Formulae and Tables_Math

for use in the State Examinations Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Formulae and Tables PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Draft for consultation Observations are invited on this draft booklet of Formulae and Tables, which is intended to replace the Mathematics Tables for use in the state examinations In 2007, the State Examinations Commission convened a working group to review and update the Mathematics Tables booklet, which is provided to candidates for use in the state examinations The Department of Education and Science and the National Council for Curriculum and Assessment are represented on the working group The group has carried out some consultation and now presents this draft for wider consultation This draft is available in English only The final booklet will be provided in both Irish and English Anyone receiving this document should note its draft status, and should not assume that content currently in the draft will be included in the final version The final version will be circulated to all schools in advance of its introduction in the state examinations The working group recognises that the Mathematics Tables booklet is used by other institutions in their own examinations and by various individuals for other purposes Whereas the needs of such users will be noted, final decisions will be taken in the context of the booklet’s primary purpose as a reference for candidates taking examinations conducted by the State Examinations Commission Comments should be forwarded by e-mail to tables@examinations.ie before 31 May 2008 Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Contents Length and area Surface area and volume .5 Area approximations Trigonometry Co-ordinate geometry 10 Geometry 11 Algebra .12 Sequences and series 12 Number sets – notation .13 Calculus 14 Financial mathematics 16 Statistics and probability 18 Units of measurement 26 Common quantities, their symbols and units of measurement 28 Frequently used constants 33 Particle physics 34 Mechanics 36 Heat and temperature 41 Waves 41 Electricity 43 Modern physics 45 Electrical circuit symbols 46 The elements .51 Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Length and area Parallelogram b h C Arc / Sector Triangle c θ h r a Perimeter = 2a + 2b Area = ah = ab sin C When θ is in radians: Length of arc = rθ Circle / Disc Area of sector = r Length of circle = π r Area of disc = πr 2 r 2θ a Perimeter = a + b + c Area = ah = ab sin C When θ is in degrees: ⎛ θ ⎞ Length of arc = π r ⎜ ⎟ ⎝ 360° ⎠ ⎛ θ ⎞ ⎟ ⎝ 360° ⎠ Area of sector = π r ⎜ b = C s ( s − a)( s − b)( s − c) , where s = a+b+c Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Surface area and volume Cylinder Sphere Solid of uniform cross-section (prism) r r h h Curved surface area = π rh Volume = π r h A Surface area = π r Volume = πr Volume = Ah , where A is the area of the base Pyramid on any base Cone Frustum of a cone l h r πr h h R Curved surface area = π rl Volume = h r Volume = ( π h R + Rr + r A ) Volume = Ah , where A is the area of the base Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Area approximations Trapezoidal rule: Area ≈ h [ y1 + y n + 2( y + y3 + y + L + y n−1 )] or ≈ h [first + last + twice the rest ] y1 y2 y3 y4 … yn h Simpson’s rule: Area ≈ h [ y1 + y n + 2( y3 + y5 + L + y n−2 ) + 4( y + y + L + y n−1 )] , (where n is odd) or ≈ h [first + last + twice the odds + four times the evens] Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Trigonometry Definitions sin A tan A = cos A cot A = tan A sec A = cos A cosec A = sin A –1 (cos A, sin A) A –1 Trigonometric ratios of certain angles A (degrees) 0° 90° 180° 270° 30° 45° 60° A (radians) π π 3π π cos A –1 π π sin A –1 tan A not defined not defined 2 2 3 Basic identities cos A + sin A = cos(− A) = cos A sin( − A) = − sin A tan(− A) = − tan A Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Compound angle formulae Double angle formulae cos( A + B ) = cos A cos B − sin A sin B cos A = cos A − sin A cos( A − B ) = cos A cos B + sin A sin B sin A = sin A cos A sin( A + B ) = sin A cos B + cos A sin B tan A = tan A − tan A cos A = − tan A + tan A sin A = tan A + tan A cos A = (1 + cos A) sin A = (1 − cos A) sin( A − B ) = sin A cos B − cos A sin B tan( A + B) = tan A + tan B − tan A tan B tan( A − B) = tan A − tan B + tan A tan B Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Products to sums and differences Trigonometry of the triangle cos A cos B = cos( A + B) + cos( A − B ) sin A cos B = sin( A + B ) + sin( A − B ) sin A sin B = cos( A − B) − cos( A + B) cos A + cos B = cos A+ B A− B cos 2 cos A − cos B = −2 sin A+ B A− B sin 2 b Area = B cos A sin B = sin( A + B ) − sin( A − B ) Sums and differences to products A c ab sin C C a Sine rule: a b c = = sin A sin B sin C Cosine rule: a = b + c − 2bc cos A In a right-angled triangle, A+ B A− B sin A + sin B = sin cos 2 sin = opposite hypotenuse A+ B A− B sin A − sin B = cos sin 2 cos = adjacent hypotenuse tan = opposite adjacent Page PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Co-ordinate geometry Line Slope: m = Distance = y − y1 x − x1 ( x1 , y1 ) ( x − x1 ) + ( y − y1 ) ( x2 , y ) ⎛ x + x y1 + y ⎞ , Midpoint = ⎜ ⎟ ⎠ ⎝ Equation of a line: y − y1 = m( x − x1 ) or y = mx + c Area of a triangle with one vertex at the origin = x1 y − x y1 ⎛ bx + ax by1 + ay ⎞ , Point dividing a line segment in the ratio a : b = ⎜ ⎟ a+b ⎠ ⎝ a+b Distance from a point to a line = ax1 + by1 + c a2 + b2 Page 10 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem F A Pressure p= Pressure in a fluid p = ρgh Thrust on an immersed plane surface T = A pcentroid Boyle’s law pV = constant Heat and temperature Celsius temperature t / ˚C = T /K – 273.15 Energy needed to change temperature ∆ E = mc ∆ θ ∆E = C∆θ Energy needed to change state ∆ E = ml ∆E = L Waves Velocity of a wave c = fλ Doppler effect f ′= Fundamental frequency of a stretched string f = fc c±u T 2l μ Page 41 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Mirror and lens formula f = u + v v u Magnification m= Power of a lens P= Two lenses in contact P = P1 + P2 Refractive index n= n= Diffraction grating f sin i real depth = sin r apparent depth sin C = c1 c2 nλ = d sin θ Page 42 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Electricity Coulomb’s law F= q1 q2 πε d Electric field strength E= F q Potential difference V= W q Ohm’s law V = IR Resistivity ρ= Resistors in series R = R1 + R2 Resistors in parallel R = RA l R1 + Wheatstone bridge R R3 = R2 R4 Joule’s law P = RI R2 Page 43 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Force on a current-carrying conductor F = IlB Force on a charged particle F = qvB Induced emf E=− Alternating voltage and current Vrms = Capacitance C= q V Parallel-plate capacitor C= Aε d Energy stored in capacitor W= Power P = VI Magnetic flux Φ = BA Transformer Vi N p = Vo N s dΦ dt V0 I rms = I0 CV Page 44 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Modern physics Energy of a photon E = hf Einstein’s photoelectric equation h f = Φ + mvmax Law of radioactive decay A = λN Half-life T1 = Mass-energy equivalence ln λ E = mc Page 45 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Electrical circuit symbols Conductors junction of conductors conductors crossing with no connection Switches push-to-make switch push-to-break switch two-way switch (SPDP) dual on-off switch (DPST) normally open on-off switch (SPST) normally closed on-off switch (SPST) dual two-way switch (DPDT) relay (SPST) Page 46 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Power cell photovoltaic cell battery d.c supply a.c supply transformer fuse earth variable resistor (rheostat) preset variable resistor potential divider Resistors fixed resistor thermistor light-dependent resistor Page 47 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Capacitors capacitor electrolytic capacitor (polarised capacitor) variable capacitor preset variable capacitor (trimmer) diode Zener diode photodiode light-emitting diode (LED) Diodes Meters Ω voltmeter galvanometer ammeter ohmmeter oscilloscope Page 48 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Amplification pnp-junction transistor light-sensitive transistor amplifier (block symbol) microphone earphone loudspeaker bell buzzer piezoelectric transducer aerial (antenna) filament lamp signal lamp neon lamp npn-junction transistor Audio Lamps Page 49 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Other devices M motor heater inductor inductor with ferromagnetic core AND NOT (inverter) NAND Logic gates OR NOR Page 50 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem The elements Periodic table of the elements 18 H He 1.008 13 14 15 16 17 10 Li Be B C N O F Ne 6.941 9.012 10.81 12.01 14.01 16.00 19.00 20.18 11 12 13 14 15 16 17 18 4.003 Na Mg A1 Si P S Cl Ar 22.99 24.31 10 11 12 26.98 28.09 30.97 32.07 35.45 39.95 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 39.10 40.08 44.96 47.87 50.94 52.00 54.94 55.85 58.93 58.69 63.55 65.41 69.72 72.64 74.92 78.96 79.90 83.80 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 85.47 87.62 88.91 91.22 92.91 95.94 (97.90) 101.1 102.9 106.4 107.9 112.4 114.8 118.7 121.8 127.6 126.9 131.3 55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 132.9 137.3 138.9 178.5 180.9 183.8 186.2 190.2 192.2 195.1 197.0 200.6 204.4 207.2 209.0 87 88 89 104 105 106 107 108 109 110 111 112 113 114 115 Fr Ra Ac Rf Db Sg Bh Hs Mt Ds Rg (209.0) (210.0) (222.0) 116 117 118 Uub Uut* Uuq Uup* Uuh Uus* Uuo (223.0) (226.0) (227.0) (261.1) (262.1) (266.6) (264.1) (277.0) (268.1) (271.0) (272.2) (285.0) (289.0) (289.0) (293.0) * These elements have not yet been detected (2008) See page 54 for the Lanthanoid and the Actinoid Series Page 51 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem First ionisation energies of the elements (in kJ mol–1) 18 H He 1312 13 14 15 16 17 2372 10 Li Be B C N O F Ne 520.2 899.5 800.6 1087 1402 1314 1681 2081 11 12 13 14 15 16 17 18 Na Mg 495.8 737.7 10 11 12 A1 Si P S Cl Ar 577.5 786.5 1012 999.6 1251 1521 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 418.8 589.8 633.1 658.8 650.9 652.9 717.3 762.5 760.4 737.1 745.5 906.4 578.8 762.0 947.0 941.0 1140 1351 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 403.0 549.5 600.0 640.1 652.1 684.3 702.0 710.2 719.7 804.4 731.0 867.8 558.3 708.6 834.0 869.3 1008 1170 55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 375.7 502.9 538.1 658.5 761.0 770.0 760.0 840.0 880.0 870.0 890.1 1007 589.4 715.6 703.0 812.1 890±40 1037 87 88 89 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 Fr Ra Ac Rf Db Sg Bh Hs Mt Ds Rg 380.0 509.3 499.0 580.0 –– –– –– –– –– –– –– Uub Uut* Uuq Uup* Uuh Uus* Uuo –– –– –– –– * These elements have not yet been detected (2008) See page 54 for the Lanthanoid and the Actinoid Series Page 52 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Electronegativity values of the elements (Using the Pauling scale) 18 H He 2.20 13 14 15 16 17 10 Li Be B C N O F Ne 0.98 1.57 2.04 2.55 3.04 3.44 3.98 –– 11 12 13 14 15 16 17 18 –– Na Mg A1 Si P S Cl Ar 0.93 1.31 10 11 12 1.61 1.90 2.19 2.58 3.16 –– 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 0.82 1.00 1.36 1.54 1.63 1.66 1.55 1.83 1.88 1.91 1.90 1.65 1.81 2.01 2.18 2.55 2.96 –– 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 0.82 0.95 1.22 1.33 1.60 2.16 2.10 2.20 2.28 2.20 1.93 1.69 1.78 1.96 2.05 2.10 2.66 2.60 55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 0.79 0.89 1.10 1.30 1.50 1.70 1.90 2.20 2.20 2.20 2.40 1.90 1.80 1.80 1.90 2.00 2.20 –– 87 88 89 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 Fr Ra Ac Rf Db Sg Bh Hs Mt Ds Rg 0.70 0.90 1.10 –– –– –– –– –– –– –– –– Uub Uut* Uuq Uup* Uuh Uus* Uuo –– –– –– –– * These elements have not yet been detected (2008) See page 54 for the Lanthanoid and the Actinoid Series Page 53 PDF Watermark Remover DEMO : Purchase from www.PDFWatermarkRem Periodic table of the elements Lanthanoid Series Actinoid Series 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 140.1 140.9 144.2 (144.9) 150.4 152.0 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 232.0 231.0 238.0 (237.0) (244.1) (243.1) (247.1) (247.1) (251.1) (252.1) (257.1) (256.0) 67 68 69 (259.1) (262.1) First ionisation energies of the elements (in kJ mol–1) 58 Lanthanoid Series Actinoid Series 59 60 61 62 63 64 65 66 70 71 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 534.4 527.0 533.1 540.0 544.5 547.1 593.4 565.8 573.0 581.0 589.3 596.7 603.4 523.5 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 587.0 568.0 597.6 604.5 584.7 578.0 581.0 601.0 608.0 619.0 627.0 635.0 642.0 470.0 66 67 68 69 70 71 Electronegativity values of the elements (Using the Pauling scale) 58 Lanthanoid Series Actinoid Series 59 60 61 62 63 64 65 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 1.12 1.13 1.14 –– 1.17 –– 1.20 –– 1.22 1.23 1.24 1.25 –– 1.00 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 1.30 1.50 1.70 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 Page 54 PDF Watermark Remover DEMO 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Tables booklet, which is provided to candidates for use in the state examinations The Department of Education and Science and the National Council for Curriculum and Assessment are represented on... of Formulae and Tables, which is intended to replace the Mathematics Tables for use in the state examinations In 2007, the State Examinations Commission convened a working group to review and. .. the population mean μ The adjusted sample standard Σ( x − μ ) is an unbiased estimator deviation s = n −1 of the population standard deviation σ n standard error of the proportion is Hypothesis