AQA AS A Fmaths Formulae PDF

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Formulae and statistical tables for A‑level Mathematics and A‑level Further Mathematics AS Mathematics (7356) A‑level Mathematics (7357) AS Further Mathematics (7366) A‑level Further Mathematics (7367) v1.5 Issued February 2018 For the new specifications for first teaching from September 2017.

This booklet of formulae and statistical tables is required for all AS and A‑level Further Mathematics exams. Students may also use this booklet in all AS and A‑level Mathematics exams. However, there is a smaller booklet of formulae available for use in AS and A‑level Mathematics exams with only the formulae required for those examinations included.

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Contents Page Pure mathematics

4

Mechanics

10

Probability and statistics

11

Statistical tables Table 1 Percentage points of the student’s t ‑distribution

12

Table 2 Percentage points of the χ 2 distribution

13

Table 3 Critical values of the product moment correlation coefficient

14

4

Pure mathematics Binomial series  n  n  n (a + b)n = an +   an −1b +   an − 2b 2 + … +   a n −r br + … + b n  r  2  1  n where   =  r

n

Cr =

(1 + x )n = 1 + nx +

n! r!(n − r )!

n( n − 1) 2 n( n − 1)… ( n − r + 1) r x +…+ x + … ( x < 1, n ∈ ) 1.2 1.2…r

Arithmetic series Sn =

1 1 n (a + l) = n [2a + (n − 1)d] 2 2

Geometric series Sn =

a (1 − r n ) 1− r

S∞ =

a for | r | < 1 1− r

Trigonometry: small angles For small angle θ, measured in radians: sin θ ≈ θ cos θ ≈ 1 −

θ2 2

tan θ ≈ θ Trigonometric identities sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B tan (A ± B) =

(n ∈ )

sin A sin B

tan A ± tan B 1  tan A tan B

sin  A + sin B = 2 sin

(A ± B ≠ (k +

A+ B A− B cos 2 2

sin  A − sin B = 2 cos

A+ B A− B sin 2 2

cos  A + cos B = 2 cos

A+ B A− B cos 2 2

cos  A − cos B = −2 sin

A+ B A− B sin 2 2

1 )π) 2

5

Differentiation f(x)

f ′(x)

tan x

sec2 x

cosec x

−cosec x cot x

sec x

sec x tan x

cot x

−cosec2 x 1 1 − x2

sin−1 x

1 1 − x2

cos−1 x

−

tan−1 x

1 1 + x2

tanh x

sech2 x 1 1 + x2

sinh−1 x

1

cosh−1 x

x2

−1

tanh−1 x

1 1 − x2

f ( x) g (x )

f ( x ) g( x ) − f ( x ) g ( x) (g( x))2

Differentiation from first principles f (x + h ) − f (x ) →0 h

f ( x ) = lim h

6

Integration

∫ u d x dx = uv − ∫ v dx dx dv

du

f ( x)

∫ f (x ) dx = ln| f (x )| + c f(x)

∫ f(x) dx

tan x

ln |

 sec x

 | + c

cot x

ln |

 sin x

 | + c

cosec x

−ln | cosec x + cot x | = ln  tan 1 x + c

sec x

ln |

sec2 x

tan x + c

tanh x

ln cosh x + c

1 a2 − x2 1 a2 + x2

| ( )|  sec x|  t+antan ( x x| += lnπ)| + c 2

1 2

 x sin−1   + c  a

1 4

(| x | < a)

1 x tan−1   + c  a a

1 − a2

x cosh−1   or ln{x +  a

x2 − a2 } + c

1 a2 + x2

 x sinh−1   or ln{x + a

x 2 + a 2} + c

x2

1 a2 − x2 x2

1 − a2

a+x 1 1  x ln   = tanh−1   + c  a a a−x 2a

| | x −a 1 +c ln |   x +a | 2a

(x > a)

(| x | < a)

Numerical solution of equations f (x ) The Newton‑Raphson iteration for solving f (x) = 0: xn+1 = xn −  n f ( xn )

Numerical integration The trapezium rule:

b

∫a y dx ≈

b−a 1 h{( y0 + yn) + 2( y1 + y2 + … + yn−1)}, where h = 2 n

Complex numbers [r(cos θ + i sin θ)]n = r n(cos nθ + i sin nθ ) The roots of zn = 1 are given by z = e

2 πk i n ,

for k = 0, 1, 2, … , n − 1

7

Matrix transformations  cosθ Anticlockwise rotation through θ about O:   sinθ  cos2θ Reflection in the line y = (tan θ) x:   sin2θ

− sinθ cosθ 

sin2θ   −cos2θ 

The matrices for rotations (in three dimensions) through an angle θ about one of the axes are:

0 0  1  θ −sin θ for the x‑axis 0 cos  0 sin θ cos θ   cosθ 0 sinθ    1 0  for the y‑axis  0  −sin θ 0 cos θ  cosθ  sin θ   0

− sinθ cos θ 0

0 0 for the z‑axis 1

Summations n

∑ r2 = r =1 n

∑ r3 = r =1

1 n(n + 1)(2n + 1) 6 1 2 n (n + 1)2 4

Maclaurin’s series f (x) = f (0) + x f ′(0) +

x2 x r (r) f ″(0) + … + f (0 ) + … r! 2!

ex = exp(x) = 1 + x +

xr x2 +… +…+ r! 2!

ln(1 + x) = x − sin x = x − cos x = 1 −

for all x

x3 xr x2 + − … + (−1)r+1 +… 3 r 2

x5 x3 x2r +1 +… + − … + (−1)r ( 2r + 1)! 5! 3! x2r x4 x2 − … + (−1)r +… + 4! ( 2r)! 2!

(−1 < x  1) for all x for all x

8

Vectors The resolved part of a in the direction of b is i Vector product: a × b = | a || b|  sin θ nˆ = j k

a.b b

a1 a2 a3

b1 b2 b3

=

a 2b3 − a 3b 2 a b a b   3 1 − 13  a1b2 − a2b1 

If A is the point with position vector a = a1i + a2 j + a3k, then • the straight line through A with direction vector b = b1i + b2j + b3k has equation x − a1 y − a2 z − a3 = = =λ (Cartesian form) b1 b2 b3 or (r − a) × b = 0

(vector product form)

• the plane through A and parallel to b and c has vector equation r = a + sb + tc Area of a sector A = 1 ∫ r 2 dθ 2

(polar coordinates)

Hyperbolic functions cosh2 x − sinh2 x = 1 sinh 2 x = 2 sinh x cosh x cosh 2 x = cosh2 x + sinh2 x cosh−1 x = ln{x +

x 2 − 1}

sinh−1 x = ln{x +

x 2 + 1}

1+ tanh−1 x = 1 ln  1− 2

x  x

(x  1)

(| x | < 1)

Conics

Standard form Parametric form Asymptotes

Ellipse

Parabola

Hyperbola

x2 y2 + =1 a2 b2

y 2 = 4ax

x2 y2 − 2 =1 a2 b

x = a cos θ y = b sin θ

x = at2 y = 2at

x = a sec θ y = b tan θ

none

none

x y =± a b

9

Further numerical integration The mid‑ordinate rule: where h =

1 2

+ y3 +  + y

3 2

n−

2

+y

n−

1 2

)

b− a n

Simpson’s rule: where h =

b

∫a y d x » h ( y

1 ∫a y dx » 3 h {( y0 + yn ) + 4( y1 + y3 +  + yn− 1) + 2 ( y2 + y 4 +  + y n− 2 )} b

b− a and n is even n

Numerical solution of differential equations For

dy = f (x) and small h, recurrence relations are: dx Euler’s method: yn +1 = yn + hf (xn), xn +1 = xn + h

dy = f (x, y): dx Euler’s method: yr +1 = yr + hf (xr, yr),

For

xr +1 = xr + h

Improved Euler method: yr +1 = yr–1 + 2hf (xr, yr),

xr +1 = xr + h

Arc length s=

s=

2

∫

 d y 1 +   dx  d x

∫

 dy dx   +  dt  dt dt

2

(Cartesian coordinates) 2

(parametric form)

Surface area of revolution 2

 dy  Sx = 2π ∫ y 1 +   dx  dx  2

(Cartesian coordinates) 2

 d y  dx  Sx = 2π ∫ y   +   d t  dt   dt 

(parametric form)

10

Mechanics Constant acceleration s = ut + 1 at2

s = ut + 1 at 2

s = vt − 1 at2

s = vt − 1 at 2

v = u + at

v = u + at

s = 1 (u + v)t

s=

2

2

2

2 2

1 (u + v)t 2

v2 = u2 + 2as

Centres of mass For uniform bodies: Triangular lamina:

2 along median from vertex 3

Solid hemisphere, radius r: 3 r from centre 8

Hemispherical shell, radius r: 1 r from centre 2

Circular arc, radius r, angle at centre 2α:

r sin α from centre α

Sector of circle, radius r, angle at centre 2α:

2r sin α from centre 3α

Solid cone or pyramid of height h: 1 h above the base on the line from centre of base 4 to vertex Conical shell of height h:

1 h above the base on the line from centre of base to vertex 3

11

Probability and statistics Probability P(A ∪ B) = P(A) + P(B) − P(A ∩ B) P(A ∩ B) = P(A) × P(B | A) Standard deviation Σ ( x − )x n

2

=

Σ x2 − x2 n

Discrete distributions Distribution of X

P(X = x)

Mean

 n x n −x    x p  (1 − p)

Binomial B(n, p)

e−λ

Poisson Po(λ)

np

λx x!

Variance np(1 − p)

λ

λ

Sampling distributions For a random sample X1, X2, …, Xn of n independent observations from a distribution having mean μ and variance σ 2: σ2 X is an unbiased estimator of μ, with Var (X  ) = n S 2

is an unbiased estimator of

σ 2,

where

S 2

=

∑( X i − X )

2

n−1

For a random sample of n observations from N(μ, σ 2):

X−μ ~ N(0, 1) σ n X−μ ~ tn−1 S n Distribution-free (non-parametric) tests Contingency tables:

∑

(Oi − Ei )2 Ei

is approximately distributed as χ 2

12

TABLE 1

Percentage points of the student’s t-distribution p

The table gives the values of x satisfying P(X  x) = p, where X is a random variable having the student’s t‑distribution with v degrees of freedom.

0

p v 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.9

0.95

0.975

0.99

0.995

3.078

6.314

12.706

31.821

63.657

1.886

2.920

4.303

6.965

9.925

1.638

2.353

3.182

4.541

5.841

1.533

2.132

2.776

3.747

4.604

1.476

2.015

2.571

3.365

4.032

1.440

1.943

2.447

3.143

3.707

1.415

1.895

2.365

2.998

3.499

1.397

1.860

2.306

2.896

3.355

1.383

1.833

2.262

2.821

3.250

1.372

1.812

2.228

2.764

3.169

1.363

1.796

2.201

2.718

3.106

1.356

1.782

2.179

2.681

3.055

1.350

1.771

2.160

2.650

3.012

1.345

1.761

2.145

2.624

2.977

1.341

1.753

2.131

2.602

2.947

1.337

1.746

2.121

2.583

2.921

1.333

1.740

2.110

2.567

2.898

1.330

1.734

2.101

2.552

2.878

1.328

1.729

2.093

2.539

2.861

1.325

1.725

2.086

2.528

2.845

1.323

1.721

2.080

2.518

2.831

1.321

1.717

2.074

2.508

2.819

1.319

1.714

2.069

2.500

2.807

1.318

1.711

2.064

2.492

2.797

1.316

1.708

2.060

2.485

2.787

1.315

1.706

2.056

2.479

2.779

1.314

1.703

2.052

2.473

2.771

1.313

1.701

2.048

2.467

2.763

p v 29 30 31 32 33 34 35 36 37 38 39 40 45 50 55 60 65 70 75 80 85 90 95 100 125 150 200 ∞

0.9

0.95

0.975

0.99

x 0.995

1.311

1.699

2.045

2.462

2.756

1.310

1.697

2.042

2.457

2.750

1.309

1.696

2.040

2.453

2.744

1.309

1.694

2.037

2.449

2.738

1.308

1.692

2.035

2.445

2.733

1.307

1.691

2.032

2.441

2.728

1.306

1.690

2.030

2.438

2.724

1.306

1.688

2.028

2.434

2.719

1.305

1.687

2.026

2.431

2.715

1.304

1.686

2.024

2.429

2.712

1.304

1.685

2.023

2.426

2.708

1.303

1.684

2.021

2.423

2.704

1.301

1.679

2.014

2.412

2.690

1.299

1.676

2.009

2.403

2.678

1.297

1.673

2.004

2.396

2.668

1.296

1.671

2.000

2.390

2.660

1.295

1.669

1.997

2.385

2.654

1.294

1.667

1.994

2.381

2.648

1.293

1.665

1.992

2.377

2.643

1.292

1.664

1.990

2.374

2.639

1.292

1.663

1.998

2.371

2.635

1.291

1.662

1.987

2.368

2.632

1.291

1.661

1.985

2.366

2.629

1.290

1.660

1.984

2.364

2.626

1.288

1.657

1.979

2.357

2.616

1.287

1.655

1.976

2.351

2.609

1.286

1.653

1.972

2.345

2.601

1.282

1.645

1.960

2.326

2.576

13

TABLE 2

Percentage points of the χ 2 distribution p

The table gives the values of x satisfying P(X  x) = p, where X is a random variable having the χ 2 distribution with v degrees of freedom.

x

0 p v 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 45 50 55 60 65 70 75 80 85 90 95 100

0.005

0.01

0.025

0.05

0.1

0.9

0.95

0.975

0.99

0.995

0.00004

0.0002

0.001

0.004

0.016

2.706

3.841

5.024

6.635

7.879

0.010

0.020

0.051

0.103

0.211

4.605

5.991

7.378

9.210

10.597

0.072

0.115

0.216

0.352

0.584

6.251

7.815

9.348

11.345

12.838

0.207

0.297

0.484

0.711

1.064

7.779

9.488

11.143

13.277

14.860

0.412

0.554

0.831

1.145

1.610

9.236

11.070

12.833

15.086

16.750

0.676

0.872

1.237

1.635

2.204

10.645

12.592

14.449

16.812

18.548

0.989

1.239

1.690

2.167

2.833

12.017

14.067

16.013

18.475

20.278

1.344

1.646

2.180

2.733

3.490

13.362

15.507

17.535

20.090

21.955

1.735

2.088

2.700

3.325

4.168

14.684

16.919

19.023

21.666

23.589

2.156

2.558

3.247

3.940

4.865

15.987

18.307

20.483

23.209

25.188

2.603

3.053

3.816

4.575

5.578

17.275

19.675

21.920

2...