S14 ]AL C34

1' f(x):2x3+v-19

(a) Show that the equation f(x) : 0 has a root a in the interval lt.S, Z]

The only real root of f(x) : 0 is a

The iterative formula

( t \txn*t=lt-r.,), xo=1.5

can be used to find an approximate value for a

(b) Calculate xl, xrand x' giving your answers to 4 decimal places.

Q)

(3)

(c) By choosing a suitable interval, show that a: 1.6126 correct to 4 decimal places.(2)

"*fr*)-:f'A +lz),E ;ra$ ---(O r Fr t.lt rt tn

o)-j[a:l f,+, f .6118 Iz, l.6t 2L-- L 1 = l-( l?-6-s

c)

-.000493 )o -

rJJ z-Oi0OrL66-<o b1 Srin G0rrraqr-

2. A curve C has the equation

x3 -3xY -x * !3 - 11 :0

Find an equation of the tangent to C atthe point (2, -l),giving your answer in the formax + by + c:0, where a, b and c are integers.

(6)

..l- i ,- t-z ).^ -n -

- *--+ {g{- -3*) r#-' ++gg -3:#-

3. Given that

cos20y=I + sin20

show that

dy- o ,de L + sin20

where a is a constant to be determined.

(4)

--]L=lrr,3L -- \r r g--lul2-qu1 qtlSlrlO V!, ,,LCirtO

7t^3t44

-I.'.*

'- U--E--:al tL t7€,

-( t+!rn2-O)1db;i

eo)

4. Find

(a) Irr.+ 3;12dx

(b) I#r*a)

(2)

11f(x) = 18 + 27 x,)3 , l*l . -;

J

Find the first three non-zero terms of the binomial expansion of f(r) in ascending powersof x. Give each coefficient as a simplified fraction.

--8t1LrE*I*(s)

=zT*C{?:lF

6. (a) Express5-4x

in partial fractions.(2x-l)(x+1)

(b) (i) Find a general solution of the differential equation

(3)

(2x -1)(x + 1)* (5 - 4x)y, x > 12

Given that y: 4 when x : 2,

(ii) find the particular solution of this differential equation.

Give your answer in the formY: f(r).(7)

- S.r.Y--!:lr* - - ---=?-+ . {--JL-12rffir+*) ?*-1 7rt

rL

{qq}- lntf = Ia -3ln 3 tL lrrt 9 i -ztn}tq

7. The function f is defined by

^ 3x-5t:xr--+ x+1 , xelR.,x*-1

(a) Find an expression for f-l(r)

(b) Show that

ff(x) = !::, r €JR,.r * -1.x * Ix-l'

where a is an integer to be determined.

The function g is defined by

g:x'-' x' -3x, x € JR, 0 ( x ( 5

Find the value of fg(2)

Find the range of g

(3)

(4)

(c)

(d)

(2)

(3)

'L-- "+* ( -

0f{rr; : 3(*:)-s

+l-'- 3 -S:) I .x s &+J

t' !''a'l-I -.--{- -ry{(

tH)''

-f'ae-a1x.-ts - S(x+d-

tx-y<\

iz-s + (Y-tr)1< (>(t)

d z-s-.e*7-\

' (bt-) t -y-l

4w -?D1>t- 'l+

c) (t

*,

(z) g ( (ta-tt 1)

(x-:)*L

Y

sll

d)g(r's) .grs) ,

-a*

tot6

).xI lllttrrr rr r,r.. ---

g S 1tx-) S ,l

8. The volum e V of a spherical balloon is increasin g at a constant rate of 250 cm3 s-r.

Find the rate of increase of the radius of the balloon, in cm s-1, at the instant when the

volume of the balloon is 12 000 cm3.

Give your answer to 2 significant figures'(s)

lYou may assume that the volume V of a sphere of radius r is given by the4

formula V = axrt.fJ'

J

- N :-}So- $",' * trJ^tuL V-a lz(flX}dA s'\v

-W*-4.4l.: -il__

n@{n(3

*-*a1lc;,,tsez

9.

Figure 1

Figure 1 shows a sketch of part of the curve with equation y = eG, x ) 0

The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and thelinesx:4andx:9

(a) Use the trapeziumrule, with 5 strips of equal width, to obtain an estimate for the areaof R, giving your answer to 2 decimal places.

(4)

(b) Use the substitutiq! , = J; to find, by integrating, the exact value for the area of R.z^D h'l (7)

rg')

tW'2rq v ,u el- rz?,o

10. (a) Use the identity for sin(l + B) to prove that

sin2A =2sinA cosA(2)

(b) Show that

d-i [t"(t*1;r))] : cosec,dJf'

(4)

A curve C has the equation

y: tn(tan(]x)) - :slnr, o < x < it

(c) Find the x coordinates of the points on C where $ = Odr

Give your answers to 3 decimal places.

(Solutions based entirely on graphical or numerical methods are not acceptable.)(6)

4\ let i:6-- Sr^ {&B) aJ,n{f * 8 + ho *S,^B

S'r*2I- t }&n*(nA- &

*-WA;y- _x@

- furA

Smltw S&A2t

31,^r talx=-t tl

;l{,n2rc *- =r-1i:,s{r:r(t'F-O-?L171 ...i Or4p.

I 2Srnx tioX r?_

0, ?,6s*-z +

11.

X'igure 2

Figure 2 shows a sketch of part of the curve c with equation

y : ga_3x _ 3e_r, .X € lR

where a is aconstant and a > ln4

The curve c has a turning point p and crosses the x-axis at theFigure 2.

(a) Find, in terms of a, the coordinates of the point p.

(b) Find, in terms of a, thex coordinate of the point e.

(c) Sketch the curve with equation

poifi Q as shown in

(6)

(3)

y :ls'-z*- 3.,1, .r € lR, a ) ln4

Show on your sketch the exact coordinates, in terms of a, ofthe points at which thecurve meets or cuts the coordinate axes.

(3)W- j-- :-oc r-lf,;;3x:---

b) 1,0a -Sx-e,- =

[n g^-r* ?

a -3r s

& -lx-,

F' -7'r5e-

[n3;''. -x-

ln 3 + l{r{,

ln3 *7,-. .'' L* t 4-ln3L z Lrt*-lY'3)

2-

tr lgo'ix' -i<-" \

aso y= {-Z

C)

*6-,"")

t2.

Figure 3

Figure 3 shows a sketch of part of the curve C with parametric equations

x = tant, y = 2sin2t, 0 < I <:

The finite region S, shown shaded in Figure 3, is bound "OO:the

curve C, the line x = rEand the x-axis. This shaded region is rotated through 2tr rudians about the x-axis to forma solid of revolution.

(a) Show that the volume of the solid of revolution formed is given by

.,+o l' (tan2 r - sin2l)dr

Jo'

(b) Hence use integration to find the exact value for this volume.

(6)

(6)

rC''[i

Vo[uruo- ' n f,y"tr ' "J t' # oo tnnt ' di '' t'5wa b"nt 'o ; t=o

a bant 3' , 4 S,n+k E

: , Serlt .'" r/oturua- , ArrJ:'^* v tItt' *L

,q, Yrr t s@ dt't' ?ttifgt*':"'ros2€ )*t

, r.lf Can't, t $br'(o't tL 3 wJ

otat{?-srr2t 'tU

H.-'E::*,-',-H,Ii,;1:, 3;i' : ;;Ir:E

, t,*n J' Srr]* -t - tu+Lc*zr al

o5; t*..iE g.rr* +ltoscr !*? = 2n (zsdu+ cal'? -TdP

t,Jo vs ,Lrr[ Zt*^ -+LSr* ]r -{:

, ?.rr[ c ,* .ti ' rr ) -(o) J

lIa)

7L

dufla

D

:rrrfq*[ l z +r (tJ-s-4q

13. (a) Express 2sin0 * cos 0 inthe form Rsin (0 + a),where R and a are constants,R >0and0 <a< 90o. Giveyourvalueofato2decimalplaces.

(3)

Figure 4

Figure 4 shows the design for a logo that is to be displayed on the side of a large building.The logo consists of three rectangles, C, D and, E, each of which is in contact with twohorizontal parallel lines /, and,lr. Rectangle D touches rectangles C and.E as shown inFigure 4.

Rectangles C, D and' E eachhave length 4 m and width 2 m. The acute angle d betweenthe line l, and the longer edge of each rectangle is shown in Figure 4.

Given that l, and lrarc 4 m apart,

(b) show that

2sin0*cos0:2

Given also that 0 < 0 < 45o,

(c) solve the equation

2sin0*cos0:2

giving the value of d to 1 decimal place.

Rectangles c and D atd, rectangles D and, E touch for a distanceFigure 4.

Using your answer to part (c), or otherwise,

(d) find the value of fr, giving your answer to 2 significant figures.

( 5,o ,fi>-t* ) z k",fCosol t kc,&&Suro1, _( Srne{ = \ , ,lW tt?(e 4Etrl=i_ it .rcr --L ( = o .+ 636q+- - 0r. * o 'sa" eh+Zlye.6{fs,,*[eto'k6, - ry 'G$h(q+r6.b+)

4m

(2)

(3)

shown in

2m

ix4m

{S'nO

hrn as

(3)

b)

**w

F4s'gtktkSrrg+2CorB=4

ZS,nB +CoB =L

c) G Srr^ (g* o'tkcl s Lg+ 06.sp.Srrr-- (fr) = 6 3 kr lr6.s6 . -

'i: $t 36'qoft

7-

:. h: *-Z 3

^) .L139tnn0 3

At r'^

2

* l-3--)

14. Relative to a fixed origin O,the line I has vector equation

' [-l].'[j]

(3)

(2)(b) Find the vector AB.

where ,t is a scalar parameter.

Points A and B lie on the line /, where A has coordinates (1, a, 5)

and B has coordinates (b, -1, 3).

(a) Find the value of the constant a and the value of the constant b.

The point C has coordinates (4, -3,2)

(c) Show that the size of the angle CAB is 30"(3)

(d) Find the exact area of the triangle CAB, giving your answer in the form k.6,where ft is a constant to be determined.

(2)

The point D lies on the line / so that the area of the triangle CAD is twice the area of the

triangle CAB.

(e) Find the coordinates of the two possible positions of D.

-[ +?-\

6, -t

-q-&-.=1tLt3=tiJ

--j,t'rg--=- lqF-* .E--*€'=:0"362-2

d)Arreo, f ,ftt J-( SrnSo i tZG

+3G

An* = 6'li

i -fax t-f,bl s,n3o 3 6,.8L

.: \[B] , ,S .6Ca(2*

,'l-re \\/l- I

("

6L LAg

frb,,[fr " 3{iffi , zxfd

.-)

(+)

,

,['

r)',