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Also provides kinematics relations, vector operations (dot/cross products), and circular function identities for quick reference during 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VCAA 2024  \nMensuration  \n\n| area of a\u003Cbr>circle segment | r22~~ ~~ 􀀝 􀀞sin( ) 􀀜 | volume of a sphere | 43 π r3 |\n| --- | --- | --- | --- |\n| volume of\u003Cbr>a cylinder | π r 2h | area of a triangle | 12 bcsin(A) |\n| volume ofa cone | 13 π r2 h | sine rule | a b c\u003Cbr>~~ ~~ = ~~ ~~ = ~~  ~~sin(A) sin(B) sin(C) |\n| volume\u003Cbr>of a pyramid | 1 ~~ ~~ Ah 3 | cosine rule | c2 = a2 + b2 − 2abcos(C) |\n\nAlgebra, number and structure (complex numbers)  \n\n| z x 􀀞 iy r 􀀝 cos(􀀛) 􀀞 isin (􀀛) 􀀜 rcis (􀀛) | z  x2 􀀞 y2 r\u003Cbr>|  |\n| --- | --- | --- |\n| −π \u003C Arg(z) ≤ π | z1z2 r1r2 cis 􀀝 􀀛1 􀀞 􀀛2 􀀜 |  |\n| zz~~1 ~~rr~~1 ~~ cis 􀀝 􀀛1 􀀞􀀛2 􀀜\u003Cbr>2 2 | de Moivre’s theorem | z n = r n cis (n θ) |\n\nData analysis, probability and statistics  \n\n| for independent random variables X1 , X2 …Xn | E 􀀞 aX1\u003Cbr>b 􀀝 􀀜 a E 􀀞 X1 􀀝\u003Cbr>b\u003Cbr>E 􀀞 a1X1\u003Cbr>a X 2 2\u003Cbr>􀀡\u003Cbr>a Xn n\u003Cbr>􀀝\u003Cbr>􀀜 a1E 􀀞 X1 􀀝\u003Cbr>a2 E 􀀞 X2 􀀝\u003Cbr>􀀡\u003Cbr>anE 􀀞 Xn 􀀝 |  |\n| --- | --- | --- |\n|  | Var 􀀞 aX1 b 􀀝 􀀜 a2Var 􀀞 X1 􀀝 Var􀀞 a1X1 a2X2 􀀡 an Xn 􀀝\u003Cbr>􀀜 a12Var 􀀞 X1 􀀝\u003Cbr>a22Var 􀀞 X2 􀀝\u003Cbr>􀀡 an2Var 􀀞 Xn 􀀝 |  |\n| for independent identically distributed variables\u003Cbr>X1 , X2…Xn | E 􀀞 X1 X2 􀀡 Xn 􀀝 􀀜 n􀀛 |  |\n|  | Var􀀞 X X 􀀡 X 􀀝 􀀜 n􀀛 2\u003Cbr>1 2 n |  |\n| approximate confidence interval for µ | 􀀝   s   s 􀀚\u003Cbr>􀀛 x z ~~ ~~ , x 􀀞 z ~~ ~~ 􀀘\u003Cbr>􀀜   􀀙 |  |\n| distribution of \u003Cbr>sample mean X | mean | E X  µ\u003Cbr>\u003Cbr>|\n|  | variance | Var X 􀀞 􀀝 􀀜n2 |\n\nCalculus  \n\n| ddx xn 􀀞 􀀝 nxn􀀜1 | xn dx 􀀞 n~~ ~~1􀀝~~ ~~1 xn􀀝 1 􀀝 c, n 􀀜 􀀛1 |\n| --- | --- |\n| ddx eax 􀀞 􀀝 aeax | eax dx 􀀞  1 eax 􀀝 c\u003Cbr>a |\n| ddx loge (x)􀀞 􀀝  | 1\u003Cbr>~~ ~~ dx 􀀞 loge x  􀀝 c\u003Cbr>x |\n| ddx sin(ax) 􀀞 􀀝 acos(ax) | sin(ax) dx 􀀞􀀝 ~~ ~~ cos(ax) 􀀜 c |\n| ddx cos(ax) 􀀞 􀀝􀀜 asin (ax) | cos(ax) dx 􀀞  sin (ax) 􀀝 c |\n| ddx tan(ax) 􀀞 􀀝 asec2 (ax) | sec2 (ax) dx 􀀞  tan (ax) 􀀝 c |\n| ddx cot(ax) 􀀞 􀀝􀀜 acosec2 (ax) | cosec2 (ax)dx 􀀞􀀝 ~~ ~~ cot(ax) 􀀜 c |\n| ddx sec(ax) 􀀞 􀀝 asec(ax) tan(ax) | sec(ax)tan (ax) dx 􀀞  sec(ax) 􀀝 c |\n| ddx cosec(ax) 􀀞 􀀝􀀜 acosec(ax)cot (ax) | cosec(ax) cot(ax) dx 􀀞􀀝 ~~ ~~ cosec(ax) 􀀜 c |\n| ddx 􀀞 sin 1 (ax) 􀀝 􀀜 ~~ ~~1~~ ~~a(ax)2~~ ~~ | 􀀕 ~~ ~~1~~ ~~ dx 􀀞 sin 1 􀀜􀀛􀀝 ~~ ~~xa~~ ~~ 􀀚􀀙􀀘 􀀗 c, a 􀀖 0 |\n| ddx 􀀞 cos 1 (ax) 􀀝 􀀜 aax~~ ~~ | 􀀕 ~~ ~~1~~ ~~ dx 􀀞 cos 1 􀀜􀀛􀀝 ~~ ~~xa~~ ~~ 􀀚􀀙􀀘 􀀗 c, a 􀀖 0 |\n| ddx 􀀞 tan 1 (ax) 􀀝 􀀜 1~~ ~~􀀛~~ ~~(aax)2 | 􀀗 ~~ ~~a2~~ ~~a~~ ~~x2~~ ~~ dx 􀀞 tan􀀖1 􀀛􀀝􀀜 ~~ ~~xa~~ ~~ 􀀚􀀙􀀘 c |\n|  | 􀀛 (ax b)n dx 􀀞 a(n1~~ ~~1) (ax b)n 1 c, n 􀀝 􀀜 1 |\n|  | 􀀝 ~~ ~~ax~~ ~~1~~ ~~b dx 􀀞  loge ax b c |\n\nCalculus – continued  \n\n| product rule | d dv du\u003Cbr>~~ ~~ (uv) u ~~ ~~ 􀀞 v ~~ ~~\u003Cbr>dx dx dx |\n| --- | --- |\n| quotient rule | du dvd u 􀀜 v  dx 􀀘 u  dx ~~ ~~dx~~ ~~􀀞~~􀀝 ~~v~~ ~~ 􀀛􀀚 􀀙  v2  |\n| chain rule | dy dy du ~~ ~~ 􀀞 ~~  ~~dx du dx |\n| integration by parts | u ~~ ~~dvdx dx 􀀞 uv 􀀝 v ~~ ~~dudx~~ ~~ dx |\n| Euler’s method | dy\u003Cbr>If dx = f (x, y), x0 = a andy0 = b,\u003Cbr>then x = x + h and\u003Cbr>n + 1 n\u003Cbr>yn 1 􀀞 yn h 􀀝 f 􀀜 xn , yn 􀀛 . |\n| arc length parametric | 􀀘tt12 􀀞􀀝 dxdt~~ ~~ 􀀜􀀛􀀚 2 􀀙 􀀞􀀝 dydt~~ ~~ 􀀜􀀛􀀚 2 dt |\n| surface area Cartesian about x-axis | 􀀗xx12 2 y 1 􀀞 􀀜􀀛􀀝 ~~ ~~dydx~~ ~~ 􀀚􀀙􀀘 2 dx\u003Cbr>|\n| surface area Cartesian about y-axis | 􀀗yy12 2 x 1 􀀞  ~~ ~~dxdy~~ ~~ 􀀚􀀙􀀘 2 dy\u003Cbr>|\n| surface area parametric\u003Cbr>about x-axis | 􀀗tt12 2 y 􀀞􀀝􀀜 dxdt~~ ~~ 􀀛􀀚􀀙 2 􀀘 􀀜􀀞􀀝 ~~ ~~dydt~~ ~~ 􀀛􀀚􀀙 2 dt |\n| surface area parametric about y-axis | 􀀗tt12 2 x 􀀞􀀝􀀜 dxdt~~ ~~ 􀀛􀀚􀀙 2 􀀘 􀀜􀀞􀀝 ~~ ~~dydt~~ ~~ 􀀛􀀚􀀙 2 dt |\n\nKinematics  \n\n| acceleration | d 2x dv dv d 􀀞 1 􀀛 a  dt 2 dt  v  dx dx 􀀝􀀜 2 v2 􀀚􀀙 |  |\n| --- | --- | --- |\n| constant acceleration formulas | v = u + at | s ut 􀀞 12 at2 |\n|  | v2 = u2 + 2as | s ~~ ~~12 (u 􀀞 v) t |\n\nVectors in two and three dimensions  \n\n| (t) x(t) + y(t) j 􀀞 z(t)􀀄 |  |  | (t)  x(t)2 + y(t)2 􀀞 z(t)2  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  | 􀀞 dr dx dy dz\u003Cbr>r (t) ~~ ~~ i + ~~ ~~ j 􀀞 ~~ ~~ k\u003Cbr>dt dt dt dt |  |  |  |  |  |  |\n| for\u003Cbr>and | 12 | x1  y1 􀀞 z1\u003Cbr>x2  y2  􀀞 z2 | vector scalar product |  |  |  |  |  |  |\n|  |  |  |  1 .  2 |  |  1 | |  2 ","cbCaidqZw0vEvtNS","https://ap.wps.com/l/cbCaidqZw0vEvtNS","pdf",800932,"English","# Mensuration\n# Algebra, number and structure (complex numbers)\n# Data analysis, probability and statistics\n# Calculus\n## Calculus – continued\n# Kinematics\n# Vectors in two and three dimensions\n# Circular functions","[{\"question\":\"What topics are included in this Specialist Mathematics 2024 formula sheet?\",\"answer\":\"It covers mensuration, complex number algebra, data analysis and statistics, calculus, kinematics, vectors in 2D/3D, and circular (trigonometric) function identities.\"},{\"question\":\"Which statistical formulas are provided for independent random variables?\",\"answer\":\"It provides rules for expected value and variance of linear combinations of independent variables, plus results for independent identically distributed variables.\"},{\"question\":\"What calculus result types are listed on the sheet?\",\"answer\":\"It includes differentiation rules (including products, quotients and chain rule) plus integration by parts, Euler’s method, and formulas for arc length and surface area in Cartesian and parametric forms.\"}]","Specialist Mathematics Examination 2 - 2024 Formula Sheet | PDF"]