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Chigwell School

C4 by topic

1447 min1206 marks

1. f(x) = )21)(1(

141

xx

x

++

, 2

1

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2. The function f is given by

f(x) = )1)(2(

)1(3

+

+

xx

x

,x ,x 2,x 1.

(a) Express f(x) in partial fractions.(3)

(b) Hence, or otherwise, prove that f (x) < 0 for all values ofx in the domain.(3)

(Total 6 marks)

3. (a) Express )1()32(

213

+

xx

x

in partial fractions.(4)

(b) Given thaty = 4 atx = 2, use your answer to part (a) to find the solution of the differentialequation

x

y

d

d

= )1()32(

)213(

+

xx

xy

, x > 1.5

Express your answer in the formy = f(x).(7)

(Total 11 marks)

4. Given that

,131)1)(31(

53

x

B

x

A

xx

x

+

+

++

(a) find the values of the constantsA andB.(3)

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(b) Hence, or otherwise, find the series expansion in ascending powers ofx, up to andincluding the term inx2, of

)1)(31(

53

xx

x

++

.(5)

(c) State, with a reason, whether your series expansion in part (b) is valid forx = 21

.(2)

(Total 10 marks)

5.

.)21(

13)(f

21

2

+=+= t

tytx

The finite regionR between the curve Cand thex-axis, bounded by the lines with equationsx = 1n 2 andx = 1n 4, is shown shaded in the diagram above.

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(a) Show that the area ofR is given by the integral

t

tt

d

)2)(1(

12

0

++ .

(4)

(b) Hence find an exact value for this area.(6)

(c) Find a cartesian equation of the curve C, in the formy = f(x).(4)

(d) State the domain of values forx for this curve.(1)

(Total 15 marks)

19.

xO

y

P

l

C

4

R

The diagram above shows the curve Cwith parametric equations

20,2sin4,cos8

== ttytx

.

The pointPlies on Cand has coordinates (4, 23).

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(a) Find the value of tat the pointP.(2)

The line lis a normal to CatP.

(b) Show that an equation forlisy = x3 + 63.(6)

The finite regionR is enclosed by the curve C, thex-axis and the linex = 4, as shown shaded inthe diagram above.

(c) Show that the area ofR is given by the integral 2

3

2

dcossin64

ttt.

(4)

(d) Use this integral to find the area ofR, giving your answer in the form a + b3, where aand b are constants to be determined.

(4)

(Total 16 marks)

20. The binomial expansion of4

3

)121( x+ in ascending powers ofx up to and including theterm inx3 is

1 + 9x +px2 + qx3, 12x< 1.

(a) Find the value of p and the value ofq.(4)

(b) Use this expansion with your values ofp and q together with an appropriate value ofx

to obtain an estimate of4

3

)6.1( .(2)

(c) Obtain4

3

)6.1( from your calculator and hence make a comment on the accuracy of theestimate you obtained in part (b).

(2)

(Total 8 marks)

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21. (a) Expand (1 + 3x)2, x< 31

, in ascending powers ofx up to and including the terminx3, simplifying each term.

(4)

(b) Hence, or otherwise, find the first three terms in the expansion of

2)31(

4

x

x

++

as a series inascending powers ofx.(4)

(Total 8 marks)

22. f(x) = )1()23(

252 xx + , x< 1.

(a) Express f(x) as a sum of partial fractions.(4)

(b) Hence find xx d)(f .(5)

(c) Find the series expansion of f(x) in ascending powers ofx up to and including the term inx

2. Give each coefficient as a simplified fraction.(7)

(Total 16 marks)

23. When (1 + ax)n is expanded as a series in ascending powers ofx, the coefficients ofx andx2

are 6 and 27 respectively.

(a) Find the value of a and the value ofn.(5)

(b) Find the coefficient ofx3.(2)

(c) State the set of values ofx for which the expansion is valid.(1)

(Total 8 marks)

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24. (a) Show that=

++n

r

rr1

)5)(1(

= 6

1

n(n+ 7)(2n + 7).(4)

(b) Hence calculate the value of=

++40

10

)5)(1(r

rr

(2)

(Total 6 marks)

25. f(x) =)1(

1

x(1 +x), 1

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( ) .12,200100 21

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31. (a) Use the binomial theorem to expand

3

8,)38( 3

1

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(a) Prove that x

y

d

d

= a

b

cosec t.(4)

(b) Find the equation in the formy =px + q of the tangent to Cat the point where t= 4

.(4)

(Total 8 marks)

34. A curve has equation

x32xy4x +y351 = 0.

Find an equation of the normal to the curve at the point (4, 3), giving your answer in the formax + by + c = 0, where a, b and c are integers.

(Total 8 marks)

35.

y

O x

M

The diagram above shows the curve with equationy =

2

1

xe

2x

.(a) Find thex-coordinate ofM, the maximum point of the curve.

(5)

The finite region enclosed by the curve, thex-axis and the linex = 1 is rotated through 2aboutthex-axis.

(b) Find, in terms ofand e, the volume of the solid generated.(7)

(Total 12 marks)

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36. (a) Use the identity for cos (A +B) to prove that cos 2A = 2 cos2A 1.(2)

(b) Use the substitutionx = 22 sin to prove that

6

2

2d)8( xx

= 31 (+ 33 6).(7)

A curve is given by the parametric equations

x = sec , y = ln(1 + cos 2), 0 < 2

.

(c) Find an equation of the tangent to the curve at the point where = 3

.

(5)(Total 14 marks)

37.

y

x

M

O

P

N

R

The curve Cwith equationy = 2ex + 5 meets they-axis at the pointM, as shown in the diagramabove.

(a) Find the equation of the normal to CatMin the form ax + by = c, where a, b and c areintegers.

(4)

This normal to CatMcrosses thex-axis at the pointN(n, 0).

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(b) Show that n = 14.(1)

The pointP(ln 4, 13) lies on C. The finite regionR is bounded by C, the axes and the linePN, asshown in the diagram above.

(c) Find the area of R, giving your answer in the formp + q ln 2, wherep and q are integersto be found.

(7)

(Total 12 marks)

38.

A

O x

y

The diagram above shows a graph ofy =x sinx, 0

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(a) Find the gradient of the curve atP.(5)

(b) Find the equation of the normal to the curve CatP, in the formy = ax + b, where a and bare constants.

(3)(Total 8 marks)

40. f(x) =x + 5

ex

, x .

(a) Find f (x).

(2)

The curve C, with equationy = f(x), crosses they-axis at the pointA.

(b) Find an equation for the tangent to CatA.(3)

(c) Complete the table, giving the values of

+

5

exx

to 2 decimal places.

x 0 0.5 1 1.5 2

+

5

exx

0.45 0.91

(2)

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(d) Use the trapezium rule, with all the values from your table, to find an approximation forthe value of

xxx

d5e

2

0

+

.(4)

(Total 11 marks)

41. A drop of oil is modelled as a circle of radius r. At time t

r= 4(1 et

), t> 0,

where is a positive constant.

(a) Show that the areaA of the circle satisfies

t

A

d

d

= 32(et e2

t).(5)

In an alternative model of the drop of oil its areaA at time tsatisfies

2

2

3

d

d

t

A

t

A=

, t> 0.

Given that the area of the drop is 1 at t= 1,

(b) find an expression forA in terms oftfor this alternative model.(7)

(c) Show that, in the alternative model, the value ofA cannot exceed 4.

(1)

(Total 13 marks)

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42. The curve Cwith equationy = k+ ln 2x, where kis a constant, crosses thex-axis at the pointA

0,e2

1

.

(a) Show that k= 1.(2)

(b) Show that an equation of the tangent to CatA isy = 2ex 1.(4)

(c) Complete the table below, giving your answers to 3 significant figures.

x 1 1.5 2 2.5 3

1 + ln 2x 2.10 2.61 2.79(2)

(d) Use the trapezium rule, with four equal intervals, to estimate the value of

+

3

1 d)2ln1( xx .(4)

(Total 12 marks)

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43.

y

xO

C

P

Q

R

The diagram shows a sketch of part of the curve Cwith parametric equations

x = t2 + 1, y = 3(1 + t).

The normal to Cat the pointP(5, 9) cuts thex-axis at the point Q, as shown in the diagram.

(a) Find thex-coordinate ofQ.(6)

(b) Find the area of the finite regionR bounded by C, the linePQ and thex-axis.(9)

(Total 15 marks)

44. The value Vof a cartyears after the 1st January 2001 is given by the formula

V= 10 000 (1.5)t.

(a) Find the value of the car on 1st January 2005.(2)

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(b) Find the value of t

V

d

d

when t= 4.(3)

(c) Explain what the answer to part (b) represents.(1)

(Total 6 marks)

45. A curve has equation

x2 + 2xy 3y2 + 16 = 0.

Find the coordinates of the points on the curve where x

y

d

d

= 0.(Total 7 marks)

46. A curve Cis described by the equation

3x2 2y2 + 2x 3y+ 5 = 0.

Find an equation of the normal to Cat the point (0, 1), giving your answer in the formax+by+c= 0, where a, b and c are integers.

(Total 7 marks)

47. f(x) = (x2

+ 1) lnx, x > 0.(a) Use differentiation to find the value of f'(x) atx = e, leaving your answer in terms of e.

(4)

(b) Find the exact value of

e

1d)(f xx

(5)

(Total 9 marks)

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48. A curve Cis described by the equation

3x2 + 4y2 2x + 6xy 5 = 0.

Find an equation of the tangent to Cat the point (1, 2), giving your answer in the formax + by + c = 0, where a, b and c are integers.

(Total 7 marks)

49. The volume of a spherical balloon of radius rcm is Vcm3, where V= 34

r3.

(a) Find r

V

d

d

(1)

The volume of the balloon increases with time tseconds according to the formula

.0,)12(

1000

d

d2

+

= ttt

V

(b) Using the chain rule, or otherwise, find an expression in terms ofrand tfor.

d

d

t

r

(2)

(c) Given that V = 0 when t= 0, solve the differential equation2

)12(

1000

d

d

+=

tt

V

, to obtain Vin terms oft.

(4)

(d) Hence, at time t= 5,

(i) find the radius of the balloon, giving your answer to 3 significant figures,(3)

(ii) show that the rate of increase of the radius of the balloon is approximately2.90 102 cm s1.

(2)

(Total 12 marks)

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50. A curve has parametric equations

x = 7cos t cos7t,y = 7 sin t sin 7t, 38

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(b) Find the gradient of the curve with equation)(

2

2 xy = at the point with coordinates(2,16).

(4)

(Total 6 marks)

53. A curve is described by the equation

x3 4y2 = 12xy.

(a) Find the coordinates of the two points on the curve wherex = 8.(3)

(b) Find the gradient of the curve at each of these points.(6)

(Total 9 marks)

54.

5 x

x

The diagram above shows a right circular cylindrical metal rod which is expanding as it isheated. Aftertseconds the radius of the rod isx cm and the length of the rod is 5x cm. Thecross-sectional area of the rod is increasing at the constant rate of 0.032 cm2 s1.

(a) Find t

x

d

d

when the radius of the rod is 2 cm, giving your answer to 3 significant figures.(4)

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(b) Find the rate of increase of the volume of the rod whenx = 2.(4)

(Total 8 marks)

55. A curve has equation 3x2y2 +xy = 4. The pointsPand Q lie on the curve. The gradient of the

tangent to the curve is 3

8

atPand at Q.

(a) Use implicit differentiation to show thaty 2x = 0 atPand at Q.(6)

(b) Find the coordinates ofPand Q. (3)(Total 9 marks)

56. Find the volume generated when the region bounded by the curve with equation

y = 2 + x

1

, thex-axis and the linesx = 21

andx = 4 is rotated through 360aboutthex-axis.

Give your answer in the form(a + b ln 2), where a and b are rational constants.

.....(Total 7 marks)

57. Given thaty = 1 atx = , solve the differential equation

x

y

d

d

= yx2 cosx, y > 0.

(Total 9 marks)

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58.

g(x) =)2)(41(

85

xx

x

++

.

(a) Express g(x) in the form )2()41( x

B

x

A

+

+ , whereA andB are constants to be found.(3)

The finite regionR is bounded by the curve with equationy = g(x), the coordinate axes and the

linex = 21

.

(b) Find the area of R, giving your answer in the form a ln 2 + b ln 3.(7)

(Total 10 marks)

59. Use the substitution u2 = (x 1) to find

x

x

xd

)1(

2

,

giving your answer in terms ofx.(Total 10 marks)

60. Use the substitution u = 4 + 3x2 to find the exact value of

xx

xd

)34(

22

0

22

+ . (Total 6 marks)

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61.

x

y

R

O 2

The diagram above shows the curve with equation

y =x2 sin ( 21

x), 0

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(ii) atx = , x = 4

5

, x = 2

3

, x = 4

7

andx = 2 to find an improvedapproximation for the areaR, giving your answer to 4 significant figures.

(5)(Total 13 marks)

62.

y

xO 41

R

The diagram above shows part of the curve with equationy = 1 + x2

1

. The shaded regionR,bounded by the curve, thex-axis and the linesx = 1 andx = 4, is rotated through 360about the

x-axis. Using integration, show that the volume of the solid generated is (5 + 21

ln 2).(Total 8 marks)

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63.

x

y

O 1 2

R

The diagram above shows part of the curve with equationy = 1 + x

c

, where c is a positiveconstant.

The pointPwithx-coordinatep lies on the curve. Given that the gradient of the curve atPis 4,

(a) show that c = 4p2.(2)

Given also that they-coordinate ofPis 5,

(b) prove that c = 4.(2)

The regionR is bounded by the curve, thex-axis and the linesx = 1 andx = 2, as shown in thediagram above. The regionR is rotated through 360about thex-axis.

(c) Show that the volume of the solid generated can be written in the form (k+ q ln 2),where kand q are constants to be found.

(7)

(Total 11 marks)

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64.

y

xO

C

A1

R

The diagram above shows the curve Cwith equationy = f(x), where

f(x) = x

8

x2, x > 0.

Given that Ccrosses thex-axis at the pointA,

(a) find the coordinates ofA.(3)

The finite regionR, bounded by C, thex-axis and the linex = 1, is rotated through 2radiansabout thex-axis.

(b) Use integration to find, in terms ofthe volume of the solid generated.(7)

(Total 10 marks)

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65.

O x

y

R

1 4

C

The diagram above shows parts of the curve Cwith equation

y = x

x

+ 2

.

The shaded regionR is bounded by C, thex-axis and the linesx = 1 andx = 4.

This region is rotated through 360 about thex-axis to form a solid S.

(a) Find, by integration, the exact volume ofS.

(7)

The solid Sis used to model a wooden support with a circular base and a circular top.

(b) Show that the base and the top have the same radius.(1)

Given that the actual radius of the base is 6 cm,

(c) show that the volume of the wooden support is approximately 630 cm3.(2)

(Total 10 marks)

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66. Use the substitution u = 1 + sinx and integration to show that

sinx cosx (1 + sinx)5

dx = 42

1

(1 + sinx)6

[6 sinx 1] + constant.(Total 8 marks)

67.

y

O

R

x2 4

The diagram above shows part of the curve with equation

y = 4x x

6

, x > 0.

The shaded regionR is bounded by the curve, thex-axis and the lines with equationsx = 2 andx = 4. This region is rotated through 2radians about thex-axis.

Find the exact value of the volume of the solid generated.(Total 8 marks)

68. (a) Use the formulae for sin (A B), withA = 3x andB =x, to show that 2 sinx cos 3xcan be written as sinpx sin qx, wherep and q are positive integers.

(3)

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(b) Hence, or otherwise, find dxxx 3cossin2 .(2)

(c) Hence find the exact value of65

2

3cossin2

dxxx

(2)

(Total 7 marks)

69. (a) Use integration by parts to show that

+ xxx d

6cosec2

= x cot

+

6

x

+ ln

+

6sin

x

+ c, 6

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70.

x

yC

1 3

( 1 , 2 )

y =x

x+ 1

O

Figure 1

Figure 1 shows part of the curve Cwith equationy = x

x 1+

, x > 0.

The finite region enclosed by C, the linesx = 1,x = 3 and thex-axis is rotated through 360about thex-axis to generate a solid S.

(a) Using integration, find the exact volume ofS.(7)

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x

yC

1 3

( 1 , 2 )

y =x

x+ 1

O

R

T

Figure 2

The tangent Tto Cat the point (1, 2) meets thex-axis at the point (3, 0). The shaded regionR isbounded by C, the linex = 3 and T, as shown in Figure 2.

(b) Using your answer to part (a), find the exact volume generated byR when it is rotatedthrough 360about thex-axis.

(3)

(Total 10 marks)

71. (a) Use integration by parts to find

xxx d2cos

.(4)

(b) Hence, or otherwise, find

xxx dcos2

.(3)

(Total 7 marks)

72. (a) Express )2)(32(

35

+

+

xx

x

in partial fractions.(3)

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(b) Hence find the exact value ofx

xx

xd

)2)(32(

356

2

++

, giving your answer as a singlelogarithm.

(5)

(Total 8 marks)

73. Use the substitutionx = sin to find the exact value of

xx

d)1(

121

2

3

02

.(Total 7 marks)

74.

x

y

R

0 0 . 2 0 . 4 0 . 6 0 . 8 1

The diagram shows the graph of the curve with equation

y =xe

2x

, x 0.The finite regionR bounded by the linesx = 1, thex-axis and the curve is shown shaded in thediagram.

(a) Use integration to find the exact value of the area forR.(5)

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(b) Complete the table with the values ofy corresponding tox = 0.4 and 0.8.

x 0 0.2 0.4 0.6 0.8 1y =xe2x 0 0.29836 1.99207 7.38906

(1)

(c) Use the trapezium rule with all the values in the table to find an approximate value forthis area, giving your answer to 4 significant figures.

(4)

75.

y

O 2 x

The curve with equation,20,

2sin3 = x

xy

, is shown in the figure above. The finiteregion enclosed by the curve and thex-axis is shaded.

(a) Find, by integration, the area of the shaded region.(3)

This region is rotated through 2radians about thex-axis.

(b) Find the volume of the solid generated.(6)

(Total 9 marks)

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76.

y

O x1

The figure above shows a sketch of the curve with equationy= (x 1) lnx, x > 0.

(a) Complete the table with the values ofy corresponding tox= 1.5 andx= 2.5.

x 1 1.5 2 2.5 3

y 0 ln 2 2ln 3(1)

Given that =

3

1dln)1( xxxI

(b) use the trapezium rule

(i) with values of y atx= 1, 2 and 3 to find an approximate value forIto 4 significantfigures,

(ii) with values ofy atx= 1, 1.5, 2, 2.5 and 3 to find another approximate value forIto4 significant figures.

(5)

(c) Explain, with reference to the figure above, why an increase in the number of valuesimproves the accuracy of the approximation.

(1)

(d) Show, by integration, that the exact value of

3

123 3lnisdln)1( xxx

.(6)

(Total 13 marks)

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77. Using the substitution u2 = 2x 1, or otherwise, find the exact value of

5

1d

)12(

3x

x

x

(Total 8 marks)

78.

y x= ex

R

O x

y

1 3

The figure above shows the finite shaded region,R, which is bounded by the curvey = xex, the linex = 1, the linex = 3 and thex-axis.

The regionR is rotated through 360 degrees about thex-axis.

Use integration by parts to find an exact value for the volume of the solid generated.(Total 8 marks)

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79.

Figure 1

y

14

12

x0

The curve with equation ( ) 2

1,

213

1>

+= x

xy

, is shown in Figure 1.

The region bounded by the lines 2

1,

4

1 == xx, thex-axis and the curve is shown shaded in

Figure 1.

This region is rotated through 360 degrees about thex-axis.

(a) Use calculus to find the exact value of the volume of the solid generated.(5)

Figure 2

B

A

Figure 2 shows a paperweight with axis of symmetryAB whereAB = 3 cm.A is a point on thetop surface of the paperweight, andB is a point on the base of the paperweight.The paperweight is geometrically similar to the solid in part (a).

(b) Find the volume of this paperweight.(2)

(Total 7 marks)

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80.

( ).de

5

0

13xI

x +=

(a) Given thaty = e(3x+1), complete the table with the values ofy corresponding tox = 2,3 and 4.

x 0 1 2 3 4 5y e1 e2 e4

(2)

(b) Use the trapezium rule, with all the values ofy in the completed table, to obtain anestimate for the original integralI, giving your answer to 4 significant figures.

(3)

(c) Use the substitution t= (3x + 1) to show thatImay be expressed as b

a

t tkte d, giving the

values ofa, b and k.(5)

(d) Use integration by parts to evaluate this integral, and hence find the value ofIcorrect to 4significant figures, showing all the steps in your working.

(5)

(Total 15 marks)

81. Use the substitution u = 2x to find the exact value of

( ) +1

0 2.d

12

2x

x

x

(Total 6 marks)

82. (a) Find .d2cos xxx .(4)

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(b) Hence, using the identity = xxxxx dcosdeduce,1cos22cos22

.(3)

(Total 7 marks)

83.

( )( )( ) ( ) ( )12121212

142 2

+

++

++

x

C

x

BA

xx

x

(a) Find the values of the constantsA,B and C.(4)

(b) Hence show that the exact value of

( )( )( )

xxx

xd

1212

1422

1

2

++

is 2 + ln k, giving the value ofthe constant k.

(6)

(Total 10 marks)

84.

y

1

O a b x

The curve shown in the diagram above has equation )12(

1

+=

xy

. The finite region bounded bythe curve, thex-axis and the linesx = a andx = b is shown shaded in the diagram. This region isrotated through 360 about thex-axis to generate a solid of revolution.

Find the volume of the solid generated. Express your answer as a single simplified fraction, interms ofa and b.

(Total 5 marks)

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85. (i) Find

xx

d2

1n

.(4)

(ii) Find the exact value of

24

2 dsin

x

.(5)

(Total 9 marks)

86. (a) Use integration by parts to findxx x de .

(3)

(b) Hence findxx x de

2 .(3)

(Total 6 marks)

87. On separate diagrams, sketch the curves with equations

(a) y = arcsinx, 1 x 1,

(b) y = secx, 3

x 3

, stating the coordinates of the end pointsof your curves in each case.

(4)

Use the trapezium rule with five equally spaced ordinates to estimate the area of the

region bounded by the curve with equationy = secx, thex-axis and the linesx = 3

and

x = 3

, giving your answer to two decimal places.(4)

(Total 8 marks)

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88. In an experiment a scientist considered the loss of mass of a collection of picked leaves. ThemassMgrams of a single leaf was measured at times tdays after the leaf was picked.

The scientist attempted to find a relationship betweenMand t. In a preliminary model she

assumed that the rate of loss of mass was proportional to the massMgrams of the leaf.

(a) Write down a differential equation for the rate of change of mass of the leaf, usingthis model.

(2)

(b) Show, by differentiation, thatM= 10(0.98)tsatisfies this differential equation.(2)

Further studies implied that the massMgrams of a certain leaf satisfied a modified differential

equation

10 t

M

d

d

= k(10M1), (I)

where kis a positive constant and t 0.

Given that the mass of this leaf at time t= 0 is 10 grams, and that its mass at time t= 10 is 8.5grams,

(c) solve the modified differential equation (I) to find the mass of this leaf at time t= 15.(9)

(Total 13 marks)

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89.

y

O x 2 1 0

1 0

1 2

The diagram above shows the cross-section of a road tunnel and its concrete surround. The

curved section of the tunnel is modelled by the curve with equationy = 8

10sin

x

, in theinterval 0 x 10. The concrete surround is represented by the shaded area bounded by thecurve, thex-axis and the linesx = 2,x = 12 andy = 10. The units on both axes are metres.

(a) Using this model, copy and complete the table below, giving the values ofy to 2 decimalplaces.

x 0 2 4 6 8 10y 0 6.13 0

(2)

The area of the cross-section of the tunnel is given byxy d

10

0

.

(b) Estimate this area, using the trapezium rule with all the values from your table. (4)

(c) Deduce an estimate of the cross-sectional area of the concrete surround.(1)

(d) State, with a reason, whether your answer in part (c) over-estimates or under-estimatesthe true value.

(2)

(Total 9 marks)

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90.

y

xO

R

2

The diagram above shows part of the curve with equation

y = ex cosx, 0 x 2

.

The finite regionR is bounded by the curve and the coordinate axes.

(a) Calculate, to 2 decimal places, they-coordinates of the points on the curve wherex = 0,

6

, 3

and 2

.(3)

(b) Using the trapezium rule and all the values calculated in part (a), find an approximationfor the area ofR.

(4)

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(c) State, with a reason, whether your approximation underestimates or overestimates thearea ofR.

(2)

(Total 9 marks)

91.

x

y

O

C

5

The figure above shows part of the curve Cwith equation .1e206.0 = xy The shaded region

bounded by C, thex-axis and the line with equationx = 5 represents the cross-section of a

skateboarding ramp. The units on each axis are in metres.

(a) Complete the table, showing the heighty of the ramp. Give the values ofy to 3 decimalplaces.

x 0 1 2 3 4 5

y 0 0.062 0.716(3)

(b) Use the trapezium rule, with all the values from your table, to estimate the area of cross-section of the ramp.

(4)

The ramp is made of concrete and is 6 m wide.

(c) Calculate an estimate for the volume of concrete required to make the ramp.(1)

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(d) A builder makes the amount of concrete calculated in part (c). State, with a reason,whether or not there is enough concrete to make the ramp.

(2)

(Total 10 marks)

92. (a) Given thaty = secx, complete the table with the values ofy corresponding to

.4

and8

,16

=x

x 0

16

8

16

3

4

y 1 1.20269(2)

(b) Use the trapezium rule, with all the values fory in the completed table, to obtain an

estimate for40 dsec

xx. Show all the steps of your working, and give your answer to 4

decimal places.(3)

The exact value of40 dsec

xxis ln(1 + 2).

(c) Calculate the % error in using the estimate you obtained in part (b).(2)

(Total 7 marks)

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93.

y

xO

R

4

The diagram above shows part of the curve with equationy = (tanx). The finite regionR,

which is bounded by the curve, thex-axis and the line 4

=x, is shown shaded in the diagram.

(a) Given thaty = (tanx), complete the table with the values ofy corresponding to

16

3and

8,

16

=x, giving your answers to 5 decimal places.

x 016

8

16

3

4

y 0 1

(3)

(b) Use the trapezium rule with all the values ofy in the completed table to obtain anestimate for the area of the shaded regionR, giving your answer to 4 decimal places.

(4)

The regionR is rotated through 2radians around thex-axis to generate a solid of revolution.

(c) Use integration to find an exact value for the volume of the solid generated.(4)

(Total 11 marks)

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94.

R

xO

y

The curve shown in the diagram above has equationy = ex (sinx), 0 x . The finite regionRbounded by the curve and thex-axis is shown shaded in the diagram.

(a) Complete the table below with the values ofy corresponding to 2and

4

=x, giving

your answers to 5 decimal places.

x 04

2

4

3

y 0 8.87207 0

(2)

(b) Use the trapezium rule, with all the values in the completed table, to obtain an estimatedfor the area of the regionR. Give your answers to 4 decimal places.

(4)

(Total 6 marks)

95. The pointsA,B and Chave position vectors 2i +j + k, 5i + 7j + 4kand ij respectively,relative to a fixed origin O.

(a) Prove that the pointsA,B and Clie on a straight line l.(4)

The pointD has position vector 2i +j 3 k.

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(b) Find the cosine of the acute angle between land the line OD.(3)

The pointEhas position vector 3j k.

(c) Prove thatElies on land that OEis perpendicular to OD.(4)

(Total 11 marks)

96. The line 1l has vector equation

+

=

4

2

4

6

5

11

r

, where is a parameter.

The line 2l has vector equation

+

=

5

1

7

13

4

24

r

, whereis a parameter.

(a) Show that the lines l1and l2 intersect.(4)

(b) Find the coordinates of their point of intersection.(2)

Given that is the acute angle between l1and l2,

(c) Find the value of cos . Give your answer in the form k 3, where kis a simplifiedfraction.

(4)

(Total 10 marks)

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97. Referred to a fixed origin O, the pointsA andB have position vectors (i + 2j 3k) and(5i 3j) respectively.

(a) Find, in vector form, an equation of the line l1 which passes throughA andB.(2)

The line l2 has equation r = (4i 4j + 3k) +(i 2j + 2k), whereis a scalar parameter.

(b) Show thatA lies on l2.(1)

(c) Find, in degrees, the acute angle between the lines l1 and l2.(4)

The point Cwith position vector (2i k) lies on l2.

(d) Find the shortest distance from Cto the line l1.(4)

98. Relative to a fixed origin O, the pointA has position vector 3i + 2j k, the pointB has positionvector 5i +j + k, and the point Chas position vector 7i j.

(a) Find the cosine of angleABC.

(4)

(b) Find the exact value of the area of triangleABC.(3)

The pointD has position vector 7i + 3k.

(c) Show thatACis perpendicular to CD.(2)

(d) Find the ratioAD :DB.(2)

(Total 11 marks)

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99. The equations of the lines l1 and l2 are given by

l1: r = i + 3j + 5k+ (i + 2j k),

l2: r = 2i + 3j 4k+(2i +j + 4k),

where andare parameters.

(a) Show that l1 and l2 intersect and find the coordinates ofQ, their point of intersection.(6)

(b) Show that l1 is perpendicular to l2. (2)

The pointPwithx-coordinate 3 lies on the line l1 and the pointR withx-coordinate 4 lies on theline l2.

(c) Find, in its simplest form, the exact area of the trianglePQR.(6)

(Total 14 marks)

100. Relative to a fixed origin O, the vector equations of the two lines l1 and l2 are

l1: r = 9i + 2j + 4k+ t(8i 3j + 5k),

and

l2: r = 16i + j + 10k+s(i 4j + 9k),

where is a constant.

The two lines intersect at the pointA.

(a) Find the value of .(6)

(b) Find the position vector of the pointA.(1)

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(c) Prove that the acute angle between l1 and l2 is 60.(5)

PointB lies on l1 and point Clies on l2. The triangleABCis equilateral with sides of length142.

(d) Find one of the possible position vectors for the pointB and the corresponding positionvector for the point C.

(4)

(Total 16 marks)

101. Relative to a fixed origin O, the pointA has position vector 5j + 5kand the pointB has positionvector 3i + 2j k.

(a) Find a vector equation of the lineL which passes throughA andB.(2)

The point Clies on the lineL and OCis perpendicular toL.

(b) Find the position vector ofC.(5)

The points O,B andA, together with the pointD, lie at the vertices of parallelogram OBAD.

(c) Find, the position vector ofD.(2)

(d) Find the area of the parallelogram OBAD.

(4)(Total 13 marks)

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102. The pointsA andB have position vectors 5j + 11kand ci + dj + 21krespectively, where c and dare constants.

The line l, through the pointsA andB, has vector equation r = 5j + 11k+ (2i +j + 5k), whereis a parameter.

(a) Find the value of c and the value ofd.(3)

The pointPlies on the line l, and OPis perpendicular to l, where O is the origin.

(b) Find the position vector ofP.(6)

(c) Find the area of triangle OAB, giving your answer to 3 significant figures.(4)

(Total 13 marks)

103. The line l1 has vector equation

r =

2

1

3

+

4

1

1

and the line l2 has vector equation

r =

2

4

0

+

0

1

1

,

where andare parameters.

The lines l1 and l2 intersect at the pointB and the acute angle between l1 and l2 is .

(a) Find the coordinates ofB.(4)

(b) Find the value of cos , giving your answer as a simplified fraction. (4)

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The pointA, which lies on l1, has position vectora = 3i +j + 2k.The point C, which lies on l2, has position vectorc = 5i j 2k.The pointD is such thatABCD is a parallelogram.

(c) Show that AB = BC .(3)

(d) Find the position vector of the pointD.(2)

(Total 13 marks)

104. The pointA, with coordinates (0, a, b) lies on the line l1, which has equation

r = 6i + 19jk + (i + 4j 2k).

(a) Find the values ofa and b.(3)

The pointPlies on l1 and is such that OPis perpendicular to l1, where O is the origin.

(b) Find the position vector of pointP.(6)

Given thatB has coordinates (5, 15, 1),

(c) show that the pointsA,PandB are collinear and find the ratioAP:PB.(4)

(Total 13 marks)

105. The pointsA andB have position vectors i j +pkand 7i + qj + 6krespectively, wherep and qare constants.

The line l1, passing through the pointsA andB, has equation

r = 9i + 7j + 7k+ 7k+(2i + 2j + k), where is a parameter.

(a) Find the value of p and the value ofq.(4)

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(b) Find a unit vector in the direction ofAB .(2)

A second line l2 has vector equation

r = 3i + 2j + 3k+(2i +j + 2k), whereis a parameter.

(c) Find the cosine of the acute angle between l1 and l2.(3)

(d) Find the coordinates of the point where the two lines meet.

(5)(Total 14 marks)

106. The line l1 has vector equation

r = 8i + 12j + 14k+ (i + jk),

where is a parameter.

The pointA has coordinates (4, 8, a), where a is a constant. The point B has coordinates(b, 13, 13), where b is a constant. PointsA andB lie on the line l1.

(a) Find the values ofa and b.(3)

Given that the point O is the origin, and that the pointPlies on l1 such that OPis perpendicularto l1

(b) find the coordinates ofP.(5)

(c) Hence find the distance OP, giving your answer as a simplified surd.(2)

(Total 10 marks)

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107. The pointA has position vectora = 2i +2j + kand the pointB has position vectorb = i +j 4k,relative to an origin O.

(a) Find the position vector of the point C, with position vectorc, given by

c = a + b.(1)

(b) Show that OACB is a rectangle, and find its exact area.(6)

The diagonals of the rectangle,AB and OC, meet at the pointD.

(c) Write down the position vector of the pointD.(1)

(d) Find the size of the angleADC.(6)

(Total 14 marks)

108. The line l1 has equation

.

0

1

1

1

0

1

+

= r

The line l2 has equation

+

=

1

1

2

6

3

1

r

.

(a) Show that l1 and l2 do not meet.(4)

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The pointA is on l1 where = 1, and the pointB is on l2 where= 2.

(b) Find the cosine of the acute angle betweenAB and l1.(6)

(Total 10 marks)

109. The pointsA andB have position vectors 2i + 6j kand 3i + 4j + krespectively.

The line l1 passes through the pointsA andB.

(a) Find the vector AB

(2)

(b) Find a vector equation for the line l1.(2)

A second line l2 passes through the origin and is parallel to the vectori + k. The line l1 meetsthe line l2 at the point C.

(c) Find the acute angle between l1 and l2.(3)

(d) Find the position vector of the point C.(4)

(Total 11 marks)

110. With respect to a fixed origin O, the lines l1 and l2 are given by the equations

l1: r = (9i + 10k) +(2i +j k)

l2: r = (3i +j + 17k) +(3i j + 5k)

where and are scalar parameters.

(a) Show that l1 and l2 meet and find the position vector of their point of intersection.(6)

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(b) Show that l1 and l2 are perpendicular to each other.(2)

The pointA has position vector 5i + 7j + 3k.

(c) Show thatA lies on l1.(1)

The pointB is the image ofA after reflection in the line l2.

(d) Find the position vector ofB.(3)

(Total 12 marks)

111. Liquid is poured into a container at a constant rate of 30 cm3 s1. At time tseconds liquid is

leaking from the container at a rate of 152

Vcm3 s1, where Vcm3 is the volume of liquid in the

container at that time.

(a) Show that

15 t

V

d

d

= 2V 450.(3)

Given that V= 1000 when t= 0,

(b) find the solution of the differential equation, in the form V= f(t).(7)

(c) Find the limiting value ofVas t.

(1)(Total 11 marks)

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112. Fluid flows out of a cylindrical tank with constant cross section. At time tminutes, t 0, the

volume of fluid remaining in the tank is Vm3. The rate at which the fluid flows, in m3 min1, is

proportional to the square root ofV.

(a) Show that the depth h metres of fluid in the tank satisfies the differential equation

t

h

d

d

= kh, where kis a positive constant.(3)

(b) Show that the general solution of the differential equation may be written as

h = (A Bt)2, whereA andB are constants.(4)

Given that at time t= 0 the depth of fluid in the tank is 1 m, and that 5 minutes later the depth offluid has reduced to 0.5 m,

(c) find the time, Tminutes, which it takes for the tank to empty.(3)

(d) Find the depth of water in the tank at time 0.5Tminutes.(2)

(Total 12 marks)

113. A spherical balloon is being inflated in such a way that the rate of increase of its volume, Vcm3,with respect to time tseconds is given by

t

V

d

d

= V

k

, where kis a positive constant.

Given that the radius of the balloon is rcm, and that V=34

r

3

,(a) prove that rsatisfies the differential equation

t

r

d

d

=5r

B

, whereB is a constant.(4)

(b) Find a general solution of the differential equation obtained in part (a).(3)

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When t= 0 the radius of the balloon is 5 cm, and when t= 2 the radius is 6 cm.

(c) Find the radius of the balloon when t= 4. Give your answer to 3 significant figures.(5)

(Total 12 marks)

114. Liquid is pouring into a container at a constant rate of 20 cm3 s1 and is leaking out at a rateproportional to the volume of the liquid already in the container.

(a) Explain why, at time tseconds, the volume, Vcm3, of liquid in the container satisfies thedifferential equation

t

V

d

d

= 20 kV,

where kis a positive constant.(2)

The container is initially empty.

(b) By solving the differential equation, show that

V=A +Bekt,

giving the values ofA andB in terms ofk.(6)

Given also that t

V

d

d

= 10 when t= 5,

(c) find the volume of liquid in the container at 10 s after the start. (5)(Total 13 marks)

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115.

At time tseconds the length of the side of a cube isx cm, the surface area of the cube

is Scm2, and the volume of the cube is Vcm3.

The surface area of the cube is increasing at a constant rate of 8 cm2 s1.

Show that

(a),

d

d

x

k

t

x =where kis a constant to be found,

(4)

(b)3

1

2d

dV

t

V =

(4)

Given that V= 8 when t= 0,

(c) solve the differential equation in part (b), and find the value oftwhen V= 162.(7)

(Total 15 marks)

116. The rate of decrease of the concentration of a drug in the blood stream is proportional to theconcentration Cof the drug, which is present at that time. The time tis measured in hours fromthe administration of the drug and Cis measured in micrograms per litre.

(a) Show that this process is described by the differential equation,

d

dkC

t

C =

explaining why kis a positive constant.(1)

(b) Find the general solution of the differential equation, in the form C= f(t).(3)

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After 4 hours, the concentration of the drug in the blood stream is reduced to 10% of its startingvalue C0.

(c) Find the exact value ofk.(4)

(Total 8 marks)

117. (a) Express ( )( )321

12

xx

x

in partial fractions.(3)

(b) Given thatx 2, find the general solution of the differential equation

(2x3)(x1) x

y

d

d

= (2x1)y.(5)

(c) Hence find the particular solution of this differential equation that satisfiesy = 10 atx =2, giving your answer in the formy = f(x).

(4)

(Total 12 marks)

118. A population growth is modelled by the differential equation

,d

dkP

t

P=

wherePis the population, tis the time measured in days and kis a positive constant.

Given that the initial population isP0,

(a) solve the differential equation, givingPin terms ofP0, kand t.(4)

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Given also that k= 2.5,

(b) find the time taken, to the nearest minute, for the population to reach 2P0.(3)

In an improved model the differential equation is given as

,cosd

dtP

t

P=

wherePis the population, tis the time measured in days and is a positive constant.

Given, again, that the initial population isP0 and that time is measured in days,

(c) solve the second differential equation, givingPin terms ofP0, and t.(4)

Given also that = 2.5,

(d) find the time taken, to the nearest minute, for the population to reach 2P0 for the firsttime, using the improved model.

(3)

(Total 14 marks)

119. Liquid is pouring into a large vertical circular cylinder at a constant rate of 1600 cm3 s1 and isleaking out of a hole in the base, at a rate proportional to the square root of the height of theliquid already in the cylinder. The area of the circular cross section of the cylinder is 4000 cm2.

(a) Show that at time tseconds, the height h cm of liquid in the cylinder satisfies thedifferential equation

hkt

h= 4.0

d

d

, where kis a positive constant.(3)

When h = 25, water is leaking out of the hole at 400 cm3 s1.

(b) Show that k = 0.02(1)

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(c) Separate the variables of the differential equation

ht

h02.04.0

d

d =

,to show that the time taken to fill the cylinder from empty to a height of 100 cm is given

by

100

0.d

20

50h

h

(2)

Using the substitution h = (20 x)2, or otherwise,

(d) find the exact value of 100

0.d

20

50h

h

(6)

(e) Hence find the time taken to fill the cylinder from empty to a height of 100 cm, givingyour answer in minutes and seconds to the nearest second.

(1)

(Total 13 marks)

120. (a) Express2

4

2

y in partial fractions.(3)

(b) Hence obtain the solution of

)4(ddcot2 2yxyx =

for whichy = 0 at 3

=x, giving your answer in the form sec2x = g(y).