Fist Issued Septembe 07 Fo the ew specifictios fo fist techig fom Septembe 07 SPECIMEN MATERIAL Fomule d Sttisticl Tbles fo A-level Mthemtics AS MATHEMATICS (7356) A-LEVEL MATHEMATICS (7357) AS FURTHER MATHEMATICS (7366) A-LEVEL FURTHER MATHEMATICS (7367)
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Cotets Pge 4 Pue Mthemtics 0 Mechics 0 Pobbility d Sttistics Sttisticl Tbles Tble Cumultive Poisso Distibutio Fuctio 4 Tble Pecetge Poits of the Studet s t-distibutio 5 Tble 3 Pecetge Poits of the χ Distibutio 6 Tble 4 Citicl Vlues of the Poduct Momet Coeltio Coefficiet 3
PURE MATHEMATICS Biomil seies ( + b) = + b+ b + + b + + b ( )! whee = C =!( )! ( ) ( ) ( + ) ( + ) = + + + + +.. ( <, ) Aithmetic seies S= ( + l) = + ( ) d Geometic seies ( ) S = S = fo < Tigoomety: smll gles Fo smll gle θ, siθ θ cos θ θ tθ θ [ ] Tigoometic idetities si( A± B) = si Acos B± cos Asi B cos( A± B) = cos Acos B si Asi B t( A± B) = t A± t B ± ( + ) t At B si A+ si B = si A+ B cos A B si A si B = cos A+ B si A B cos A+ cos B = cos A+ B cos A B cos A cos B = si A+ B si A B ( A B k π) 4
Diffeetitio f( ) f ( ) t k k sec k cosec cosec cot sec sec t cot f ( ) g ( ) si cos t cosec f ( ) g ( ) f() g () ( g ( )) th sih cosh th + sech + 5
Itegtio u dv d = uv v du d d d (+ costt; > 0 whee elevt) f( ) f( ) d t cot l sec l si cosec l cosec + cot = l t ( ) sec l sec + t = l t ( + π) sec k th + + t k k l cosh si ( < ) t Numeicl solutio of equtios 4 { } cosh o l + ( > ) sih o l{ } + + + l th = ( < ) l + The Newto-Rphso itetio fo solvig f( ) = 0 : Numeicl itegtio b f( ) = + f( ) The tpezium ule: y d h {( y 0 + y ) + ( y + y y + + )}, whee Comple umbes [ ( cos θ + i si θ)] = ( cos θ + i si θ) The oots of z = e give by z e π = ki, fo k = 0,,,, b h = 6
Mti tsfomtios Aticlockwise ottio though θ bout O: Reflectio i the lie y = (t θ ) : cos θ siθ cos θ si θ si θ cos θ siθ cos θ The mtices fo ottios (i thee dimesios) though gle θ bout oe of the es e Summtios = = 0 0 0 cos θ siθ fo the -is 0 siθ cos θ cos θ 0 siθ 0 0 fo the y-is siθ 0 cos θ cos θ siθ 0 siθ cos θ 0 fo the z-is 0 0 = ( + )( + ) 6 4 = ( + ) 3 Mclui s seies = + + + + +!! ( ) f( ) f( 0) f( 0) f ( 0) f ( 0) e = ep( ) = + + + + + fo ll!! 3 + l( + ) = + + ( ) + ( < ) 3 3 5 + si = + + ( ) + fo ll 3! 5! ( + )! 4 cos = + + ( ) + fo ll! 4! ( )! 7
Vectos The esolved pt of i the diectio of b is.b b i b b 3 b 3 Vecto poduct: b = b si θ ˆ = j b = b 3 b 3 k 3 b 3 b b If A is the poit with positio vecto = i+ j+ 3k, the the stight lie though A with diectio vecto b = bi+ bj+ b3k hs equtio y z 3 = = = λ (Ctesi fom) b b b3 o ( ) b = 0 (vecto poduct fom) the ple though A d pllel to b d c hs vecto equtio = + sb+ tc Ae of secto A= dθ (pol coodites) Hypebolic fuctios Coics sih = sih cosh cosh sih = cosh cosh sih = + { } { } cosh = l + ( ) sih = l + + + th = l ( ) < Stdd fom Pmetic fom Asymptotes Ecceticity Ellipse Pbol Hypebol y y + = y = 4 = b b = cos θ = t = sec θ y = bsiθ y = t y = btθ oe oe y = ± b b b 8
Futhe umeicl itegtio b The mid-odite ule: y d hy ( + y 3 + + y + y 3 ) whee b h = b Simpso s ule: d {( 0 + ) + 4 ( + 3 +... + ) + ( + 4 +... + ) } y h y y y y y y y y 3 Numeicl solutio of diffeetil equtios Fo d y = f( ) d smll h, ecuece eltios e: d Eule s method: y = y + + hf ( ); = + + h dy Fo f(, y) : d = Eule s method: y = y + + hf(, y ) whee b h = d is eve Impoved Eule method: y = y + ( k + k ), whee k = hf, y ), k = + hf( + h, y + k ) Ac legth s= dy + d d (Ctesi coodites) s = d dy + dt dt dt (pmetic fom) Sufce e of evolutio dy S = π y + d d (Ctesi coodites) d d π y S = y + dt (pmetic fom) dt dt Tigoomety: t substitutio Witig t = t θ, si θ = t + t cos θ t t = + ( 9
MECHANICS Costt cceletio s = ut + t s= ut+ t s = vt t s= vt t v = u + t v= u+ t = ( + ) s= ( u+ v ) s u vt v = u + s Cetes of mss Fo uifom bodies Tigul lmi: 3 t log medi fom vete Solid hemisphee, dius : 3 fom cete 8 Hemispheicl shell, dius : fom cete Cicul c, dius, gle t cete α : si α fom cete α Secto of cicle, dius, gle t cete α : si α fom cete 3α Solid coe o pymid of height h: h bove the bse o the lie fom cete of bse to vete 4 Coicl shell of height h: h bove the bse o the lie fom cete of bse to vete 3 PROBABILITY d STATISTICS Pobbility P( A B) = P( A) + P( B) P( A B) P( A B) = P( A) P( B A) Discete distibutios Stdd discete distibutios: Distibutio of X P( X = ) Me Vice Biomil B(, p ) p ( p ) p p( p) Poisso Po( λ ) λ λ e λ λ! 0
Smplig distibutios Fo dom smple X, X,, X of idepedet obsevtios fom distibutio hvig me µ d vice σ : X is ubised estimto of µ, with V( X ) = σ S is ubised estimto of σ, whee S Fo dom smple of obsevtios fom N( µ, σ ) X µ ~ N( 0, ) σ X µ S t ~ Distibutio-fee (o-pmetic) tests = ( Xi X) ( Oi Ei) Cotigecy tbles: is ppoimtely distibuted s χ Ei
TABLE CUMULATIVE POISSON DISTRIBUTION FUNCTION The tbulted vlue is P( X ), whee X hs Poisso distibutio with me λ. λ 0.0 0.0 0.30 0.40 0.50 0.60 0.70 0.80 0.90.0..4.6.8 λ 0 0.9048 0.887 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 0.3679 0.30 0.466 0.09 0.653 0 0.9953 0.985 0.963 0.9384 0.9098 0.878 0.844 0.8088 0.775 0.7358 0.666 0.598 0.549 0.468 0.9998 0.9989 0.9964 0.99 0.9856 0.9769 0.9659 0.956 0.937 0.997 0.8795 0.8335 0.7834 0.7306 3.0000 0.9999 0.9997 0.999 0.998 0.9966 0.994 0.9909 0.9865 0.980 0.966 0.9463 0.9 0.893 3 4.0000.0000 0.9999 0.9998 0.9996 0.999 0.9986 0.9977 0.9963 0.993 0.9857 0.9763 0.9636 4 5.0000.0000.0000 0.9999 0.9998 0.9997 0.9994 0.9985 0.9968 0.9940 0.9896 5 6.0000.0000.0000 0.9999 0.9997 0.9994 0.9987 0.9974 6 7.0000.0000 0.9999 0.9997 0.9994 7 8.0000.0000 0.9999 8 9.0000 9 λ.0..4.6.8 3.0 3. 3.4 3.6 3.8 4.0 4.5 5.0 5.5 λ 0 0.353 0.08 0.0907 0.0743 0.0608 0.0498 0.0408 0.0334 0.073 0.04 0.083 0.0 0.0067 0.004 0 0.4060 0.3546 0.3084 0.674 0.3 0.99 0.7 0.468 0.57 0.074 0.096 0.06 0.0404 0.066 0.6767 0.67 0.5697 0.584 0.4695 0.43 0.3799 0.3397 0.307 0.689 0.38 0.736 0.47 0.0884 3 0.857 0.894 0.7787 0.7360 0.699 0.647 0.605 0.5584 0.55 0.4735 0.4335 0.343 0.650 0.07 3 4 0.9473 0.975 0.904 0.8774 0.8477 0.853 0.7806 0.744 0.7064 0.6678 0.688 0.53 0.4405 0.3575 4 5 0.9834 0.975 0.9643 0.950 0.9349 0.96 0.8946 0.8705 0.844 0.856 0.785 0.709 0.660 0.589 5 6 0.9955 0.995 0.9884 0.988 0.9756 0.9665 0.9554 0.94 0.967 0.909 0.8893 0.83 0.76 0.6860 6 7 0.9989 0.9980 0.9967 0.9947 0.999 0.988 0.983 0.9769 0.969 0.9599 0.9489 0.934 0.8666 0.8095 7 8 0.9998 0.9995 0.999 0.9985 0.9976 0.996 0.9943 0.997 0.9883 0.9840 0.9786 0.9597 0.939 0.8944 8 9.0000 0.9999 0.9998 0.9996 0.9993 0.9989 0.998 0.9973 0.9960 0.994 0.999 0.989 0.968 0.946 9 0.0000.0000 0.9999 0.9998 0.9997 0.9995 0.999 0.9987 0.998 0.997 0.9933 0.9863 0.9747 0.0000.0000 0.9999 0.9999 0.9998 0.9996 0.9994 0.999 0.9976 0.9945 0.9890.0000.0000 0.9999 0.9999 0.9998 0.9997 0.999 0.9980 0.9955 3.0000.0000.0000 0.9999 0.9997 0.9993 0.9983 3 4.0000 0.9999 0.9998 0.9994 4 5.0000 0.9999 0.9998 5 6.0000 0.9999 6 7.0000 7
λ 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 0.0.0.0 3.0 4.0 5.0 λ 0 0.005 0.005 0.0009 0.0006 0.0003 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0.074 0.03 0.0073 0.0047 0.0030 0.009 0.00 0.0008 0.0005 0.000 0.000 0.0000 0.0000 0.0000 0.060 0.0430 0.096 0.003 0.038 0.0093 0.006 0.004 0.008 0.00 0.0005 0.000 0.000 0.0000 3 0.5 0.8 0.088 0.059 0.044 0.030 0.0 0.049 0.003 0.0049 0.003 0.00 0.0005 0.000 3 4 0.85 0.37 0.730 0.3 0.0996 0.0744 0.0550 0.0403 0.093 0.05 0.0076 0.0037 0.008 0.0009 4 5 0.4457 0.3690 0.3007 0.44 0.9 0.496 0.57 0.0885 0.067 0.0375 0.003 0.007 0.0055 0.008 5 6 0.6063 0.565 0.4497 0.378 0.334 0.56 0.068 0.649 0.30 0.0786 0.0458 0.059 0.04 0.0076 6 7 0.7440 0.678 0.5987 0.546 0.4530 0.3856 0.339 0.687 0.0 0.43 0.0895 0.0540 0.036 0.080 7 8 0.847 0.796 0.79 0.660 0.595 0.53 0.4557 0.398 0.338 0.30 0.550 0.0998 0.06 0.0374 8 9 0.96 0.8774 0.8305 0.7764 0.766 0.6530 0.5874 0.58 0.4579 0.3405 0.44 0.658 0.094 0.0699 9 0 0.9574 0.933 0.905 0.86 0.859 0.7634 0.7060 0.6453 0.5830 0.4599 0.347 0.57 0.757 0.85 0 0.9799 0.966 0.9467 0.908 0.888 0.8487 0.8030 0.750 0.6968 0.5793 0.466 0.353 0.600 0.848 0.99 0.9840 0.9730 0.9573 0.936 0.909 0.8758 0.8364 0.796 0.6887 0.5760 0.463 0.3585 0.676 3 0.9964 0.999 0.987 0.9784 0.9658 0.9486 0.96 0.898 0.8645 0.783 0.685 0.5730 0.4644 0.363 3 4 0.9986 0.9970 0.9943 0.9897 0.987 0.976 0.9585 0.9400 0.965 0.8540 0.770 0.675 0.5704 0.4657 4 5 0.9995 0.9988 0.9976 0.9954 0.998 0.986 0.9780 0.9665 0.953 0.9074 0.8444 0.7636 0.6694 0.568 5 6 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.983 0.9730 0.944 0.8987 0.8355 0.7559 0.664 6 7 0.9999 0.9998 0.9996 0.999 0.9984 0.9970 0.9947 0.99 0.9857 0.9678 0.9370 0.8905 0.87 0.7489 7 8.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.9957 0.998 0.983 0.966 0.930 0.886 0.895 8 9.0000.0000 0.9999 0.9997 0.9995 0.9989 0.9980 0.9965 0.9907 0.9787 0.9573 0.935 0.875 9 0.0000 0.9999 0.9998 0.9996 0.999 0.9984 0.9953 0.9884 0.9750 0.95 0.970 0.0000 0.9999 0.9998 0.9996 0.9993 0.9977 0.9939 0.9859 0.97 0.9469.0000 0.9999 0.9999 0.9997 0.9990 0.9970 0.994 0.9833 0.9673 3.0000 0.9999 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 3 4.0000.0000 0.9998 0.9993 0.9980 0.9950 0.9888 4 5 0.9999 0.9997 0.9990 0.9974 0.9938 5 6.0000 0.9999 0.9995 0.9987 0.9967 6 7 0.9999 0.9998 0.9994 0.9983 7 8.0000 0.9999 0.9997 0.999 8 9.0000 0.9999 0.9996 9 30 0.9999 0.9998 30 3.0000 0.9999 3 3.0000 3 3
TABLE PERCENTAGE POINTS OF THE STUDENT'S t-distribution The tble gives the vlues of stisfyig P( X ) = p, whee X is dom vible hvig the studet's t-distibutio with v degees of feedom. 0 p 0.9 0.95 0.975 0.99 0.995 p 0.9 0.95 0.975 0.99 0.995 v v 3.078 6.34.706 3.8 63.657 9.3.699.045.46.756.886.90 4.303 6.965 9.95 30.30.697.04.457.750 3.638.353 3.8 4.54 5.84 3.309.696.040.453.744 4.533.3.776 3.747 4.604 3.309.694.037.449.738 5.476.05.57 3.365 4.03 33.308.69.035.445.733 6.440.943.447 3.43 3.707 34.307.69.03.44.78 7.45.895.365.998 3.499 35.306.690.030.438.74 8.397.860.306.896 3.355 36.306.688.08.434.79 9.383.833.6.8 3.50 37.305.687.06.43.75 0.37.8.8.764 3.69 38.304.686.04.49.7.363.796.0.78 3.06 39.304.685.03.46.708.356.78.79.68 3.055 40.303.684.0.43.704 3.350.77.60.650 3.0 45.30.679.04.4.690 4.345.76.45.64.977 50.99.676.009.403.678 5.34.753.3.60.947 55.97.673.004.396.668 6.337.746..583.9 60.96.67.000.390.660 7.333.740.0.567.898 65.95.669.997.385.654 8.330.734.0.55.878 70.94.667.994.38.648 9.38.79.093.539.86 75.93.665.99.377.643 0.35.75.086.58.845 80.9.664.990.374.639.33.7.080.58.83 85.9.663.998.37.635.3.77.074.508.89 90.9.66.987.368.63 3.39.74.069.500.807 95.9.66.985.366.69 4.38.7.064.49.797 00.90.660.984.364.66 5.36.708.060.485.787 5.88.657.979.357.66 6.35.706.056.479.779 50.87.655.976.35.609 7.34.703.05.473.77 00.86.653.97.345.60 8.33.70.048.467.763.8.645.960.36.576 4
TABLE 6 PERCENTAGE POINTS OF THE χ DISTRIBUTION The tble gives the vlues of stisfyig P( X ) = p, whee X is dom vible hvig the χ distibutio with v degees of feedom. p 0.005 0.0 0.05 0.05 0. 0.9 0.95 0.975 0.99 0.995 p v v 0.00004 0.000 0.00 0.004 0.06.706 3.84 5.04 6.635 7.879 0.00 0.00 0.05 0.03 0. 4.605 5.99 7.378 9.0 0.597 3 0.07 0.5 0.6 0.35 0.584 6.5 7.85 9.348.345.838 3 4 0.07 0.97 0.484 0.7.064 7.779 9.488.43 3.77 4.860 4 5 0.4 0.554 0.83.45.60 9.36.070.833 5.086 6.750 5 6 0.676 0.87.37.635.04 0.645.59 4.449 6.8 8.548 6 7 0.989.39.690.67.833.07 4.067 6.03 8.475 0.78 7 8.344.646.80.733 3.490 3.36 5.507 7.535 0.090.955 8 9.735.088.700 3.35 4.68 4.684 6.99 9.03.666 3.589 9 0.56.558 3.47 3.940 4.865 5.987 8.307 0.483 3.09 5.88 0.603 3.053 3.86 4.575 5.578 7.75 9.675.90 4.75 6.757 3.074 3.57 4.404 5.6 6.304 8.549.06 3.337 6.7 8.300 3 3.565 4.07 5.009 5.89 7.04 9.8.36 4.736 7.688 9.89 3 4 4.075 4.660 5.69 6.57 7.790.064 3.685 6.9 9.4 3.39 4 5 4.60 5.9 6.6 7.6 8.547.307 4.996 7.488 30.578 3.80 5 6 5.4 5.8 6.908 7.96 9.3 3.54 6.96 8.845 3.000 34.67 6 7 5.697 6.408 7.564 8.67 0.085 4.769 7.587 30.9 33.409 35.78 7 8 6.65 7.05 8.3 9.390 0.865 5.989 8.869 3.56 34.805 37.56 8 9 6.844 7.633 8.907 0.7.65 7.04 30.44 3.85 36.9 38.58 9 0 7.434 8.60 9.59 0.85.443 8.4 3.40 34.70 37.566 39.997 0 8.034 8.897 0.83.59 3.40 9.65 3.67 35.479 38.93 4.40 8.643 9.54 0.98.338 4.04 30.83 33.94 36.78 40.89 4.796 3 9.60 0.96.689 3.09 4.848 3.007 35.7 38.076 4.638 44.8 3 4 9.886 0.856.40 3.848 5.659 33.96 36.45 39.364 4.980 45.559 4 5 0.50.54 3.0 4.6 6.473 34.38 37.65 40.646 44.34 46.98 5 6.60.98 3.844 5.379 7.9 35.563 38.885 4.93 45.64 48.90 6 7.808.879 4.573 6.5 8.4 36.74 40.3 43.95 46.963 49.645 7 8.46 3.565 5.308 6.98 8.939 37.96 4.337 44.46 48.78 50.993 8 9 3. 4.56 6.047 7.708 9.768 39.087 4.557 45.7 49.588 5.336 9 30 3.787 4.953 6.79 8.493 0.599 40.56 43.773 46.979 50.89 53.67 30 3 4.458 5.655 7.539 9.8.434 4.4 44.985 48.3 5.9 55.003 3 3 5.34 6.36 8.9 0.07.7 4.585 46.94 49.480 53.486 56.38 3 33 5.85 7.074 9.047 0.867 3.0 43.745 47.400 50.75 54.776 57.648 33 34 6.50 7.789 9.806.664 3.95 44.903 48.60 5.996 56.06 58.964 34 35 7.9 8.509 0.569.465 4.797 46.059 49.80 53.03 57.34 60.75 35 36 7.887 9.3.336 3.69 5.643 47. 50.998 54.437 58.69 6.58 36 37 8.586 9.960.06 4.075 6.49 48.363 5.9 55.668 59.89 6.883 37 38 9.89 0.69.878 4.884 7.343 49.53 53.384 56.896 6.6 64.8 38 39 9.996.46 3.654 5.695 8.96 50.660 54.57 58.0 6.48 65.476 39 40 0.707.64 4.433 6.509 9.05 5.805 55.758 59.34 63.69 66.766 40 45 4.3 5.90 8.366 30.6 33.350 57.505 6.656 65.40 69.957 73.66 45 50 7.99 9.707 3.357 34.764 37.689 63.67 67.505 7.40 76.54 79.490 50 55 3.735 33.570 36.398 38.958 4.060 68.796 73.3 77.380 8.9 85.749 55 60 35.534 37.485 40.48 43.88 46.459 74.397 79.08 83.98 88.379 9.95 60 65 39.383 4.444 44.603 47.450 50.883 79.973 84.8 89.77 94.4 98.05 65 70 43.75 45.44 48.758 5.739 55.39 85.57 90.53 95.03 00.45 04.5 70 75 47.06 49.475 5.94 56.054 59.795 9.06 96.7 00.839 06.393 0.86 75 80 5.7 53.540 57.53 60.39 64.78 96.578 0.879 06.69.39 6.3 80 85 55.70 57.634 6.389 64.749 68.777 0.079 07.5.393 8.36.35 85 90 59.96 6.754 65.647 69.6 73.9 07.565 3.45 8.36 4.6 8.99 90 95 63.50 65.898 69.95 73.50 77.88 3.038 8.75 3.858 9.973 34.47 95 00 67.38 70.065 74. 77.99 8.358 8.498 4.34 9.56 35.807 40.69 00 0 5
TABLE 3 CRITICAL VALUES OF THE PRODUCT MOMENT CORRELATION COEFFICIENT The tble gives the citicl vlues, fo diffeet sigificce levels, of the poduct momet coeltio coefficiet,, fo vyig smple sizes,. Oe til 0% 5%.5% % 0.5% Oe til Two til 0% 0% 5% % % Two til 4 0.8000 0.9000 0.9500 0.9800 0.9900 4 5 0.6870 0.8054 0.8783 0.9343 0.9587 5 6 0.6084 0.793 0.84 0.88 0.97 6 7 0.5509 0.6694 0.7545 0.839 0.8745 7 8 0.5067 0.65 0.7067 0.7887 0.8343 8 9 0.476 0.58 0.6664 0.7498 0.7977 9 0 0.448 0.5494 0.639 0.755 0.7646 0 0.487 0.54 0.60 0.685 0.7348 0.398 0.4973 0.5760 0.658 0.7079 3 0.380 0.476 0.559 0.6339 0.6835 3 4 0.3646 0.4575 0.534 0.60 0.664 4 5 0.3507 0.4409 0.540 0.593 0.64 5 6 0.3383 0.459 0.4973 0.574 0.66 6 7 0.37 0.44 0.48 0.5577 0.6055 7 8 0.370 0.4000 0.4683 0.545 0.5897 8 9 0.3077 0.3887 0.4555 0.585 0.575 9 0 0.99 0.3783 0.4438 0.555 0.564 0 0.94 0.3687 0.439 0.5034 0.5487 0.84 0.3598 0.47 0.49 0.5368 3 0.774 0.355 0.43 0.485 0.556 3 4 0.7 0.3438 0.4044 0.476 0.55 4 5 0.653 0.3365 0.396 0.46 0.505 5 6 0.598 0.397 0.388 0.4534 0.4958 6 7 0.546 0.333 0.3809 0.445 0.4869 7 8 0.497 0.37 0.3739 0.437 0.4785 8 9 0.45 0.35 0.3673 0.497 0.4705 9 30 0.407 0.306 0.360 0.46 0.469 30 3 0.366 0.3009 0.3550 0.458 0.4556 3 3 0.37 0.960 0.3494 0.4093 0.4487 3 33 0.89 0.93 0.3440 0.403 0.44 33 34 0.54 0.869 0.3388 0.397 0.4357 34 35 0.0 0.86 0.3338 0.396 0.496 35 36 0.87 0.785 0.39 0.386 0.438 36 37 0.56 0.746 0.346 0.380 0.48 37 38 0.6 0.709 0.30 0.3760 0.48 38 39 0.097 0.673 0.360 0.37 0.4076 39 40 0.070 0.638 0.30 0.3665 0.406 40 4 0.043 0.605 0.308 0.36 0.3978 4 4 0.08 0.573 0.3044 0.3578 0.393 4 43 0.993 0.54 0.3008 0.3536 0.3887 43 44 0.970 0.5 0.973 0.3496 0.3843 44 45 0.947 0.483 0.940 0.3457 0.380 45 46 0.95 0.455 0.907 0.340 0.376 46 47 0.903 0.49 0.876 0.3384 0.37 47 48 0.883 0.403 0.845 0.3348 0.3683 48 49 0.863 0.377 0.86 0.334 0.3646 49 50 0.843 0.353 0.787 0.38 0.360 50 60 0.678 0.44 0.54 0.997 0.330 60 70 0.550 0.98 0.35 0.776 0.3060 70 80 0.448 0.85 0.99 0.597 0.864 80 90 0.364 0.745 0.07 0.449 0.70 90 00 0.9 0.654 0.966 0.34 0.565 00 6