Printable algebra formula sheet, Cheat Sheet of Algebra

Download Printable algebra formula sheet and more Cheat Sheet Algebra in PDF only on Docsity! List MF19 List of formulae and statistical tables Cambridge International AS & A Level Mathematics (9709) and Further Mathematics (9231) For use from 2020 in all papers for the above syllabuses. CST319 *2508709701* 2 PURE MATHEMATICS Mensuration Volume of sphere = 34 3 rπ Surface area of sphere = 24 rπ Volume of cone or pyramid = 1 3 base area height× × Area of curved surface of cone = slant heightrπ × Arc length of circle rθ= (θ in radians) Area of sector of circle 21 2 r θ= (θ in radians) Algebra For the quadratic equation 2 0ax bx c+ + = : 2 4 2 b b acx a − ± − = For an arithmetic series: ( 1)nu a n d= + − , 1 1 2 2( ) {2 ( 1) }nS n a l n a n d= + = + − For a geometric series: 1n nu ar −= , (1 ) ( 1) 1 n n a rS r r − = ≠ − , ( ) 1 1 aS r r∞ = < − Binomial series: 1 2 2 3 3( ) 1 2 3 n n n n n nn n n a b a a b a b a b b− − −      + = + + + + +            K , where n is a positive integer and ! !( )! n n r r n r   =  −  2 3( 1) ( 1)( 2)(1 ) 1 2! 3! n n n n n nx nx x x− − − + = + + + +K , where n is rational and 1x < 5 FURTHER PURE MATHEMATICS Algebra Summations: 1 2 1 ( 1) n r r n n = = +∑ , 2 1 6 1 ( 1)(2 1) n r r n n n = = + +∑ , 3 2 21 4 1 ( 1) n r r n n = = +∑ Maclaurin’s series: 2 ( )f( ) f(0) f (0) f (0) f (0) 2! ! r rx xx x r ′ ′′= + + + + +K K 2 e exp( ) 1 2! ! r x x xx x r = = + + + + +K K (all x) 2 3 1ln(1 ) ( 1) 2 3 r rx x xx x r ++ = − + − + − +K K (–1 < x ⩽ 1) 3 5 2 1 sin ( 1) 3! 5! (2 1)! r rx x xx x r + = − + − + − + + K K (all x) 2 4 2 cos 1 ( 1) 2! 4! (2 )! r rx x xx r = − + − + − +K K (all x) 3 5 2 1 1tan ( 1) 3 5 2 1 r rx x xx x r + − = − + − + − + + K K (–1 ⩽ x ⩽ 1) 3 5 2 1 sinh 3! 5! (2 1)! rx x xx x r + = + + + + + + K K (all x) 2 4 2 cosh 1 2! 4! (2 )! rx x xx r = + + + + +K K (all x) 3 5 2 1 1tanh 3 5 2 1 rx x xx x r + − = + + + + + + K K (–1 < x < 1) Trigonometry If 1 2tant x= then: 2 2sin 1 tx t = + and 2 2 1cos 1 tx t − = + Hyperbolic functions 2 2cosh sinh 1x x− ≡ , sinh 2 2sinh coshx x x≡ , 2 2cosh 2 cosh sinhx x x≡ + 1 2sinh ln 1( )x x x− = + + 1 2cosh ln 1( )x x x− = + − (x ⩾ 1) 1 1 2 1tanh ln (| | 1) 1 xx x x − + = < −  6 Differentiation f( )x ′f ( )x 1sin x− 2 1 1 x− 1cos x− 2 1 1 x − − sinh x cosh x cosh x sinh x tanh x 2sech x 1sinh x− 2 1 1 x+ 1cosh x− 2 1 1x − 1tanh x− 2 1 1 x− Integration (Arbitrary constants are omitted; a denotes a positive constant.) f( )x ∫ f( ) dx x sec x 1 1 2 4ln | sec tan | ln | tan( ) |x x x+ = + π ( ) 1 2x < π cosec x 1 2ln | cosec cot | ln | tan |( )x x x− + = (0 )x< < π sinh x cosh x cosh x sinh x 2sech x tanh x 2 2 1 a x− 1sin x a −       ( ) x a< 2 2 1 x a− 1cosh x a −       ( )x a> 2 2 1 a x+ 1sinh x a −       7 MECHANICS Uniformly accelerated motion v u at= + , 1 2 ( )s u v t= + , 21 2s ut at= + , 2 2 2v u as= + FURTHER MECHANICS Motion of a projectile Equation of trajectory is: 2 2 2tan 2 cos gxy x V θ θ = − Elastic strings and springs xT l λ = , 2 2 xE l λ = Motion in a circle For uniform circular motion, the acceleration is directed towards the centre and has magnitude 2rω or 2v r Centres of mass of uniform bodies Triangular lamina: 2 3 along median from vertex Solid hemisphere of radius r: 3 8 r from centre Hemispherical shell of radius r: 1 2 r from centre Circular arc of radius r and angle 2α: sinr α α from centre Circular sector of radius r and angle 2α: 2 sin 3 r α α from centre Solid cone or pyramid of height h: 3 4 h from vertex 10 THE NORMAL DISTRIBUTION FUNCTION If Z has a normal distribution with mean 0 and variance 1, then, for each value of z, the table gives the value of Φ(z), where Φ(z) = P(Z ⩽ z). For negative values of z, use Φ(–z) = 1 – Φ(z). z 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 ADD 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 4 8 12 16 20 24 28 32 36 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 4 8 12 16 20 24 28 32 36 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 4 8 12 15 19 23 27 31 35 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 4 7 11 15 19 22 26 30 34 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 4 7 11 14 18 22 25 29 32 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 3 7 10 14 17 20 24 27 31 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 3 7 10 13 16 19 23 26 29 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 3 6 9 12 15 18 21 24 27 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 3 5 8 11 14 16 19 22 25 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 3 5 8 10 13 15 18 20 23 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 2 5 7 9 12 14 16 19 21 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 2 4 6 8 10 12 14 16 18 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 2 4 6 7 9 11 13 15 17 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 2 3 5 6 8 10 11 13 14 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1 3 4 6 7 8 10 11 13 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1 2 4 5 6 7 8 10 11 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1 2 3 4 5 6 7 8 9 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1 2 3 4 4 5 6 7 8 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1 1 2 3 4 4 5 6 6 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 1 1 2 2 3 4 4 5 5 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 0 1 1 2 2 3 3 4 4 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 0 1 1 2 2 2 3 3 4 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 0 1 1 1 2 2 2 3 3 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 0 1 1 1 1 2 2 2 2 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 0 0 1 1 1 1 1 2 2 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 0 0 0 1 1 1 1 1 1 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 0 0 0 0 1 1 1 1 1 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 0 0 0 0 0 1 1 1 1 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 0 0 0 0 0 0 0 1 1 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0 0 0 0 0 0 0 0 0 Critical values for the normal distribution If Z has a normal distribution with mean 0 and variance 1, then, for each value of p, the table gives the value of z such that P(Z ⩽ z) = p. p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 z 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291 11 CRITICAL VALUES FOR THE t-DISTRIBUTION If T has a t-distribution with ν degrees of freedom, then, for each pair of values of p and ν, the table gives the value of t such that: P(T ⩽ t) = p. p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 ν = 1 1.000 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.6 2 0.816 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.60 3 0.765 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.92 4 0.741 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610 5 0.727 1.476 2.015 2.571 3.365 4.032 4.773 5.894 6.869 6 0.718 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959 7 0.711 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408 8 0.706 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041 9 0.703 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781 10 0.700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587 11 0.697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437 12 0.695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318 13 0.694 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221 14 0.692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140 15 0.691 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073 16 0.690 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015 17 0.689 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965 18 0.688 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922 19 0.688 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883 20 0.687 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850 21 0.686 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819 22 0.686 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792 23 0.685 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.768 24 0.685 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745 25 0.684 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725 26 0.684 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707 27 0.684 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.689 28 0.683 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674 29 0.683 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.660 30 0.683 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646 40 0.681 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.551 60 0.679 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460 120 0.677 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3.373 ∞ 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291 12 CRITICAL VALUES FOR THE 2χ -DISTRIBUTION If X has a 2χ -distribution with ν degrees of freedom then, for each pair of values of p and ν, the table gives the value of x such that P(X ⩽ x) = p. p 0.01 0.025 0.05 0.9 0.95 0.975 0.99 0.995 0.999 ν = 1 0.031571 0.039821 0.023932 2.706 3.841 5.024 6.635 7.879 10.83 2 0.02010 0.05064 0.1026 4.605 5.991 7.378 9.210 10.60 13.82 3 0.1148 0.2158 0.3518 6.251 7.815 9.348 11.34 12.84 16.27 4 0.2971 0.4844 0.7107 7.779 9.488 11.14 13.28 14.86 18.47 5 0.5543 0.8312 1.145 9.236 11.07 12.83 15.09 16.75 20.51 6 0.8721 1.237 1.635 10.64 12.59 14.45 16.81 18.55 22.46 7 1.239 1.690 2.167 12.02 14.07 16.01 18.48 20.28 24.32 8 1.647 2.180 2.733 13.36 15.51 17.53 20.09 21.95 26.12 9 2.088 2.700 3.325 14.68 16.92 19.02 21.67 23.59 27.88 10 2.558 3.247 3.940 15.99 18.31 20.48 23.21 25.19 29.59 11 3.053 3.816 4.575 17.28 19.68 21.92 24.73 26.76 31.26 12 3.571 4.404 5.226 18.55 21.03 23.34 26.22 28.30 32.91 13 4.107 5.009 5.892 19.81 22.36 24.74 27.69 29.82 34.53 14 4.660 5.629 6.571 21.06 23.68 26.12 29.14 31.32 36.12 15 5.229 6.262 7.261 22.31 25.00 27.49 30.58 32.80 37.70 16 5.812 6.908 7.962 23.54 26.30 28.85 32.00 34.27 39.25 17 6.408 7.564 8.672 24.77 27.59 30.19 33.41 35.72 40.79 18 7.015 8.231 9.390 25.99 28.87 31.53 34.81 37.16 42.31 19 7.633 8.907 10.12 27.20 30.14 32.85 36.19 38.58 43.82 20 8.260 9.591 10.85 28.41 31.41 34.17 37.57 40.00 45.31 21 8.897 10.28 11.59 29.62 32.67 35.48 38.93 41.40 46.80 22 9.542 10.98 12.34 30.81 33.92 36.78 40.29 42.80 48.27 23 10.20 11.69 13.09 32.01 35.17 38.08 41.64 44.18 49.73 24 10.86 12.40 13.85 33.20 36.42 39.36 42.98 45.56 51.18 25 11.52 13.12 14.61 34.38 37.65 40.65 44.31 46.93 52.62 30 14.95 16.79 18.49 40.26 43.77 46.98 50.89 53.67 59.70 40 22.16 24.43 26.51 51.81 55.76 59.34 63.69 66.77 73.40 50 29.71 32.36 34.76 63.17 67.50 71.42 76.15 79.49 86.66 60 37.48 40.48 43.19 74.40 79.08 83.30 88.38 91.95 99.61 70 45.44 48.76 51.74 85.53 90.53 95.02 100.4 104.2 112.3 80 53.54 57.15 60.39 96.58 101.9 106.6 112.3 116.3 124.8 90 61.75 65.65 69.13 107.6 113.1 118.1 124.1 128.3 137.2 100 70.06 74.22 77.93 118.5 124.3 129.6 135.8 140.2 149.4 15 BLANK PAGE 16 BLANK PAGE