c4 june 2011 unofficial ms

8/6/2019 C4 June 2011 Unofficial MS

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rl 9,rsa l\o-2

. (x-1) '(2x+l) (x-1) (;- l) 'z (2x+1)

Find hevalues f theconstants. B andC.

--)Q>c' A(r-rXzx+\)+g(zz+r) c(r-t)2

)c=l -)qe38aB--3

O : -A +ts* C O :Ar3t!iO_,?A=-L



r(')=----l- ' l! a1r / (s++* ' ) 2

Find the first threenon-zero erms of thebinomial expansionof f(;r) in ascending

of -r. Give eachcoefficientas a simplified fraction.

"t --?t-



Figure 1

A hollow hemisphericalbowl is shown n Figure 1. Water s flowing into the bowl

When the depth of the water is /'rm, the volume Zm3 is given by

v =Lxh' (3-4h) , o<f t<0.2512

^dv{a) Find. n terms f r" : when r= 0.dh

Water lows nto thebowl at a rateof a -tr-'.800

(b) Find he rateof change f ft, in msr,

when ft = 0.1

+dv=-ILdl" ZS



Figure2

Figure shows sketch f thecurvewith equationv:xr ln(r'?+2), x20.

The initeregionR, shown hadedn Figure2, s bounded y the curue,he r-axisa

l inex=r /2 .

The able elowshows offespondingalues fn and for y-x3 ln(x2+2).

x 0

^lz4

^lz2

3"'1

4 ".12

v 0 o'0333 0.3240 r.35q6 3.9210

(a) Complete he table abovegiving the missing valuesofy to 4 decimalplaces.

(b) Use the trapeziumrule, with all the valuesofy in the completed able, to ob

estimate or the areaof R, giving your answer o 2 decimalplaces.

(c.) Use he substitution : -rt+2to show that the areaofR is

r f o

; I t , - 2l lnudu



'o33t O.3Zq l'3sq5

1= :3O {zo\p)

d, U = ?cz+2-,-, ?L--o - \l=6a+1- = Z --ahA .,-. 2C r-rlh ur,=(fz)t+Z = t

E_='L)c

31,- t ry2

- t

-- nu



Find hegradient fthe curvewith equation

lny=Va1n* , x> 0,Y> 0

at thepoint onthe curvewhere :2. Giveyour answer san exactvalue'

v t i , . /+ rav'



With respecto a fixed originO, the ines , and , ategivenby the equations

f ul f-') r:l f 1lt,: =l-3l.,tl zl. t,, =l tsl-ul-,l-l-2) |.3J \31 \rr

where1andp arescalar arameters.

(a) Show hat /, and /, meetand ind theposition ectorof theirpointof intersec

(b) Find, to the nearest0.1', the acuteanglebetween , and lr.

( 5lThepointB hasposition ector|-1

l.

I t l '

(c) Show hatB lies on /,.

(d) Find the shortest istanceromB to the line /r, givingyour answero 3 sign

figures'

-3+2\

z+3

5-tr=-5+2r,,r i-i+Zf =\S:3a-)-?.+j \= j+n-t^



- ^ \ : \ = ) A=\-j+2tr=-\ =)2tr= --rA=-2+3)r ' \ =) SX=] zy A=



Figure 3

Figure3 shows arl of thecurveC with parametric quattons

y: sin O<e lx - I an0 ,

The oint ies nC and as oordinatesI v' ' iV'l\ rJ

(a) Find the value of d at the PointP

The line / is a normal Io C at'P. The normal cuts the r-axis at the point Q'

(b) Show that Q has coordinates k-{3, 0)' giving the valueof the constant r'

The finite shaded egion ,Sshown in Figure 3 is boundedby the cume C' the line

andthex_axis.This ihaded region s rotated hrough2z radiansabout he.r-axis o

solid of revolution'

(c) Find the volume of the solid of revolution,giving your answer n the form

prtr3+qf,wherep andq arcconstants.



S at P

-q =-8

a\. . v t 5

tri-ze +t =S

-\- otg-o

rr

r - -2= 1Tri3 -+N



(a)

(b)

ri,'aJ(+r+:)-la:,

Given hat y=I.5 atx:-2, solvehedifferentialquation

dY

giving your answer in the form y = f(x).

=)+ (vqt

t.i-E- f =\

c4 june 2011 unofficial ms

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