FP2 Questions

8/10/2019 FP2 Questions

1/44

FP2 questions from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002

The following pages contain questions from past papers which could conceivably appear onEdexcels new FP2 papers from June 2! onwards"

#ar$ schemes are available on a separate document% originally sent with this one"

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + ,ersion 2 + #arch 2! )


2/44

1! Find the set of values for which

x+ )- (x+ )"(5

.P& January 22 /n 20

2! 1a Find the general solution of the differential equation

tt

v

d

dv 3 t% t-

and hence show that the solution can be written in the form v3 t1ln t4 c% where cis anarbitrary constant"

(6

1b This differential equation is used to model the motion of a particle which has speedv m s)at time ts" 5hen t3 2 the speed of the particle is * m s )" Find% to * significantfigures% the speed of the particle when t3 &"

(4

.P& January 22 /n (0

3! 1a 6how thaty3 2) x2exis a solution of the differential equation

2

2

d

d

x

y2

x

y

d

d4y3 ex"

(4

1b 6olve the differential equation

2

2

d

d

x

y2

x

y

d

d4y3 ex"

given that atx3 %y3 ) andx

y

d

d3 2"

("

.P& January 22 /n 70

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! 2


3/44

4! The curve Chas polar equation r3 *acos % 2

2

" The curveDhas polar equation

r 3 a1) 4 cos % 9 " :iven that ais a positive constant%

1a s$etch% on the same diagram% the graphs of CandD% indicating where each curve cuts theinitial line"

(4The graphs of C intersect at the pole Oand at the pointsPand Q"

1b Find the polar coordinates ofPand Q"(3

1c ;se integration to find the exact value of the area enclosed by the curve Dand the lines

3 and 3*

"

(#

The regionRcontains all points which lie outsideDand inside C.

:iven that the value of the smaller area enclosed by the curve Cand the line 3*

is

)(

* 2a12**%

1d show that the area ofRis a2"(4

.P& January 22 /n


4/44

2 yt

y

t

y*

d

d7

d

d2

2

++ 3 *t24 ))t"

($

1b Find the particular solution of this differential equation for which y3 ) andt

y

d

d3 )

when t3 "(5

1c For this particular solution% calculate the value ofywhen t3 )"(1

.P& June 22 /n 70

$! Fi%ure 1

The curve C shown in Fig" ) has polar equation

r3 a1* 4 ' cos % 9 "

1a Find the polar coordinates of the points Pand Qwhere the tangents to Care parallel tothe initial line"

(6

The curve Crepresents the perimeter of the surface of a swimming pool" The direct distancefromPto Qis 2 m"

1b >alculate the value of a"(3

1c Find the area of the surface of the pool"(6

.P& June 22 /n


5/44

"! 1a The pointPrepresents a complex numberzin an ?rgand diagram" :iven that

z+ 2i3 2z4 i%

1i find a cartesian equation for the locus ofP% simplifying your answer"(2

1ii s$etch the locus ofP"(3

1b ? transformation Tfrom thez=plane to the w=plane is a translation 7 4 ))i followed byan enlargement with centre the origin and scale factor *"

5rite down the transformation Tin the form

w3 az4 b% a% b"(2

.P( June 22 /n *0

10!2

2

2

d

d

d

d

+x

y

x

yy 4y3 "

1a Find an expression for*

*

d

d

x

y"

(5

:iven thaty3 ) andx

y

d

d3 ) atx3 %

1b find the series solution fory% in ascending powers ofx% up to an including the term inx*"(5

1c >omment on whether it would be sensible to use your series solution to give estimatesforyatx3 "2 and atx3 '"

(2

.P( June 22 /n &0

11! z3

+

&sini

&cos&

% and w3

+

*

2sini

*

2cos*

"

Expresszwin the form r1cos 4 i sin % r- % 9 9 "(3

.P& January 2* /n )0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! '


6/44

12! 1a Express*1)1

2

++ rr in partial fractions"

(2

1b @ence prove that = ++

+

++

n

r nn

nn

rr) *121(

)*'1

*1)1

2"

(5

.P& January 2* /n *0

13! 1a 6$etch% on the same axes% the graphs with equation y 3 2x + *% and the line withequationy 3 'x+ )"

(2

1b 6olve the inequality 2x+ *9 'x+ )"(3

.P& January 2* /n 20

14! 1a ;se the substitutiony3 vxto transform the equation

x

y

d

d3 2

1&1

x

yxyx ++%x- 1A

into the equation x

x

v

d

d3 12 4 v2" 1AA

(4

1b 6olve the differential equation AA to find vas a function ofx"(5

1c @ence show that

y3 2xcx

x

+ln% where cis an arbitrary constant%

is a general solution of the differential equation A"

(1

.P& January 2* /n '0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! (


7/44

15! 1a Find the value of for which xcos *xis a particular integral of the differential equation

2

2

d

d

x

y4 !y3 )2 sin *x"

(4

1b @ence find the general solution of this differential equation"(4

The particular solution of the differential equation for which y3 ) andx

y

d

d32 atx3 % is

y 3 g1x"

1c Find g1x"(4

1d 6$etch the graph ofy3 g1x% x"

(2

.P& January 2* /n 70



8/44

16! Fi%ure 1

Figure ) shows a s$etch of the cardioid Cwith equation r3 a1) 4 cos % 9 " ?lsoshown are the tangents to C that are parallel and perpendicular to the initial line" Thesetangents form a rectangle WXYZ"

1a Find the area of the finite region% shaded in Fig" )% bounded by the curve C"(6

1b Find the polar coordinates of the pointsA andBwhere WZtouches the curve C"(5

1c @ence find the length of WX"(2

:iven that the length of WZis2

** a %

1d find the area of the rectangle WXYZ"(1

? heart=shape is modelled by the cardioid C% where a3 ) cm" The heart shape is cut fromthe rectangular card WXYZ% shown in Fig" )"

1e Find a numerical value for the area of card wasted in ma$ing this heart shape"(2

.P& January 2* /n


9/44

1#! 1a Express as a simplified fraction( )

2 2

) )

)r r

"

(2

1b Prove% by the method of differences% that

=

=n

r nrr

r

2222

))

)1

)2"

(3

.P& June 2* /n )0

1$! 6olve the inequality2*)2

)>+ x

xx

"

(6

.P& June 2* /n 20

1"! 1a ;sing the substitution t3x2% or otherwise% find

xx x de2

*

"

(6

1b Find the general solution of the differential equation

2

e*d

d xxyx

yx =+ % x- "

(4

.P& June 2* /n (0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! !


10/44

20! Fi%ure 1

D

O B A Anitial line

C

? logo is designed which consists of two overlapping closed curves"

The polar equations of these curves are

r3 a1* 4 2cos and

r3 a1' + 2 cos % 9 2"

Figure ) is a s$etch 1not to scale of these two curves"

1a 5rite down the polar coordinates of the pointsAandBwhere the curves meet the initialline"

(2

1b Find the polar coordinates of the points CandDwhere the two curves meet"(4

1c 6how that the area of the overlapping region% which is shaded in the figure% is

*

2a

1&!&


11/44

21!t

y

t

y

d

d(

d

d2

2

& !y3 &e*t% t"

1a 6how thatKt2e*tis a particular integral of the differential equation% whereKis a constantto be found"

(4

1b Find the general solution of the differential equation"(3

:iven that a particular solution satisfiesy3 * andt

y

d

d3 ) when t3 %

1c find this solution"(4

?nother particular solution which satisfiesy3 ) and tydd

3 when t3 % has equation

y3 1) + *t4 2t2e*t"

1d For this particular solution draw a s$etch graph ofyagainst t% showing where the graphcrosses the t=axis" Betermine also the coordinates of the minimum of the point on thes$etch graph"

(5

.P& June 2* /n


12/44

22! 1i 1a Cn the same?rgand diagram s$etch the loci given by the following equations"

"2

)arg1

%)2

)arg1

%))

=+

=+

=

z

z

z

(4

1b 6hade on your diagram the region for which

"2

)arg1)2

and)) + zz

(1

1ii 1a 6how that the transformation

%%) = z

z

zw

maps )) =z in thez=plane onto )= ww in the w=plane"(3

The region )) z in thez=plane is mapped onto the region Tin the w=plane"

1b 6hade the region Ton an ?rgand diagram"

(2

.P( June 2* /n &0

23. 1a ;se de #oivres theorem to show that

"cos'cos2cos)('cos *' +=(6

1b @ence find * distinct solutions of the equation %)'2)( *' =++ xxx giving youranswers to * decimal places where appropriate"

(4

.P( June 2* /n '0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + ,ersion 2 + #arch 2! )*


13/44

24! Prove by the method of differences that =

n

r

r)

23 (

)n1n4 )12n4 )% n- )"

(6

.P& January 2& /n )0

25! 2)*

)d

d

xxy

x

y=

++ % x- "

1a ,erify thatx*exis an integrating factor for the differential equation"(3

1b Find the general solution of the differential equation"(4

1c :iven thaty3 ) atx3)% findyatx3 2"(3

.P& January 2& /n &0

26! 1a 6$etch% on the same axes% the graph ofy3 |1x+ 21x+ &|% and the line with equationy 3 ( + 2x"

(4

1b Find the exact values ofxfor which |1x+ 21x+ &|3 ( + 2x" (5

1c @ence solve the inequality |1x+ 21x+ &|< ( + 2x"

(2

.P& January 2& /n '0

2#!2

2

d

d

x

y4 &

x

y

d

d4 'y3 (' sin 2x% x- "

1a Find the general solution of the differential equation"("

1b 6how that for large values of x this general solution may be approximated by a sinefunction and find this sine function"

(2

.P& January 2& /n (0

2$! 1a 6$etch the curve with polar equation

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! )&


14/44

r3 * cos 2% +&

9&

"

(2

1b Find the area of the smaller finite region enclosed between the curve and the half=line

3 (

"(6

1c Find the exact distance between the two tangents which are parallel to the initial line"($

.P& January 2& /n 70

2"! Find the complete set of values ofxfor which

|x22|- 2x"(#

.P& June 2& /n &0


x

y

d

d4 2y3x"

(5

:iven thaty3 ) atx3 %

1b find the exact values of the coordinates of the minimum point of the particular solutioncurve%

(4

1c draw a s$etch of this particular solution curve"(2

.P& June 2& /n (0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! )'


15/44


2

2

d

d

t

y4 2

t

y

d

d4 2y3 2et"

(6

1b Find the particular solution that satisfiesy3 ) andt

y

d

d3 ) at t3 "

(6

.P& June 2& /n 70

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! )(


16/44

32! Fi%ure 1

Figure ) is a s$etch of the two curves C)and C2with polar equations

C)D r3 *a1) cos %


17/44

33! :iven thaty3 tanx%

1a findx

y

d

d%

2

2

d

d

x

yand

*

*

d

d

x

y"

(3

1b Find the Taylor series expansion of tan xin ascending powers of

&

x up to and

including the term in*

&

x "

(3

1c @ence show that tan*2)

))

* *2 +++ "

(2

.P( June 2& /n 20

34! 1a Prove by induction thatn

x

n

n

xx

2)

2cose1d

d = excos 1x4 &)

n% n)"

($

1b Find the #aclaurin series expansion of excosx% in ascending powers ofx% up to andincluding the term inx&"

(3

.P( June 2& /n &0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! )


18/44

35! The transformation Tfrom the complexz=plane to the complex w=plane is given by

w3i

)

+

+

z

z% zi"

1a 6how that Tmaps points on the half=line arg1z 3 &

in thez=plane into points on thecirclew3 ) in the w=plane"

(4

1b Find the image under Tin the w=plane of the circle z3 ) in thez=plane"(6

1c 6$etch on separate diagrams the circle z3 ) in thez=plane and its image under Tin the

w=plane"(2

1d #ar$ on your s$etches the pointP% wherez3 i% and its image Qunder Tin the w=plane"(2

.P( June 2& /n 70

36! 1a 6$etch the graph of y3 x+ 2a% given that a- "(2

1b 6olve x+ 2a- 2x4 a% where a- "(3

.FP)P& January 2' /n )0

3#! Find the general solution of the differential equation

x

y

d

d4 2ycot 2x3 sinx% 9x9

2

%

giving your answer in the formy3 f1x"(#

.FP)P& January 2' /n *0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! )!


19/44

3$! 1a Express21

)

+rr in partial fractions"

(2

1b @ence prove% by the method of differences% that

= +n

r rr) 21

&3

21)1

'*1

+++nn

nn"

(5

1c Find the value of = +

)

' 21

&

r rr% to & decimal places"

(3

.FP)P& January 2' /n '0

3"! 1a 6how that the transformationy3xvtransforms the equation

x22

2

d

d

x

y+ 2x

x

y

d

d4 12 4 !x2y3x'% A

into the equation

2

2

d

d

x

v4 !v3x2" AA

(5

1b 6olve the differential equation AA to find vas a function ofx"(6

1c @ence state the general solution of the differential equation A"(1

.FP)P& January 2' /n (0



20/44

40! The curve Chas polar equation r3 ( cos % +2

92

%

and the lineDhas polar equation r3 * sec

*% +

(

9(

'"

1a Find a cartesian equation of Cand a cartesian equation ofD"(5

1b 6$etch on the same diagram the graphs of C and D% indicating where each cuts theinitial line"

(3

The graphs of CandDintersect at the pointsPand Q"

1c Find the polar coordinates ofPand Q"(5

.FP)P& January 2' /n 70

41! 1a y expressing)&

22 r

in partial fractions% or otherwise% prove that

=

n

r r)2 )&

23 ) +

)2

)

+n"

(3

1b @ence find the exact value of =

2

))2 )&

2

r r"

(2

.FP)P& June 2' /n )0

42! Find the general solution of the differential equation

1x4 ) xy

d

d

4 2y3 x)

% x- "

giving your answer in the formy3 f1x"(#

.FP)P& June 2' /n *0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! 2)


21/44

43! 1a Cn the same diagram% s$etch the graphs of y3 x2+ & andy3 2x+ ) % showing thecoordinates of the points where the graphs meet the axes"

(4

1b 6olve x2+ & 3 2x+ ) % giving your answers in surd form where appropriate"(5

1c @ence% or otherwise% find the set of values ofxfor which of x2+ & - 2x+ ) "(3

.FP)P& June 2' /n (0


22

2

ddtx 4 '

txdd 4 2x3 2t4 !"

(6

1b Find the particular solution of this differential equation for which x3 * andt

x

d

d3 +)

when t3 "(4

The particular solution in part 1b is used to model the motion of a particlePon thex=axis" ?ttime tseconds 1t%Pisxmetres from the origin O"

1c 6how that the minimum distance between OandPis 2) 1' 4 ln 2 m and Gustify that the

distance is a minimum"(4

.FP)P& June 2' /n 70



22/44

45! Fi%ure 1

The curve Cwhich passes through Ohas polar equation

r3 &a1) 4 cos % + 9 "

The line lhas polar equation

r3 *asec % +2

9 92

"

The line lcuts Cat the pointsPand Q% as shown in Figure )"

1a Prove thatPQ3 (*a"(6

The regionR% shown shaded in Figure )% is bounded by land C"

1b ;se calculus to find the exact area ofR"(#

.FP)P& June 2' /n


23/44

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! 2&


24/44

46! ? complex numberzis represented by the pointPin the ?rgand diagram" :iven that

z+ *i 3 *%

1a s$etch the locus ofP"

(2

1b Find the complex numberzwhich satisfies both z+ *i 3 * and arg 1z+ *i 3 &* "

(4

The transformation Tfrom thez=plane to the w=plane is given by

w3w

i2"

1c 6how that Tmaps z+ *i 3 * to a line in the w=plane% and give the cartesian equation of

this line"(5

.FP*P( June 2' /n &0

4#! 1a :iven thatz3 ei% show that

zn+ nz

)3 2i sin n%

where nis a positive integer" (2

1b 6how that

sin' 3)(

)1sin ' + ' sin * 4 ) sin "

(5

1c @ence solve% in the interval 9 2%

sin '+ ' sin * 4 ( sin 3 "

(5

.FP*P( June 2' /n '0

FP2 questions from old P&% P'% P( and FP)% FP2% FP* papers + 8 ,ersion #arch 2! 2'


25/44

4$! Find the set of values ofxfor which

2

2

xx

- 2x"

(6

.FP)P& January 2( /n 20

4"! 1a Find the general solution of the differential equation

2

2

d

d

t

x4 2

t

x

d

d4 'x3 "

(4

1b :iven that x3 ) andt

x

d

d3 ) at t3 % find the particular solution of the differential

equation% giving your answer in the formx3 f1t"(5

1c 6$etch the curve with equation x3 f1t% t% showing the coordinates% as multiplesof % of the points where the curve cuts the t=axis"

(4

.FP)P& January 2( /n &0

50! 1a 6how that the substitutiony3 vxtransforms the differential equation

x

y

d

d3

yx

yx

*&

&*

+

1A

into the differential equation

xx

v

d

d3 +

&*

*


26/44

51! Fi%ure 1

? curve Chas polar equation r23 a2cos 2% &

" The line lis parallel to the initial

line% and lis the tangent to Cat the pointP% as shown in Figure )"

1a 1i 6how that% for any point on C% r2sin2can be expressed in terms of sin and aonly"(1

1ii @ence% using differentiation% show that the polar coordinates ofPare

(

%2

a"

(6

The shaded region R% shown in Figure )% is bounded by C% the line land the half=line with

equation 32

"

1b 6how that the area ofRis)(

2a

1** + &"

($

.FP)P& January 2( /n 70

52! 6olve the equation

z'3 i%

giving your answers in the form cos 4 i sin "(5

.FP*P( January 2( /n )0

53! 1) 4 2xx

y

d

d3x4 &y2"


3

C

lP

O3

R


27/44

1a 6how that

1) 4 2x2

2

d

d

x

y3 ) 4 21&y+ )

x

y

d

d" 1)

(2

1b Bifferentiate equation 1) with respect toxto obtain an equation involving

*

*

d

d

x

y%

2

2

d

d

x

y%

x

y

d

d%xandy"

(3

:iven thaty3 2) atx3 %

1c find a series solution fory% in ascending powers ofx% up to and including the term inx*"

(6

.FP*P( January 2( /n (0

54! An the ?rgand diagram the pointPrepresents the complex numberz"

:iven that arg

+

2

i2

z

z3

2

%

1a s$etch the locus ofP%(4

1b deduce the value of z4 ) + i"(2

The transformation Tfrom thez=plane to thew=plane is defined by

w32

i)12

++

z% z+2"

1c 6how that the locus ofPin thez=plane is mapped to part of a straight line in the w=plane%and show this in an ?rgand diagram"

(6

.FP*P( January 2( /n


28/44

55! Fi%ure 1

Figure ) shows a curve Cwith polar equation r3 &acos 2% &

% and a line with

polar equation 3

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