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* PURE MATHEMATICS Mensuration Volume of sphere = 4 3 πr 3 Surface area of sphere = 4πr 2 Volume of cone or pyramid = 13 × base area × height Area of curved surface of cone = πr × slant height Arc length of circle = rθ ( θ in radians) Area of sector of circle = 12 r 2θ ( θ in radians) Algebra For the quadratic equation ax 2 + bx + c = 0 : x= −b ± b 2 − 4ac 2a For an arithmetic series: un = a + (n − 1)d , S n = 12 n( a + l ) = 12 n{2a + (n − 1) d } For a geometric series: un = ar n −1 , Sn = a(1 − r n ) 1− r (r ≠ 1) , S∞ = a 1− r ( r <1 ) Binomial series: n n n (a + b) n = a n + a n −1b + a n − 2b 2 + a n −3b3 + K + b n , where n is a positive integer 1 2 3 n n! and = r r!(n − r )! (1 + x) n = 1 + nx + n(n − 1) 2 n(n − 1)(n − 2) 3 x + x + K , where n is rational and x < 1 2! 3! 2 Trigonometry tan θ ≡ cos 2 θ + sin 2 θ ≡ 1 , sin θ cos θ 1 + tan 2 θ ≡ sec 2 θ , cot 2 θ + 1 ≡ cosec 2 θ sin( A ± B) ≡ sin A cos B ± cos A sin B cos( A ± B) ≡ cos A cos B m sin A sin B tan( A ± B ) ≡ tan A ± tan B 1 m tan A tan B sin 2 A ≡ 2sin A cos A cos 2 A ≡ cos 2 A − sin 2 A ≡ 2cos 2 A − 1 ≡ 1 − 2sin 2 A tan 2 A ≡ 2 tan A 1 − tan 2 A Principal values: − 12 π ⩽ sin −1 x ⩽ 12 π , 0 ⩽ cos −1 x ⩽ π , Differentiation f( x ) f ′( x ) xn nx n −1 ln x 1 x ex ex sin x cos x cos x − sin x tan x sec 2 x sec x sec x tan x cosec x − cosec x cot x cot x − cosec 2 x tan −1 x 1 1 + x2 uv v v u v If x = f(t ) and y = g(t ) then − 12 π < tan −1 x < 12 π dy dy dx = ÷ dx dt dt 3 du dv +u dx dx du dv −u dx dx v2 Integration (Arbitrary constants are omitted; a denotes a positive constant.) f( x ) ∫ f( x ) dx xn x n +1 n +1 1 x ln x ex ex sin x − cos x cos x sin x sec 2 x tan x 1 x + a2 1 2 x − a2 1 x tan −1 a a 1 x−a ln 2a x + a 1 a − x2 1 a+x ln 2a a − x 2 2 dv du ∫ u dx dx = uv −∫ v dx dx f ′( x) ∫ f ( x) dx = ln f ( x) Vectors If a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then a.b = a1b1 + a2b2 + a3b3 = a b cos θ 4 (n ≠ −1) ( x > a) ( x < a) FURTHER PURE MATHEMATICS Algebra Summations: n ∑ r =1 n ∑ r = 12 n(n + 1) , r =1 n ∑r r 2 = 16 n(n + 1)(2n + 1) , 3 r =1 = 14 n 2 (n + 1) 2 Maclaurin’s series: f( x) = f(0) + x f ′(0) + x2 xr f ′′(0) + K + f ( r ) (0) + K r! 2! e x = exp( x) = 1 + x + (all x) x 2 x3 xr + − K + (−1) r +1 + K 2 3 r (–1 < x ⩽ 1) x3 x5 x 2 r +1 + − K + (−1) r +K 3! 5! (2r + 1)! (all x) x2 x4 x2r + − K + (−1) r +K 2! 4! (2r )! (all x) ln(1 + x) = x − sin x = x − x2 xr +K + +K r! 2! cos x = 1 − tan −1 x = x − x3 x5 x 2 r +1 + − K + (−1) r +K 3 5 2r + 1 sinh x = x + (–1 ⩽ x ⩽ 1) x3 x5 x 2 r +1 + +K + +K 3! 5! (2r + 1)! (all x) x2 x4 x2r + +K+ +K 2! 4! (2r )! (all x) cosh x = 1 + tanh −1 x = x + x3 x5 x 2 r +1 + +K + +K 3 5 2r + 1 (–1 < x < 1) Trigonometry If t = tan 12 x then: sin x = Hyperbolic functions cosh 2 x − sinh 2 x ≡ 1 , 2t 1+ t2 cos x = and 1− t2 1+ t2 cosh 2 x ≡ cosh 2 x + sinh 2 x sinh 2 x ≡ 2sinh x cosh x , sinh −1 x = ln( x + x 2 + 1) cosh −1 x = ln( x + x 2 − 1) 1+ x tanh −1 x = 12 ln 1− x 5 (x ⩾ 1) (| x | < 1) Differentiation f( x ) f ′( x ) sin −1 x 1 1 − x2 cos −1 x − 1 1 − x2 sinh x cosh x cosh x sinh x tanh x sech 2 x sinh −1 x 1 1 + x2 1 cosh −1 x x2 − 1 1 1 − x2 tanh −1 x Integration (Arbitrary constants are omitted; a denotes a positive constant.) f( x ) ∫ f( x ) dx sec x ln| sec x + tan x | = ln| tan( 12 x + 14 π) | ( cosec x − ln| cosec x + cot x | = ln| tan( 12 x) | (0 < x < π) sinh x cosh x cosh x sinh x sech 2 x tanh x 1 x sin −1 a 2 a −x 2 x cosh −1 a 1 2 x −a 2 x sinh −1 a 1 2 a +x 2 6 x < 12 π ) ( x < a) ( x > a) MECHANICS Uniformly accelerated motion v = u + at , v 2 = u 2 + 2as s = ut + 12 at 2 , s = 12 (u + v)t , FURTHER MECHANICS Motion of a projectile Equation of trajectory is: y = x tan θ − gx 2 2V 2 cos 2 θ Elastic strings and springs T= λx l E= , λ x2 2l Motion in a circle For uniform circular motion, the acceleration is directed towards the centre and has magnitude ω 2r v2 r or Centres of mass of uniform bodies Triangular lamina: 23 along median from vertex Solid hemisphere of radius r: 83 r from centre Hemispherical shell of radius r: 1 2 r from centre Circular arc of radius r and angle 2α: r sin α Circular sector of radius r and angle 2α: Solid cone or pyramid of height h: 3 4 α from centre 2r sin α from centre 3α h from vertex 7 PROBABILITY & STATISTICS Summary statistics For ungrouped data: x= Σx , n x= Σxf , Σf Σ( x − x ) 2 Σx 2 = − x2 n n standard deviation = For grouped data: Σ( x − x ) 2 f Σx 2 f = − x2 Σf Σf standard deviation = Discrete random variables Var( X ) = Σx 2 p − {E( X )}2 E( X ) = Σxp , For the binomial distribution B(n, p) : n pr = p r (1 − p) n − r , r µ = np , σ 2 = np(1 − p ) For the geometric distribution Geo(p): pr = p(1 − p) r −1 , µ= 1 p For the Poisson distribution Po(λ ) pr = e − λ λr r! , Continuous random variables E( X ) = x f( x) dx , ∫ σ2 =λ µ =λ , ∫ Var( X ) = x 2 f( x) dx − {E( X )}2 Sampling and testing Unbiased estimators: x= Σx , n s2 = Σ( x − x ) 2 1 2 ( Σx ) 2 = Σx − n −1 n −1 n Central Limit Theorem: σ2 X ~ N µ, n Approximate distribution of sample proportion: p (1 − p) N p, n 8 FURTHER PROBABILITY & STATISTICS Sampling and testing Two-sample estimate of a common variance: s2 = Probability generating functions G X (t ) = E(t X ) , Σ( x1 − x1 ) 2 + Σ( x2 − x2 ) 2 n1 + n 2 − 2 E( X ) = G ′X (1) , 9 Var( X ) = G ′′X (1) + G ′X (1) − {G ′X (1)}2 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). 1 2 3 4 6 7 8 9 20 20 19 19 18 24 24 23 22 22 28 28 27 26 25 32 32 31 30 29 36 36 35 34 32 14 13 12 11 10 17 16 15 14 13 20 19 18 16 15 24 23 21 19 18 27 26 24 22 20 31 29 27 25 23 7 6 6 5 4 9 8 7 6 6 12 10 9 8 7 14 12 11 10 8 16 14 13 11 10 19 16 15 13 11 21 18 17 14 13 2 2 2 1 1 4 3 3 2 2 5 4 4 3 2 6 5 4 4 3 7 6 5 4 4 8 7 6 5 4 10 11 8 9 7 8 6 6 5 5 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 2 2 1 1 1 2 2 2 1 1 3 2 2 2 1 3 3 2 2 1 4 3 3 2 2 4 4 3 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 z 0 1 2 3 4 5 6 7 8 9 0.0 0.1 0.2 0.3 0.4 0.5000 0.5398 0.5793 0.6179 0.6554 0.5040 0.5438 0.5832 0.6217 0.6591 0.5080 0.5478 0.5871 0.6255 0.6628 0.5120 0.5517 0.5910 0.6293 0.6664 0.5160 0.5557 0.5948 0.6331 0.6700 0.5199 0.5596 0.5987 0.6368 0.6736 0.5239 0.5636 0.6026 0.6406 0.6772 0.5279 0.5675 0.6064 0.6443 0.6808 0.5319 0.5714 0.6103 0.6480 0.6844 0.5359 0.5753 0.6141 0.6517 0.6879 4 4 4 4 4 8 8 8 7 7 12 12 12 11 11 16 16 15 15 14 0.5 0.6 0.7 0.8 0.9 0.6915 0.7257 0.7580 0.7881 0.8159 0.6950 0.7291 0.7611 0.7910 0.8186 0.6985 0.7324 0.7642 0.7939 0.8212 0.7019 0.7357 0.7673 0.7967 0.8238 0.7054 0.7389 0.7704 0.7995 0.8264 0.7088 0.7422 0.7734 0.8023 0.8289 0.7123 0.7454 0.7764 0.8051 0.8315 0.7157 0.7486 0.7794 0.8078 0.8340 0.7190 0.7517 0.7823 0.8106 0.8365 0.7224 0.7549 0.7852 0.8133 0.8389 3 3 3 3 3 7 7 6 5 5 10 10 9 8 8 1.0 1.1 1.2 1.3 1.4 0.8413 0.8643 0.8849 0.9032 0.9192 0.8438 0.8665 0.8869 0.9049 0.9207 0.8461 0.8686 0.8888 0.9066 0.9222 0.8485 0.8708 0.8907 0.9082 0.9236 0.8508 0.8729 0.8925 0.9099 0.9251 0.8531 0.8749 0.8944 0.9115 0.9265 0.8554 0.8770 0.8962 0.9131 0.9279 0.8577 0.8790 0.8980 0.9147 0.9292 0.8599 0.8810 0.8997 0.9162 0.9306 0.8621 0.8830 0.9015 0.9177 0.9319 2 2 2 2 1 5 4 4 3 3 1.5 1.6 1.7 1.8 1.9 0.9332 0.9452 0.9554 0.9641 0.9713 0.9345 0.9463 0.9564 0.9649 0.9719 0.9357 0.9474 0.9573 0.9656 0.9726 0.9370 0.9484 0.9582 0.9664 0.9732 0.9382 0.9495 0.9591 0.9671 0.9738 0.9394 0.9505 0.9599 0.9678 0.9744 0.9406 0.9515 0.9608 0.9686 0.9750 0.9418 0.9525 0.9616 0.9693 0.9756 0.9429 0.9535 0.9625 0.9699 0.9761 0.9441 0.9545 0.9633 0.9706 0.9767 1 1 1 1 1 2.0 2.1 2.2 2.3 2.4 0.9772 0.9821 0.9861 0.9893 0.9918 0.9778 0.9826 0.9864 0.9896 0.9920 0.9783 0.9830 0.9868 0.9898 0.9922 0.9788 0.9834 0.9871 0.9901 0.9925 0.9793 0.9838 0.9875 0.9904 0.9927 0.9798 0.9842 0.9878 0.9906 0.9929 0.9803 0.9846 0.9881 0.9909 0.9931 0.9808 0.9850 0.9884 0.9911 0.9932 0.9812 0.9854 0.9887 0.9913 0.9934 0.9817 0.9857 0.9890 0.9916 0.9936 2.5 2.6 2.7 2.8 2.9 0.9938 0.9953 0.9965 0.9974 0.9981 0.9940 0.9955 0.9966 0.9975 0.9982 0.9941 0.9956 0.9967 0.9976 0.9982 0.9943 0.9957 0.9968 0.9977 0.9983 0.9945 0.9959 0.9969 0.9977 0.9984 0.9946 0.9960 0.9970 0.9978 0.9984 0.9948 0.9961 0.9971 0.9979 0.9985 0.9949 0.9962 0.9972 0.9979 0.9985 0.9951 0.9963 0.9973 0.9980 0.9986 0.9952 0.9964 0.9974 0.9981 0.9986 5 ADD 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 10 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 ν=1 1.000 3.078 6.314 2 3 4 0.816 0.765 0.741 1.886 1.638 1.533 2.920 2.353 2.132 5 6 7 8 9 0.727 0.718 0.711 0.706 0.703 1.476 1.440 1.415 1.397 1.383 2.015 1.943 1.895 1.860 1.833 10 11 12 13 14 0.700 0.697 0.695 0.694 0.692 1.372 1.363 1.356 1.350 1.345 15 16 17 18 19 0.691 0.690 0.689 0.688 0.688 20 21 22 23 24 0.975 0.99 0.995 0.9975 0.999 0.9995 12.71 31.82 63.66 127.3 318.3 636.6 4.303 3.182 2.776 6.965 4.541 3.747 9.925 5.841 4.604 14.09 7.453 5.598 22.33 10.21 7.173 31.60 12.92 8.610 2.571 2.447 2.365 2.306 2.262 3.365 3.143 2.998 2.896 2.821 4.032 3.707 3.499 3.355 3.250 4.773 4.317 4.029 3.833 3.690 5.894 5.208 4.785 4.501 4.297 6.869 5.959 5.408 5.041 4.781 1.812 1.796 1.782 1.771 1.761 2.228 2.201 2.179 2.160 2.145 2.764 2.718 2.681 2.650 2.624 3.169 3.106 3.055 3.012 2.977 3.581 3.497 3.428 3.372 3.326 4.144 4.025 3.930 3.852 3.787 4.587 4.437 4.318 4.221 4.140 1.341 1.337 1.333 1.330 1.328 1.753 1.746 1.740 1.734 1.729 2.131 2.120 2.110 2.101 2.093 2.602 2.583 2.567 2.552 2.539 2.947 2.921 2.898 2.878 2.861 3.286 3.252 3.222 3.197 3.174 3.733 3.686 3.646 3.610 3.579 4.073 4.015 3.965 3.922 3.883 0.687 0.686 0.686 0.685 0.685 1.325 1.323 1.321 1.319 1.318 1.725 1.721 1.717 1.714 1.711 2.086 2.080 2.074 2.069 2.064 2.528 2.518 2.508 2.500 2.492 2.845 2.831 2.819 2.807 2.797 3.153 3.135 3.119 3.104 3.091 3.552 3.527 3.505 3.485 3.467 3.850 3.819 3.792 3.768 3.745 25 26 27 28 29 0.684 0.684 0.684 0.683 0.683 1.316 1.315 1.314 1.313 1.311 1.708 1.706 1.703 1.701 1.699 2.060 2.056 2.052 2.048 2.045 2.485 2.479 2.473 2.467 2.462 2.787 2.779 2.771 2.763 2.756 3.078 3.067 3.057 3.047 3.038 3.450 3.435 3.421 3.408 3.396 3.725 3.707 3.689 3.674 3.660 30 40 60 120 0.683 0.681 0.679 0.677 1.310 1.303 1.296 1.289 1.697 1.684 1.671 1.658 2.042 2.021 2.000 1.980 2.457 2.423 2.390 2.358 2.750 2.704 2.660 2.617 3.030 2.971 2.915 2.860 3.385 3.307 3.232 3.160 3.646 3.551 3.460 3.373 ∞ 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291 11 CRITICAL VALUES FOR THE χ 2 -DISTRIBUTION If X has a χ 2 -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 2 3 4 0.031571 0.02010 0.1148 0.2971 0.039821 0.05064 0.2158 0.4844 0.023932 0.1026 0.3518 0.7107 2.706 4.605 6.251 7.779 3.841 5.991 7.815 9.488 5.024 7.378 9.348 11.14 6.635 9.210 11.34 13.28 7.879 10.60 12.84 14.86 10.83 13.82 16.27 18.47 5 6 7 8 9 0.5543 0.8721 1.239 1.647 2.088 0.8312 1.237 1.690 2.180 2.700 1.145 1.635 2.167 2.733 3.325 9.236 10.64 12.02 13.36 14.68 11.07 12.59 14.07 15.51 16.92 12.83 14.45 16.01 17.53 19.02 15.09 16.81 18.48 20.09 21.67 16.75 18.55 20.28 21.95 23.59 20.51 22.46 24.32 26.12 27.88 10 11 12 13 14 2.558 3.053 3.571 4.107 4.660 3.247 3.816 4.404 5.009 5.629 3.940 4.575 5.226 5.892 6.571 15.99 17.28 18.55 19.81 21.06 18.31 19.68 21.03 22.36 23.68 20.48 21.92 23.34 24.74 26.12 23.21 24.73 26.22 27.69 29.14 25.19 26.76 28.30 29.82 31.32 29.59 31.26 32.91 34.53 36.12 15 16 17 18 19 5.229 5.812 6.408 7.015 7.633 6.262 6.908 7.564 8.231 8.907 7.261 7.962 8.672 9.390 10.12 22.31 23.54 24.77 25.99 27.20 25.00 26.30 27.59 28.87 30.14 27.49 28.85 30.19 31.53 32.85 30.58 32.00 33.41 34.81 36.19 32.80 34.27 35.72 37.16 38.58 37.70 39.25 40.79 42.31 43.82 20 21 22 23 24 8.260 8.897 9.542 10.20 10.86 9.591 10.28 10.98 11.69 12.40 10.85 11.59 12.34 13.09 13.85 28.41 29.62 30.81 32.01 33.20 31.41 32.67 33.92 35.17 36.42 34.17 35.48 36.78 38.08 39.36 37.57 38.93 40.29 41.64 42.98 40.00 41.40 42.80 44.18 45.56 45.31 46.80 48.27 49.73 51.18 25 30 40 50 60 11.52 14.95 22.16 29.71 37.48 13.12 16.79 24.43 32.36 40.48 14.61 18.49 26.51 34.76 43.19 34.38 40.26 51.81 63.17 74.40 37.65 43.77 55.76 67.50 79.08 40.65 46.98 59.34 71.42 83.30 44.31 50.89 63.69 76.15 88.38 46.93 53.67 66.77 79.49 91.95 52.62 59.70 73.40 86.66 99.61 70 80 90 100 45.44 53.54 61.75 70.06 48.76 57.15 65.65 74.22 51.74 60.39 69.13 77.93 85.53 96.58 107.6 118.5 90.53 101.9 113.1 124.3 95.02 106.6 118.1 129.6 100.4 112.3 124.1 135.8 104.2 116.3 128.3 140.2 112.3 124.8 137.2 149.4 12 WILCOXON SIGNED-RANK TEST The sample has size n. P is the sum of the ranks corresponding to the positive differences. Q is the sum of the ranks corresponding to the negative differences. T is the smaller of P and Q. For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at the level of significance indicated. Critical values of T One-tailed Two-tailed n=6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.05 0.1 2 3 5 8 10 13 17 21 25 30 35 41 47 53 60 Level of significance 0.025 0.01 0.05 0.02 0 2 0 3 1 5 3 8 5 10 7 13 9 17 12 21 15 25 19 29 23 34 27 40 32 46 37 52 43 0.005 0.01 0 1 3 5 7 9 12 15 19 23 27 32 37 For larger values of n, each of P and Q can be approximated by the normal distribution with mean and variance 1 24 n(n + 1)(2n + 1) . 13 1 4 n(n + 1) WILCOXON RANK-SUM TEST The two samples have sizes m and n, where m ⩽ n. Rm is the sum of the ranks of the items in the sample of size m. W is the smaller of Rm and m(n + m + 1) – Rm. For each pair of values of m and n, the table gives the largest value of W which will lead to rejection of the null hypothesis at the level of significance indicated. Critical values of W One-tailed Two-tailed n 3 4 5 6 7 8 9 10 One-tailed Two-tailed n 7 8 9 10 0.05 0.1 6 6 7 8 8 9 10 10 0.05 0.1 39 41 43 45 0.025 0.05 m=3 – – 6 7 7 8 8 9 0.025 0.05 m=7 36 38 40 42 0.01 0.02 – – – – 6 6 7 7 0.01 0.02 34 35 37 39 0.05 0.1 11 12 13 14 15 16 17 0.05 0.1 51 54 56 Level of significance 0.025 0.01 0.05 0.025 0.05 0.02 0.1 0.05 m=4 m=5 10 11 12 13 14 14 15 – 10 11 11 12 13 13 19 20 21 23 24 26 17 18 20 21 22 23 Level of significance 0.025 0.01 0.05 0.025 0.05 0.02 0.1 0.05 m=8 m=9 49 51 53 45 47 49 For larger values of m and n, the normal distribution with mean should be used as an approximation to the distribution of Rm. 14 66 69 1 2 62 65 0.01 0.02 16 17 18 19 20 21 0.01 0.02 59 61 0.05 0.1 28 29 31 33 35 0.05 0.1 82 m(m + n + 1) and variance 1 12 0.025 0.05 m=6 0.01 0.02 26 27 29 31 32 24 25 27 28 29 0.025 0.05 m = 10 0.01 0.02 78 74 mn(m + n + 1) BLANK PAGE 15 BLANK PAGE 16