11 june fp1 model

8/12/2019 11 June FP1 Model

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otl. f@)=3 'a3 t - 7

(a) Show that the equation f(-r) = 6 1lu' a root d between x =I and x =2'

(2)

(b) Startingwith the interval [1,2], use ntervalbisectionwiceto find an intervalof

width0.25whichcontains . (3)



.- - - ' , J. l

Thesolutionso thequadratic quation

z ' ) l0z +28= 0

ate z, and zr.

(c) Find z, and, ., giving your answersn the form p + i q' wherep andq are ntegers'

(d) Show,on an Argand diagram, he points representing our complexnumbers z" zt

andzr.e)

a)_ zrl= lz+lrL = rl Zt+\a '{5

(a) Find hemodulus f z'.

(b) Find, n radians,heargument f z',

(1)

givingyour answero 2 decimal laces(2)

--k;"(-|) ' -O''as3bt bu"E--"bO.B. ) 0rE*-? +TT ar5Ca)= 2'68"

) =>Z=l,rriG,_

:. Zz --5-r ,Cs Z3 " 5:iG

zr(-1t)



(a) Given hat

A=f J' l

[ {2- t

)

(i) find A'?,

(ii) describeully thegeometricalransformationepresentedy A2

O) Given hat( 0 -1 )

"=[.-' o l

describeul1y hegeometricalransformationepresentedy B'

(c) Given hat

/k+ l 12.c=l r o I

\ "' )

where is a constant,ind thevalueofft for which he matrixC is singular'

(4)

(2)

(3)

.t 3 CcntrQ-ryrSrn

=-X-



f ( : r ) - ;6 '?+] - : t -1 ' x+o2x

(a) Usedifferentiationo find f'(.;r).(2)

Therootaof heequation("r)=6 1;., n the nterval[0.2,O.e].

(b) Taking0.8asa first approximationo a, apply heNewton-Raphsonrocess nce of(:r) to obtaina second pproximationo a. Giveyour answero 3 decimalplaces.

(4)

trtr; .:-:---*r-.-sT-,;O'8 T-a= O:8

/2-z O. s61 crap)



I - A , ' \

A=l ; ; l . whereaandare ons tan ls '\ b -z )

Given that the matrix A maps he point with coordinatesa, 6)

coordinates2, -8),

(a) find thevalueofa and he valueof6.

A quadrilateral hasarea30 square nits.

It ii transformednto another uadrilateralSby thematrixA'

Usingyour valuesof a andb,

(b) find theareaof quadrilateral .

onto the point with

-t$+ s =̂2 ') a;3jb -tz ---8ab=\

E-3=5

(4)

(4)

-c t 4

b-Ll(+

fivo-=513,0 =lS0 sq.r^,e u-..1-



hatz = x + iy, find the valueofx and he valueofy such hat

z+3 i z . = -1+ l3 i

wherez. is the complex onjugate fz.(7)

,t*U)l3i(r.-.ig) = rgrisx {i'J



, F - ,+ ,(af USe ne esultsor

Lrand

Zr ' to Snow nat

\ i l) (2r - l l ' = - n(2n+l$2n - l )

J

for all positive ntegers .

(b) Hence how hat

. s l ^ , / ,

\ ' ( 2 " - f t t = n (an '+b \/ - J

- '1 ' \

- l

where a and b are ntegers o be found.

(6)

+1-6n -6+3

2^-$ #

:trltzJ\ tD(t-.)f zll* I

' - l) - t+n2 \



TheparabolaC hasequation' = 48x.

Thepoint P(12f , 24t) is a general oint on C.

(a) Find theequation fthe directrixofC

(b) Showhat heequatlonfthe tangento C at P(12f,24t) is

x - t y+12 t2=0(4)

The angento C at thepoint (3, 12) meetshedirectrixof C at thepointX

(c) Find hecoordinatesfX(4)

4 '5'cra8?q ;- La:[- 2 q--\z drrectcu< f*-\z:o

'5$eD =)agfr=+e=)9g t

(2)

a.F* =L-\Z* =) ;g--t$t\Ztz- =OtL

Il t{,rzl + 3 -\z+- b. . 2trrtc.:\zb'+

*Ln a }'3 :-19 * a ,, { $= 2lc+6

gzrr)f,aluedrx A-r:tL i,3:4-*th=\8 -X



ProvebY induction, that fot neZ',

/3 o\ ' ( t " o)ra)lo rJ

=[lr:, rr rJ

(b) f (n)=7'?"+5 is divisible Y 12'

) n:l o'i

36

_- J,,, lneP4or *\

tA',llt;ftI,..t)=tru ?t ;':. i-.""-a-. a-^r \gV: bslltf{ttl1

b\ tr,) =?FL 5 = z=

lzrl -" tru+ eqn=\

fi"*D-.Trnro.lsl. +"11is t f-.-;a

a(+ltr.-r) f

q,-.[ir)_r4r PtD-t tPsv6=\ , tfrrc- {of n=LL+\ tkJC99- {oc n=G*

11 june fp1 model

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