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 mid-point 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 semi-circle is right ngle A tngent is line tht meets the circle t one point only The ngle etween tngent nd rdius is 90 o

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.

= n n 1 n n 3 3 1 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

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 n-1 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 x-y 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

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the