math paper 3 practice

1. (a) Find the first three terms of the Maclaurin series for ln (1 + ex).(6)

(b) Hence, or otherwise, determine the value of

.204ln)e1ln(2lim

xxx

x

(4)(Total 10 marks)

2. Consider the differential equation

where y = 1 when x = 0.22dd yx

xy

(a) Use Eulers method with step length 0.1 to find an approximate value of y when x = 0.4.(7)

(b) Write down, giving a reason, whether your approximate value for y is greater than or lessthan the actual value of y.

(1)(Total 8 marks)

3. Solve the differential equation

222 23dd xxyy

xyx

given that y = 1 when x = 1. Give your answer in the form y = f(x).(Total 11 marks)

4. The integral In is defined by In = . nxxnn x for ,dsine)1(

(a) Show that I0 =

.)e1(21

(6)

(b) By letting y = x n, show that In = enI0.(4)

(c) Hence determine the exact value of

.xxx dsine0

(5)(Total 15 marks)

5. The exponential series is given by ex =

.0 !n

n

nx

(a) Find the set of values of x for which the series is convergent.(4)

(b) (i) Show, by comparison with an appropriate geometric series, that

ex 1 <

, for 0 < x < 2.x

x22

(ii) Hence show that e <

, for n +.n

nn

1212

(6)

(c) (i) Write down the first three terms of the Maclaurin series for 1 ex and explainwhy you are able to state that

1 ex > x

, for 0 < x < 2.2

2x

(ii) Deduce that e >

+.

nnnn

n

for ,122

22

2

(4)

(d) Letting n = 1000, use the results in parts (b) and (c) to calculate the value of e correct toas many decimal places as possible.

(2)(Total 16 marks)

6. (a) (i) Find the range of values of n for which

exists.xx nd1

(ii) Write down the value of

in terms of n, when it does exist.xx nd1

(7)

(b) Find the solution to the differential equation

(cos x sin x)

+ (cos x + sin x)y = cos x + sin x,xy

dd

given that y = 1 when x =

.2

(8)(Total 15 marks)

7. (a) Find the value of

.

xxx cot1lim

0

(6)

(b) Find the interval of convergence of the infinite series

....33)2(

23)2(

13)2(

3

3

2

2

xxx

(10)

(c) (i) Find the Maclaurin series for ln(1 + sin x) up to and including the term in x3.

(ii) Hence find a series for ln(1 sin x) up to and including the term in x3.

(iii) Deduce, by considering the difference of the two series, that

ln 3

.

2161

3 2

(12)(Total 28 marks)

8. Given that

2y2 = ex and y = 1 when x = 0, use Eulers method with a step length of 0.1 toxy

dd

find an approximation for the value of y when x = 0.4. Give all intermediate values withmaximum possible accuracy.

(Total 8 marks)

9. (a) Using integration by parts, show that

.xxxx xx dsinedcose00

(5)

(b) Find the value of these two integrals.(6)

(Total 11 marks)


= y2 + xy + 4x2,xyx

dd2


11. (a) Using the Maclaurin series for (1 + x)n, write down and simplify the Maclaurin series

approximation for as far as the term in x4.2

12 )1(

x(3)

(b) Use your result to show that a series approximation for arccos x is

arccos x

.53403

61

2 xxx

(3)

(c) Evaluate

.6

22

0

)arccos(2

limx

xx

x

(5)

(d) Use the series approximation for arccos x to find an approximate value for

,xx d)(arccos2.0

0 giving your answer to 5 decimal places. Does your answer give the actual value of the

integral to 5 decimal places?(6)

(Total 17 marks)

12. (a) Consider the power series

.k

k

xk

21

(i) Find the radius of convergence.

(ii) Find the interval of convergence.(10)

(b) Consider the infinite series

.12

)1( 21

1

kk

k

k

(i) Show that the series is convergent.

(ii) Show that the sum to infinity of the series is less than 0.25.(5)

(Total 15 marks)

13. Given that

+ 2y tan x = sin x, and y = 0 when x =

, find the maximum value of y.xy

dd

3

(Total 11 marks)

14. Find

.

126

0

cos1limx

xx

(Total 7 marks)

15. (a) Using the Maclaurin series for the function ex, write down the first four terms of the

Maclaurin series for .22

ex

(3)

(b) Hence find the first four terms of the series for

.ux

u

de0

2

2

(3)

(c) Use the result from part (b) to find an approximate value for

.xx

de21 1

02

2

(3)

(Total 9 marks)


(x 1)

+ xy = (x 1)exxy

dd


17. Consider the infinite series

....5ln5

14ln4

13ln3

12ln2

1

(a) Show that the series converges.(4)

(b) Determine if the series converges absolutely or conditionally.(11)

(Total 15 marks)

18. Find

(a)

;20

tanlimxxx

x (4)

(b)

.

2sin1

ln21lim22

1 xxxx

x

(7)(Total 11 marks)

19. The variables x and y are related by

y tan x = cos x.xy

dd

(a) Find the Maclaurin series for y up to and including the term in x2 given that y = 2

when x = 0.(7)

(b) Solve the differential equation given that y = 0 when x = . Give the solution in the formy = f(x).


20. (a) Determine whether the series

is convergent or divergent.1

1sinn n

(5)

(b) Show that the series

is convergent.2 2)(ln

1

n nn(7)

(Total 12 marks)


for which y = 1 when x = 1.222

2dd

xxy

xy

(a) Use Eulers method with a step length of 0.25 to find an estimate for the valueof y when x = 2.

(7)

(b) (i) Solve the differential equation giving your answer in the form y = f(x).

(ii) Find the value of y when x = 2.(13)

(Total 20 marks)


. 1 2 239

3

n nn(6)

(b) (i) Sum the series

.0r

rx

(ii) Hence, using sigma notation, deduce a series for

(a)

;21

1x

(b) arctan x;

(c)

.6

(11)

(c) Show that

3(mod15).

100

1!

n

n

(4)(Total 21 marks)

23. (a) Assuming the series for ex, find the first five terms of the Maclaurin series for

.22

e21 x

(3)

(b) (i) Use your answer to (a) to find an approximate expression for the cumulativedistribution function of N(0, 1).

(ii) Hence find an approximate value for P(0.5 Z 0.5), where Z ~ N(0, 1).(6)

(Total 9 marks)

24. The function f(x) is defined by the series f(x) =

+ ...33)2(

23)2(

13)2(1

3

3

2

2

xxx

(a) Write down the general term.(1)

(b) Find the interval of convergence.(13)

(Total 14 marks)

25. Solve the differential equation (u + 3v3)

= 2v, giving your answer in the form u = f(v).uv

dd

(Total 8 marks)


(where x > 0)22

dd

xy

xy

xy


27. The function f is defined by f(x) = .)1(exe

(a) Assuming the Maclaurin series for ex, show that the Maclaurin series for f(x)

is 1 + x + x2 +

x3 + ...65

(5)

(b) Hence or otherwise find the value of

.1)(1)(lim

0

xfxf

x

(5)(Total 10 marks)

28. The sequence {un} is defined for n + by un =

.1

22

2

nn

(a) Find the value L of

.nn ulim

(2)

(b) Use the formal , N definition of convergence to prove that

.Lunn lim(7)

(Total 9 marks)

29. Consider the infinite series

. 1 )3(

1

n nn

(a) Using one of the standard tests for convergence, show that the series is convergent.(3)

(b) (i) Express

in partial fractions.)3(

1nn

(ii) Hence find the sum of the above infinite series.(10)

(Total 13 marks)

30. (a) Find the radius of convergence of the infinite series

...118527531

852531

5231

21 432

xxxx(7)

(b) Determine whether the series

is convergent or divergent.

1

1sinn

nn

(8)(Total 15 marks)

31. Determine whether the series

is convergent or divergent.1

10

10n nn

(Total 6 marks)

32. (a) Using lHopitals Rule, show that

= 0.xx

x elim

(2)

(b) Determine

a x xx0

.de

(5)

(c) Show that the integral

is convergent and find its value. 0 de xx x(2)

(Total 9 marks)


.1

2dd

2

3

xxy

xyx

(a) Find an integrating factor for this differential equation.(5)

(b) Solve the differential equation given that y =1 when x =1, giving your answer in the formy = f (x).

(8)(Total 13 marks)

34.

The diagram shows part of the graph of y =

together with line segments parallel to the31x

coordinate axes.

(a) Using the diagram, show that

3 3333333 5

141

31d1

61

51

41 ....x

x...

(3)

(b) Hence find upper and lower bounds for 1

3 .1

n n(12)

(Total 15 marks)

35. The function f is defined by

f (x) =

.1

1ln

x

(a) Write down the value of the constant term in the Maclaurin series for f (x).(1)

(b) Find the first three derivatives of f (x) and hence show that the Maclaurin series for f (x)

up to and including the x3 term is

.xxx32

32

(6)

(c) Use this series to find an approximate value for ln 2.(3)

(d) Use the Lagrange form of the remainder to find an upper bound for the error in thisapproximation.

(5)

(e) How good is this upper bound as an estimate for the actual error?(2)

(Total 17 marks)


.2sin

lnlim1

xx

x

(3)

(b) By using the series expansions for and cos x evaluate

2e x .cos1e1lim

2

0

x

x

x

(7)(Total 10 marks)

37. Find the exact value of

0 122

d .xx

x

(Total 9 marks)

38. A curve that passes through the point (1, 2) is defined by the differential equation

.12dd 2 yxx

xy

(a) (i) Use Eulers method to get an approximate value of y when x = 1.3, taking steps of0.1. Show intermediate steps to four decimal places in a table.

(ii) How can a more accurate answer be obtained using Eulers method?(5)

(b) Solve the differential equation giving your answer in the form y = f (x).(9)

(Total 14 marks)

39. (a) Given that y = ln cos x, show that the first two non-zero terms of the Maclaurin series for

y are

.122

42 xx (8)

(b) Use this series to find an approximation in terms of for ln 2.(6)

(Total 14 marks)

40. (a) Find the radius of convergence of the series

0.

311

nn

nn

nx

(6)

(b) Determine whether the series

is convergent or divergent.

03 3 1

n

nn

(7)(Total 13 marks)

41. Solve the following differential equation

(x + 1)(x + 2)

= x + 1yxy

dd

giving your answer in the form y = f (x).(Total 11 marks)

42. The function f is defined by f (x) = ln (1 + sin x).

(a) Show that f (x) =

.sin11

x

(4)

(b) Determine the Maclaurin series for f (x) as far as the term in x4.(6)

(c) Deduce the Maclaurin series for ln (1 sin x) as far as the term in x4.(2)

(d) By combining your two series, show that ln sec x =

....122

42

xx

(4)

(e) Hence, or otherwise, find

.seclnlim0 xx

xx

(2)(Total 18 marks)

43. Let Sn =

n

k k1.1

(a) Show that, for n 2, S2n Sn +

.21

(3)

(b) Deduce that S2 + 12 mS.

2m

(7)

(c) Hence show that the sequence is divergent. nS(3)

(Total 13 marks)

44. (a) Show that the solution of the homogeneous differential equation

, x > 0,1dd

xy

xy

given that y = 0 when x = e, is y = x(ln x 1).(5)

(b) (i) Determine the first three derivatives of the function f(x) = x(ln x 1).

(ii) Hence find the first three non-zero terms of the Taylor series for f(x) about x = 1.(7)

(Total 12 marks)

45. (a) (i) Show that

dx, p 0 is convergent if p > 1 and find its value in terms 1 )( 1 pxxof p.

(ii) Hence show that the following series is convergent.

...5.23

15.12

15.01

1 (8)

(b) Determine, for each of the following series, whether it is convergent or divergent.

(i)

1 )3(1sin

n nn

(ii)

...201

121

61

21


46. The function f(x) =

can be expanded as a power series in x, within its radius ofbxax

11

convergence R, in the form f(x) 1 +

.1n

nn xc

(a) (i) Show that cn = (b)n1(a b).

(ii) State the value of R.(5)

(b) Determine the values of a and b for which the expansion of f(x) agrees with that of ex upto and including the term in x2.

(4)

(c) Hence find a rational approximation to .31

e(3)

(Total 12 marks)

47. (a) Show that the solution of the differential equation

= cos x cos2 y,xy

dd

given that y =

when x = , is y = arctan (1 + sin x).4

(5)

(b) Determine the value of the constant a for which the following limit exists

22

2

)sin1arctan(lim

x

axx

and evaluate that limit.(12)

(Total 17 marks)

48. Calculate

.

xxx sin

11lim0

(Total 6 marks)

49. Use the integral test to show that the series

is convergent for p > 1.1

1

npn

(Total 6 marks)

50. (a) (i) Find the first four derivatives with respect to x of y = ln(1 + sin x)

(ii) Hence, show that the Maclaurin series, up to the term in x4, for y is

y = x

...121

61

21 432 xxx

(10)

(b) Deduce the Maclaurin series, up to and including the term in x4, for

(i) y = ln(1 sin x);

(ii) y = ln cosx;

(iii) y = tan x.(10)

(c) Hence calculate

.

xx

x cosln)tan(lim

2

0

(4)(Total 24 marks)


= 1, where x < 2 and y = 1 when x = 0.24dd

xxy

yy

(a) Use Eulers method with h = 0.25, to find an approximate value of y when x = 1, givingyour answer to two decimal places.

(10)

(b) (i) By first finding an integrating factor, solve this differential equation.Give your answer in the form y = f(x).

(ii) Calculate, correct to two decimal places, the value of y when x = 1.(10)

(c) Sketch the graph of y = f(x) for 0 x 1. Use your sketch to explain why yourapproximate value of y is greater than the true value of y.

(4)(Total 24 marks)

math paper 3 practice

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