Further maths edexcel notes, Study notes of Mathematics

Download Further maths edexcel notes and more Study notes Mathematics in PDF only on Docsity!

Topicstofn

☐ exponential +^ log modelling

☐ translating graphs

ooo * PURE^ MATHS^ YEARI^ CONTENTS of I ✓ ALGEBRAIC FRAMING 1 ✓ QUADRATUS 3 ✓ EQUATIONS (^) AND INEQUALITIES 4 ✓^ GRAPHS AND (^) TRANSFORMATION 5 ✓ STRAIGHT LINE^ GRAPHS 6 ✓ CIRCLES 7 ALGEBRAIC^ METHODS 8 ✓ THE BN0M EXPRESSION 9 ✓ TPIIINIMETPII RAIDS A

TRIGONOMETRIC IDENTITIES AND^ EQUATING It ✓ VECTORS 11 ✓ DIFFERENTIATION 13 ✓ INTEGRATION (^14) EN1N-0AB AND DIANTHA ⑧ ② ⑦ ⑧

⑧

f⑧f " " (^). " EXPANDING.

• multiply each^ term^ out^ one^ by one^ using the^ foil method

eg.

& (^). (^) is μμ = (^) of - Zoey + (^3) ×+20×-109 + = 4 × 2 - Zoey + (^) 23.x - toy +^15 FACTORBINII? Put (^) the terms (^) back into brackets ◦ (^) po site (^) to (^) expanding , eg. to

• OC2-5OC →^ take out →^ ✗ (^) ( x - 5)

  ◦

◦ answers are 0 or 5

• dilterence of two^ squares is^ a quadratic expression in^ the^ form -^ Y ↓ / This (^) expands into ( a^ +^ y ) ( x^ - y) / FA11-0RB / NINI'S eg. ②④ → (^) find two numbers (^) that add (^) to make - It and (^) multiply to (^) make - 60 (^60) ↓ Is -15^ and (^4) put into brackets ↓ & ( (^6) ×-15/(6×+4) ÷fs i.^ ¢ simplify ☐ (^) ( 2x - 5) (3×+2) eg.^ DC3^ +^3 ×^2 -^10 a ↓

take out a^ →^ oc^ (^ at^ +^3 ×-10)

↓

facto rise^ as^ normal

x( acts^ )^ ( x^ - 2)

IS ' 9 ' 21 AC

⑧!§^ CH I -^ s (^). 1.

• Surds (^) are irrational numbers as they cannot^ by^ written^ in^ the^ form^ % Tab =^ Ja ✗^ To § = To^ ÷^ Fb IManipulatingsurdeg.FI = (^54) × 53 to = 2 %^253 eg.^ T2^ ×^52

RATIONALISM DENOMINATORS 16 To (^) rationalise % (^) multiply (^) by %a To rationalise % + (^) To (^) multiply by

¥-5b To (^) rationalise (^) Jb (^) multiply by ◦ + %+ To eat.IE/-- % eg.^ I

× };[ˢ =^ 3-^52

" "^ "^ "

ftp.t/NfTHfffUADf

eg.^ write^208 +^12 ×+7^ in^ the^ form at oc^ +^ b)^

2 + C

◦ first take (^) out 2 - 2[ OC2-1GA +^ %] ◦ (^) half (^) the 6 a^ - 2[ +^ 3)

% ]

◦ square 3 and (^) takeaway - 2%+372-9 + (^) % ]

° multiply back by 2 - 210C + 3) 2 - I /

% ✗ =-3 or (^) - NEGATIVE DX

eg.^ -3×^2 +^6 ×+5^ in^ the^ form a -^ bloc^ +^ c)

" ◦ (^) take out -3 -^ -3^ [ OC2^ - 2x (^) ] +^5 ( 6 ÷ -3 =^ - 2) ↓ Complete the^ square as normal^ but^ rewrite in^ form a - b( a.^ + c) 2

% b- 3 ( a - 1) 2

a = & and I

COMPLETING (^) THE SQUARE TO SAVE

• This (^) can

only be^ done^ if^ the solution^ to^ solve^ is^ equal to^0

eg. Solve (^) the (^) equation 3 × 2 - Isoc + 4 =^0 to

• Complete square as^ normal^ (half^ , (^) square and^ take^ away ) LD ( ≈ - 3) (^2) _ 273 =D ( solve like (^) a (^) normal (^) equation )

( a- 3) (^2) = (^) 23/ to

a -3^ =^ I

◦ %^ a :3 ± F%

a. a.^ a mB

QUADRATIC GRAPHS c. a "

hey features of a quadratic graph :

shape.

roots when y=0 roots

+ Ve

y intercept when^ a^ =^0 I 1

• a (^) - Ve turning point (^) y intercept

eg.^ Sketch^ y= of^ +^3 ×-

• roots = factor ise it = (^) ( a + (^) 4) ( a - 1) so ✗ = (^) -4 and I

y intercept =^ oc^ =^0 ◦

◦ ◦

y =^ -^4

i

• turning point =^ complete^ the^ square

I

( ou +^ I.^ b) 2 -^ 6.

I

◦^ ◦^ ◦^ (^ -^ I.^5 ,^ -6.25) i d

' l '

(the^ a -^4

value can also^ be^ found by 1

finding the^ midpoint^ of -4^ and^ 1) I ↓ (^) _ -

(-^ I^ -5,^ -^ 6-^25 )^ /

i 4-10-21 (^) Ac μ CH^

• y < (^) floe) (^) is the area underneath the graph floor)

y

> flat is the area above the graph floor) this is

y >^ floor)

• dotted (^) line used to show^ it^ doesn't^ include^ the value
• solid (^) black line used (^) to show it does (^) include the value

≤ or^ ≥

""^ " y <^ floe)

" ' a. a Ac

SIMULTANEOUS [QUAT /Off

CH3. / +3.

eg.^2 y t^ 30C^ = (^6) →

multiply by two^ , to^ make^ the^ as^ equal

y t^ 40C^

= (^7) to subtract the two (^) equations to remove (^) the ou ↓ 60C =^12 then sub (^) a back into (^) original (^) equation to (^) get (^) y

4o ↑

2x= 5 =^ a.^ =^42 → y =^ -^ 3/ QUADRATIC (^) EQUATIONS 31

• These (^) are (^) simultaneous (^) equations containing 062 or^ y to

rearrange the^ not^ quadratic (^) equation to^ make^ either^ oc^ or^ y the^ subject^. this (^) can then be (^) subbed into the (^) quadratic and solved. I ey.^ OC2^ +^ y2 =^10 oc + y =^4 →^ re - arrange →^ a (^) :( 4 - y ) (^) quadratic simultaneous (^) equations ✗ ill (^) always have (^2) sets (^) of / μ - y) (^4 -^ y)^ +^ Y2^ =^10 Solution's 16 - by + (^) Y2-1YZ =^ to^ a-

to

ZY2 - by +^ 6= ↓ " "eat the like (^) terms (^).

I

↓ (2g - 6)ly^ - 1) ↓

y

= (^3) and I

oc =^1 and 3 →^ sub in y to find the 0C Values

" - a. " "

SIMULTANEOUS EQUATIONS ON^ f.RAPHS^

• on (^) a graph , the^ solution^ of^ a^ pair of^ simultaneous^ equations ,^ is^ where^ the^ two graphs intersect^.

eg.^ 20C^ +^ By^ =^ &^3 ×-9=

"" ' " "^ "^ "" (^) " " / to intersect at (^) μ ,^ -^ 2)

y =^ -2 i. ( (^7) ,^ -^ 2)

• This point of^ intersection^ can^ also^ be^ found

by betting^ the two^ equations equal to^ each^ other

QUADRANTS +^ THE^ DISCRIMINANT

• when (^) solving a linear (^) and quadratic equation,^ there^ can^ i
• intersect twice
• intersect once
• not (^) intersect

this can be calculated using the discriminant (B2-4AC)

↓

if it^ is^ >^0 =^ two^ solutions

if it^ is^ =^0 =^ one^ solution

if it^ is^ <^0 = no real^ solutions

eg. 2x^ +^ y^ =

y= OC^

_ 3.x + 1 to set (^) equal to each (^) other →^3 -^ 2x^ =^ of^ - 3 ×+ ↓ ← of^

• a - 2 =^0 put into^ B2-4AC ( (^) discriminant) = (^) C- 1)2-^ 4/1×-2) = 9 →^ more (^) than (^0) , so has

two real^ solutions^.

"°.^ " (^) m. ① fKffff☒ " (^) "

Y =^ aoc

y =^ a^ oc

y =^ aocs μ ex (^) sketching the (^) graph ( x^ - a) ( x^ - b) ' ( x^ - c) "

at a point of inflection , the concave

a- a ☐ ×^ (^ crosses^ as^ linear)^ part^ of^ the^ graph^ becomes^ convex "" " "μ◦nve× a- b ✓^ a

☐ ×^ (^ touches^ as^ quadratic)

✗ =^ C

/

DX (^ point^ of^ inflection)

eg.^ Cubic^ graph to stretch (^) y^ = (x^ -^ 2)^ (^ I^ -^ a)(^ I^ +^ x)

• shape = (^) (a)1- a)( =^ _ so (^) negative
• roots = (^2) , I (^) , - I
• y intercept = ( a (^) :o) = (2)(1) (^) (1) = (^) -

:Hn

°

" (^) " " (^) " m HANI GRAPHS

• in oc^ "

graphs both^ of^ the^ tails^ go either^ up or down^ E9.

( W M)

• positive W μ) - negative M (^) ( M) eg.^ stretch^ y :^ at^ a^ -13)/^ a^ -^ 2)(^ x^ -^ 3)

roots =^0 ,^ -^ I^ ,^2 ,^3

shape = positive W

y inter^ =^0 (0×-1×^2 ×3) to ; DIFFERENTIATES

° " "" " " "^ "^ " "^ ""^ "^ ""^ ""

""{ ↑ ↑

↑ makes a^ flat

creates a^ point of area ( like^ a

repeated root quadratic)

is made inflection

HANDMAID 19 - to - 21 MB - ⑥ (^) μ

CH 4.5, 4.

• example transformations

of the^ graph^ floe)^ :^ ol

' / 0C - 3)

flat 2) ' 2 to^ the left FL3A) ' squash by 1/3 fl- a) = reflection in y

-2 (^) o I^ o 3 ,^ o^3 , p -3^3 ⑧

• 4

only changes the^ OC^ values

floe) -14 ' (^4) up - f. (a) = reflection in a μ

only changes the y values

-^ " ,,!

2f( a) = ZX Steeper + Stretched

,^3

• the (^) graph of af (a) stretches (^) the (^) graph by scale (^) factor a in vertical (^) direction
• the

graph of^ flax)^ stretches^ the^ graph^ by^ scale^ factor^ Ya^ in^ horizontal^ direction

TRANSFORMATION (^) FUNCTIONS

"+^.

• Asymptotes are (^) also translated

eg.^ translate^ gloc)^ =^ You^ to^ stretch^ -^ gloc^ -11)

• glad - a. (^) g( +^ 1) = left translation (^) by / ↓

asymptotic moves^ to^ oc=^ -^ I

b. glad^ -2^ =^ down^ by 2

to

asymptote moves^ to^ y= -

ys f^ /^ a)^ transformation

flat 1) I^ to^ the tent

b. (^ f^ 2x)^ '^12 St^ in^ a^ axis g- If^ /^ a) (^3) sf in (^) y axis

y= flat^ -^ I^1 down

y= f / 74 )^ St 4 in^ a (^) axis g. =^ ff-^ a)^ reflect in (^) y axis y= -^ f^ (a)^ reflect in^ a^ axis

is.^ "^ -21 MB (^) / 1-/ /

/ CH '

you find^ the^ distance^ of 10C^ , ,Y , /^ and^1 ×2,92)^ using pythagoras :

◦^ ◦ ◦ (^) D: ✓ ( Bsc) ' +

(Ay)

'

d

' ._ %

ey. find the^ distance^ between^ 1-2,4)^ and^ (3,9) so ✓^ (3--2) ' +

( a^ -^ 4)^

2 0 (3,9^ )^ to ✓ so 0 C- 2,4)^ =^552

• if you know^3 of^ the^ points ,^ you can^ find the^ area^ of^ the^ triangle^ using^ : ' 12 ✗^ bxh II. ' %)

ey.

/ ' % 5° (^) ' " 2 "^ "^ "^ "^ %^ =^ " % (0,0)^ < > 14,0) 4

14 -^ o^ )

16-11.21 MB ① [ /// CH 55

• when (^) a linear (^) model is made (^) , we usually make (^) assumptions :
  • they value

goes up^ the^ same^ amount^ for^ each^ unit^ increase^ in^ a^1 the^ gradient^ is^ the^ same)

• ie (^) , we (^) assume it goes up by the same (^) amount each time
• can (^) also be modelled (^) as being directly (^) proportionate , when (^) the (^) gradient stays constant and it (^) passes through (^) the

origin.

y :^ hoc

ADDTHIS.tl/HB,g

. 11. (^2) , (^) mpg + / < +, g. ,

• you find the midpoint by finding the (^) average of the x x (^) y coordinates :
• ie : ×

,^ +^ ✗^ z^ Y^ ,^ +^ Y

2 I^2

• The

perpendicular bisector^ in^ a^ circle^ with^ cross^ through^ the^ midpoint of^ the^ chord^ at^ 90°^ and^ pass through^ the^ centre

of the^ circle

" I

perpendicular bisector

ia. ". "

no (^) DRUMMING an

-^. .

" . . ' '

(a. b)

-^ ' (0,0)

if the circle is^ centred^ onto^ the^ if the^ circle^ equation has^ the^ centre

origin , the^ equation is^ :^ (^ a.b)^ ,^ the^ equation is^ :

x2 +^ y2 =^ r2^ ( x^ -^ a) (^2) +

( y -^ b)

2 = r

to set^ the^ radius^ :

loc , y)

↑ use pythagoras :

I r ' so a. (^2) + (^) y2 = P

• -^ -^ -^ -^ i^ '

if the equation is^ in^ the^ form :

2 -1 y2 + ax +

by +^ C^ =D to

collect the term →^ (x2^ -1^ ax^ ) +^ (^ y^ '^ +^ by ) +^ (^ :O

complete the^ square on^ it

and collect terms into the

correct form