C4 June 2011 Unofficial MS
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Transcript of 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
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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-
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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
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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
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'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
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Find hegradient fthe curvewith equation
lny=Va1n* , x> 0,Y> 0
at thepoint onthe curvewhere :2. Giveyour answer san exactvalue'
v t i , . /+ rav'
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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^
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- ^ \ : \ = ) A=\-j+2tr=-\ =)2tr= --rA=-2+3)r ' \ =) SX=] zy A=
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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.
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S at P
-q =-8
a\. . v t 5
tri-ze +t =S
-\- otg-o
rr
r - -2= 1Tri3 -+N
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(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 =\