

S

x nx

y ny

x y nx y

xx yy

xy

ii

ii

i ii i

2

2

2

2= =

- -

-

J

LKK

J

LKK

a aN

POO

N

POO

k kR

T

SSS

V

X

WWW

//

//

///

least squares regression line of y on x is ( )y y b x x- = - where b SS

x nx

x y nx y

xx

xy

ii

i ii i

2

2= =

-

-

a k//

///

least squares regression line of x on y is ( )bx x y y- = -l where b SS

n

x y nx y

yy

xy

ii

i ii i

yy2

2= =

-

-l

a k//

///

Spearman’s coefficient of rank correlation:

( )r

n nd

11

6s

i2

2

= --

/

11


Confidence intervals

To calculate a confidence interval for a mean or difference in mean in different circumstances, use the given distribution to calculate the critical value, k.

To estimate… Confidence interval Distribution

a mean x kn

!v N(0, 1)

a mean x kns

! tn–1

difference in mean of paired populations treat differences as a single distribution

Hypothesis tests

Description Test statistic Distribution

Pearson’s product moment correlation test r

S S

S

xx yy

xy=

xn

xy

n

y

x y nx y

ii

ii

i ii i

2

2

2

2=

- -

-

J

L

KKK

J

L

KKK

a aN

P

OOO

N

P

OOO

k kR

T

SSS

V

X

WWW

//

//

///

Spearman’s rank correlation test

( )r

n nd

11

6s

i2

2

= --

/

2| testf

f fo

e

e2-^ h/

v2|

Normal test for a mean

n

xv

n-J

LKK

N

POO

N(0, 1)

t-test for a mean sn

x n-J

LKK

N

POO

tn–1

Wilcoxon single sample test A statistic T is calculated from the ranked data

Continuous distributionsX is a continuous random variable with probability density function (pdf) ( )f x

Expectation: ( ) ( )E f dX x x xn = = yVariance: ( ) ( ) ( ) ( )Var f d f dX x x x x x x2 2 2 2v n n= = - = -yyCumulative distribution function ( ) ( ) ( )F P f dx X x t tx

#= =3-y

E(X) Var(X)

Continuous uniform distribution over [a,b] a b2+ ( )b a12

1 2-

12


Cri

tical

val

ues f

or th

e pr

oduc

t mom

ent c

orre

latio

n co

effic

ient

, rC

ritic

al v

alue

s for

Spe

arm

an’s

ran

k co

rrel

atio

n co

effic

ient

, rs

5%2½

%1%

½%

1-Ta

il Te

st5%

2½%

1%½

%5%

2½%

1%½

%1-

Tail

Test

5%2½

%1%

½%

10%

5%2%

1%2-

Tail

Test

10%

5%2%

1%10

%5%

2%1%

2-Ta

il Te

st10

%5%

2%1%

nn

nn

1-

--

-31

0.30

090.

3550

0.41

580.

4556

1-

--

-31

0.30

120.

3560

0.41

850.

4593

2-

--

-32

0.29

600.

3494

0.40

930.

4487

2-

--

-32

0.29

620.

3504

0.41

170.

4523

30.

9877

0.99

690.

9995

0.99

9933

0.29

130.

3440

0.40

320.

4421

3-

--

-33

0.29

140.

3449

0.40

540.

4455

40.

9000

0.95

000.

9800

0.99

0034

0.28

690.

3388

0.39

720.

4357

41.

0000

--

-34

0.28

710.

3396

0.39

950.

4390

50.

8054

0.87

830.

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8735

0.28

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0.28

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60.

7293

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0.27

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0.38

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4238

60.

8286

0.88

570.

9429

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0.27

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4268

70.

6694

0.75

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0.70

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0.37

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100.

5494

0.63

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0.36

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5214

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0.73

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0.26

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0.36

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110.

5364

0.61

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0.22

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2636

0.31

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4451

0.48

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0.22

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2609

0.30

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0.38

280.

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0.49

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0.30

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0.37

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4372

0.47

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810.

2586

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0.37

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0.48

2858

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2589

0.30

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3115

0.36

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0.30

300.

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3061

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100.

4226

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440.

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0.29

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3301

300.

3063

0.36

240.

4251

0.46

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0.21

440.

2545

0.30

050.

3314

13


Percentage points of the 2| (chi-squared) distribution

p%

2|

p%

2|

p% 99 97.5 95 90 10 5 2.5 1 0.5

v = 1 .0001 .0010 .0039 .0158 2.706 3.841 5.024 6.635 7.8792 .0201 .0506 0.103 0.211 4.605 5.991 7.378 9.210 10.603 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.34 12.844 0.297 0.484 0.711 1.064 7.779 9.488 11.14 13.28 14.865 0.554 0.831 1.145 1.610 9.236 11.07 12.83 15.09 16.756 0.872 1.237 1.635 2.204 10.64 12.59 14.45 16.81 18.557 1.239 1.690 2.167 2.833 12.02 14.07 16.01 18.48 20.288 1.646 2.180 2.733 3.490 13.36 15.51 17.53 20.09 21.959 2.088 2.700 3.325 4.168 14.68 16.92 19.02 21.67 23.5910 2.558 3.247 3.940 4.865 15.99 18.31 20.48 23.21 25.1911 3.053 3.816 4.575 5.578 17.28 19.68 21.92 24.72 26.7612 3.571 4.404 5.226 6.304 18.55 21.03 23.34 26.22 28.3013 4.107 5.009 5.892 7.042 19.81 22.36 24.74 27.69 29.8214 4.660 5.629 6.571 7.790 21.06 23.68 26.12 29.14 31.3215 5.229 6.262 7.261 8.547 22.31 25.00 27.49 30.58 32.8016 5.812 6.908 7.962 9.312 23.54 26.30 28.85 32.00 34.2717 6.408 7.564 8.672 10.09 24.77 27.59 30.19 33.41 35.7218 7.015 8.231 9.390 10.86 25.99 28.87 31.53 34.81 37.1619 7.633 8.907 10.12 11.65 27.20 30.14 32.85 36.19 38.5820 8.260 9.591 10.85 12.44 28.41 31.41 34.17 37.57 40.0021 8.897 10.28 11.59 13.24 29.62 32.67 35.48 38.93 41.4022 9.542 10.98 12.34 14.04 30.81 33.92 36.78 40.29 42.8023 10.20 11.69 13.09 14.85 32.01 35.17 38.08 41.64 44.1824 10.86 12.40 13.85 15.66 33.20 36.42 39.36 42.98 45.5625 11.52 13.12 14.61 16.47 34.38 37.65 40.65 44.31 46.9326 12.20 13.84 15.38 17.29 35.56 38.89 41.92 45.64 48.2927 12.88 14.57 16.15 18.11 36.74 40.11 43.19 46.96 49.6428 13.56 15.31 16.93 18.94 37.92 41.34 44.46 48.28 50.9929 14.26 16.05 17.71 19.77 39.09 42.56 45.72 49.59 52.3430 14.95 16.79 18.49 20.60 40.26 43.77 46.98 50.89 53.6735 18.51 20.57 22.47 24.80 46.06 49.80 53.20 57.34 60.2740 22.16 24.43 26.51 29.05 51.81 55.76 59.34 63.69 66.7750 29.71 32.36 34.76 37.69 63.17 67.50 71.42 76.15 79.49100 70.06 74.22 77.93 82.36 118.5 124.3 129.6 135.8 140.2

14


Percentage points of the t distribution

t

%p21 %p2

1

p%v

10 5 2 1

1 6.314 12.71 31.82 63.662 2.920 4.303 6.965 9.9253 2.353 3.182 4.541 5.8414 2.132 2.776 3.747 4.6045 2.015 2.571 3.365 4.0326 1.943 2.447 3.143 3.7077 1.895 2.365 2.998 3.4998 1.860 2.306 2.896 3.3559 1.833 2.262 2.821 3.250

10 1.812 2.228 2.764 3.16911 1.796 2.201 2.718 3.10612 1.782 2.179 2.681 3.05513 1.771 2.160 2.650 3.01214 1.761 2.145 2.624 2.97715 1.753 2.131 2.602 2.94720 1.725 2.086 2.528 2.84530 1.697 2.042 2.457 2.75050 1.676 2.009 2.403 2.678

100 1.660 1.984 2.364 2.626∞ 1.645 1.960 2.326 2.576 = percentage points of the Normal distribution N(0, 1)

Percentage points of the normal distribution

z

%p21 %p2

1

p 10 5 2 1

z 1.645 1.960 2.326 2.576

15


Critical values for the Wilcoxon Single Sample test

1-tail 5% 2½% 1% ½% 1-tail 5% 2½% 1% ½%2-tail 10% 5% 2% 1% 2-tail 10% 5% 2% 1%n n

26 110 98 84 752 – – – – 27 119 107 92 833 – – – – 28 130 116 101 914 – – – – 29 140 126 110 1005 0 – – – 30 151 137 120 1096 2 0 – – 31 163 147 130 1187 3 2 0 – 32 175 159 140 1288 5 3 1 0 33 187 170 151 1389 8 5 3 1 34 200 182 162 14810 10 8 5 3 35 213 195 173 15911 13 10 7 5 36 227 208 185 17112 17 13 9 7 37 241 221 198 18213 21 17 12 9 38 256 235 211 19414 25 21 15 12 39 271 249 224 20715 30 25 19 15 40 286 264 238 22016 35 29 23 19 41 302 279 252 23317 41 34 27 23 42 319 294 266 24718 47 40 32 27 43 336 310 281 26119 53 46 37 32 44 353 327 296 27620 60 52 43 37 45 371 343 312 29121 67 58 49 42 46 389 361 328 30722 75 65 55 48 47 407 378 345 32223 83 73 62 54 48 426 396 362 33924 91 81 69 61 49 446 415 379 35525 100 89 76 68 50 466 434 397 373

16


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AS GCE (H635) A GCE (H645) Further Mathematics B (MEI