Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks


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1 Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet
2 C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1 + n n + + n r n r r + + n where (n N) nd n n n! r! n r! n n 1 x r = c r = o 1 + x n = 1 + nx + x < 1, n R 1 x + + n n 1 n r+1 xr 1 x x x r + Logrithms nd Exponentils o log x = log x log Geometric Series o u n = r n 1 o S n = (1 rn ) o S = 1 r 1 r for r < 1 Numericl Integrtion o y dx 1 y 0 + y n + (y 1 + y + + y n1 )}, were = n
3 Alger nd Functions If f(x) is polynomil nd f() = 0, then (x  ) is fctor of f(x) If f(x) is polynomil nd f = 0, then (x ) is fctor of f(x) If polynomil f(x) is divided y (x ) then the reminder is f The Sine nd Cosine Rule The sine rule is: o o = = c sin A sin B sin C sin A sin B sin C = = c You cn use the sine rule to find n unknown side in tringle if you know two ngles nd the length of one of their opposite sides You cn use the sine rule to find n unknown ngle in tringle if you know the lengths of two sides nd one of their opposite ngles The cosine rule is: o = + c c cos (A) o = + c c cos (B) o c = + cos (C) You cn use the cosine rule to find n unknown side in tringle if you know the lengths of two sides nd the ngle etween them You cn use the cosine rule to find n unknown ngle if you know the lengths of ll three sides The rerrnged form of the cosine rule used to find n unknown ngle is: o cos A = +c c o cos B = +c c o cos C = + c You cn find the re of tringle using the formul o Are = 1 sin C If you know the length of two sides ( nd ) nd the vlue of the ngle C etween them
4 Exponentils nd Logrithms A function y = x, or f(x) = x, where is constnt, is clled n exponentil function log n = x mens tht x = n, where is clled the se of the logrithm log 1 = 0 log = 1 log 10 x is sometimes written s log x The lws of logrithms re o log xy = log x + log y x (the multipliction lw) o log = log x  log y (the division lw) y o log (x) k = k log x (the power lw) From the power lw, 1 o log =  log x x You cn solve n eqution such s x = y first tking logrithms (to se 10) of ech side The chnge of se rule for logrithms cn e written s: o log x = log x log From the chnge of se rule: o log = 1 log
5 Coordinte Geometry in the (x, y) Plne The midpoint of (x 1, y 1 ) nd (x, y ) is x 1 + x, y 1+ y The distnce d etween (x 1, y 1 ) nd (x, y ) is d= [ x x 1 + y y 1 ] The eqution of the circle centre (, ) rdius r is (x ) + (y ) = r A chord is line tht joins two points on the circumference of circle The perpendiculr from the centre of circle to chord isects the chord The ngle in semicircle is right ngle A tngent is line tht meets the circle t one point only The ngle etween tngent nd rdius is 90 o
6 The Binomil Expnsion You cn use Pscl s Tringle to multiply out rcket You cn use comintions nd fctoril nottion to help you expnd inomil expressions. For lrger indices it is quicker thn using Pscl s Tringle n! = n n 1 n n The numer of wys of choosing r items from group of n items is written n n r or c r The inomil expnsion is: o ( + ) n = n + n 1 n 1 + n n + + n r n r r + + n 1 + x n = 1 + nx + n n 1 x 1 x + + n n 1 n r+1 xr 1 x x x r + Rdin Mesure nd its Applictions If the rc AB hs length r, then AOB is 1 rdin (1 c or 1 rd) A rdin is the ngle sutended t the centre of circle y n rc whose length is equl to tht of the rdius of the circle 1 rdin = 180 π The length of n rc of circle is l = r θ The re of sector is A = 1 r θ The re of segment in circle is A = 1 r (θ sin θ)
7 Geometric Sequences nd Series In geometric series you get from one term to the next y multiplying y constnt clled the common rtio The formul for the n th term = r n1 where = the first term nd r = the common rtio The formul for the sum to n terms is o S n = (1 rn ) or, 1 r o S n = (rn 1) r 1 The sum to infinity exists if r < 1 nd is S = 1 r
8 Grphs of Trigonometric Functions The xy plne is divided into qudrnts For ll vlues of Θ, the definitions of Sin (Θ), Cos (Θ) nd Tn (Θ) re tken to e... where x nd y re the coordintes of P nd r is the rdius of the circle o sin θ = y r o cos θ = x r o tn θ = y x A cst digrm tell you which ngles re positive or negtive for Sine, Cosine nd Tngent trigonometric functions: o In the first qudrnt, where Θ is cute, All trigonometric functions re positive o In the second qudrnt, where Θ is otuse, only sine is positive o In the third qudrnt, where Θ is reflex, 180 o < Θ < 70 o, only tngent is positive o In the fourth qudrnt where Θ is reflex, 70 o < Θ < 360 o, only cosine is positive o The trigonometric rtios of ngles eqully inclined to the horizontl re relted : o Sin (180 Θ) o = Sin Θ o o Sin (180 + Θ) o =  Sin Θ o o Sin (360  Θ) o =  Sin Θ o o Cos (180  Θ) o =  Cos Θ o o Cos (180 + Θ) o =  Cos Θ o o Cos (360  Θ) o = Cos Θ o o Tn (180  Θ) o =  Tn Θ o o Tn (180 + Θ) o = Tn Θ o o Tn (360  Θ) o =  Tn Θ o
9 The trigonometric rtios of 30 o, 45 o nd 60 o hve exct forms, given elow: 30 o 1 Sine (Θ) Cosine (Θ) Tngent (Θ) o 60 o The sine nd cosine functions hve period of 360 o, (or π rdins). Periodic properties re : o Sin (Θ ± 360 o ) = Sin Θ o Cos (Θ ± 360 o ) = Cos Θ The tngent function hs period of 180 o, (or π rdins). Periodic property is: o Tn (Θ ± 180 o ) = Tn Θ Other useful properties re o Sin (  Θ) =  Sin Θ o Cos (  Θ) = Cos Θ o Tn (  Θ) =  Tn Θ o Sin (90 o Θ) = Cos Θ o Cos (90 o Θ) = Sin Θ
10 Differentition For n incresing function f(x) in the intervl (, ), f (x) > 0 in the intervl x For decresing function f(x) in the intervl (, ), f (x) < 0 in the intervl x The points where f(x) stops incresing nd egins to decrese re clled mximum points The points where f(x) stops decresing nd egins to increse re clled minimum points A point of inflection is point where the grdient is t mximum or minimum vlue in the neighourhood of the point A sttionry point is point of zero grdient. It my e mximum, minimum or point of inflection To find the coordintes of sttionry point: o find dy (The grdient function) dx o Solve the eqution f (x) = 0 to find the vlue, or vlues, of x o Sustitute into y = f(x) to find the corresponding vlues of y The sttionry vlue of function is the vlue of y t the sttionry point. You cn sometimes use this to find the rnge of function You my determine the nture of sttionry point y using the second derivtive o If dy = 0 nd d y > 0, the point is minimum point dx dx o If dy = 0 nd d y <0, the point is mximum point dx dx o If dy = 0 nd d y = 0, the point is either mximum, minimum, or point of dx dx inflection o If dy = 0 nd d y = 0, ut d3 y 0, then the point is point of inflection dx dx dx 3 In prolems where you need to find the mximum or minimum vlue of vrile y, first estlish formul for y in terms of x, then differentite nd put the derived function equl to zero to then find x nd then y
11 Trigonometricl Identities nd Simple Equtions Tn θ = Sin θ Cos θ Sin θ + Cos θ = 1 (providing Cos Θ 0, when Tn Θ is not defined) A first solution of the eqution Sin x = k is your clcultor vlue, α = Sin 1 k. A second solution is (180 o α), or (π α) if you re working in rdins. Other solutions re found y dding or sutrcting multiples of 360 o or π rdins. A first solution of the eqution Cos x = k is your clcultor vlue, α = Cos 1 k. A second solution is (360 o α), or (π α) if you re working in rdins. Other solutions re found y dding or sutrcting multiples of 360 o or π rdins. A first solution of the eqution Tn x = k is your clcultor vlue, α = Tn 1 k. A second solution is (180 o + α), or (π + α) if you re working in rdins. Other solutions re found y dding or sutrcting multiples of 180 o or π rdins. Integrtion The definite integrl f x dx = f f() The re eneth curve with eqution y = f(x) nd etween the lines x = nd x = is: o Are = f x dx The re etween line (eqution y 1 ) nd curve (eqution y ) is given y: o Are = The Trpezium rule is: o y y1, y dx dx 1 y 0 + y n + (y 1 + y + + y n1 )} were = nd y i = f( + ih) n
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