OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
4776
MEI STRUCTURED MATHEMATICS
Numerical Methods
25 JANUARY 2006
Wednesday
Morning
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer all the questions.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
This question paper consists of 4 printed pages.
HN/3
© OCR 2006 [M/102/2666]
Registered Charity 1066969
[Turn over
2
Section A (36 marks)
1
Show that if
X x (1 r)
then
1
1
(1 r)
X
x
for sufficiently small r.
Explain carefully what this means in terms of relative errors. Illustrate your answer in the case
X 10, x 9.8.
[6]
2
The equation
1
sin x
x
(*)
(where x is in radians) has two roots, a and b, in the interval 0 x p .
(i) Given that a 1, use an iteration based on a simple rearrangement of (*) to find a correct to
3 significant figures.
[4]
(ii) Verify that b 2.773 is correct to 3 decimal places.
3
The integral I [3]
f(x) dx is being evaluated numerically. The following values have been
3
1
obtained.
h
mid-point rule
trapezium rule
2
2.602 42
2.448 66
1
2.569 82
Obtain 2 further trapezium rule estimates of I.
Obtain 2 Simpson’s rule estimates of I.
Give a value for I to the accuracy that appears justified.
4776 January 2006
[7]
3
4
The function f ( x ) has the values shown in the table.
x
1
2
3
4
5
f(x)
3.0
4.5
5.4
6.2
6.7
(i) Obtain two estimates of the gradient f ( 3 ) using the central difference method.
[3]
(ii) Given that the values of x are exact but that the values of f ( x ) are rounded to 1 decimal place,
determine a range of possible values of f ( 3 ) from each estimate. Comment on your results.
Give a value for f ( 3 ) to the accuracy that appears justified.
5
[5]
The function g ( x ) is quadratic. The following values are known.
x
1
3
4
g(x)
4
1
11
Use Lagrange’s method to determine g ( 2 ) .
Check your answer by drawing up a difference table for g ( x ) .
[8]
Section B (36 marks)
6
(i) Show that the curve y x 10 10x 1 has exactly one turning point.
Show that the equation
x10 10x 1 0
(*)
has exactly two real roots and that these roots both lie in the interval [0, 2].
[7]
(ii) Use the Newton-Raphson method to find the larger root correct to 4 decimal places.
[6]
(iii) Obtain another iteration based on a rearrangement of (*) and hence, without using a calculator,
show that the smaller root is almost exactly 0.1 0.111.
[5]
4776 January 2006
[Turn over
4
7
The number e satisfies the relationship log e N 5
1
dx.
log e 5 x
1
(i) Use the mid-point rule to show that loge 5 N
1
1
dx. Hence it follows, for example, that
x
1
1
1
1
.
1.5 2.5 3.5 4.5
(*)
Given that, correct to 8 decimal places, log e 5 1.609 437 91, find the error in the
approximation (*).
[5]
(ii) Correct to 8 decimal places, log e2 0.693 147 18. Hence or otherwise obtain the values of
log e 10, log e 20, log e 40 and log e 80, giving your answers correct to 6 decimal places.
[3]
It is known that, as N increases, the value of the expression
1
1
1
Ê 1
ˆ
log e N - Á
+
+
+º+
˜
Ë 1.5 2.5 3.5
( N - 0.5)¯
tends to a constant k.
(iii) Use the information in the table to obtain 4 estimates, a1, a 2, a 3, a 4, of k. Give these estimates
to 6 decimal places.
N
1
1
1
1
+
+
+º +
(N - 0.5)
1.5 2.5 3.5
10
20
40
80
2.266511 2.959346 3.652416 4.345543
[3]
(iv) Find the differences a 2 a 1, a 3 a 2, a 4 a 3. Show that the differences reduce by a factor
of approximately 4 as N is doubled. Hence obtain the best estimate you can of k, giving your
answer to an appropriate number of significant figures.
[7]
4776 January 2006
OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
4776
MEI STRUCTURED MATHEMATICS
Numerical Methods
19 JUNE 2006
Monday
Morning
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer all the questions.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
This question paper consists of 4 printed pages.
HN/5
© OCR 2006 [M/102/2666]
Registered Charity 1066969
[Turn over
2
Section A (36 marks)
1
In the diagram below, X is an approximation to x with error h. Also f ( X ) is an approximation to
f ( x ) with error k. Show, by considering the tangent at x, that k hf( x ) .
y
y = f(x)
f(X)
k
f(x)
O
Use this result, with f( x ) =
correct to 1 decimal place.
2
x
h
X
x
x , to estimate the maximum possible error in
X when X 2.5
[7]
Show that the equation
x5 5x 1 0
has a root in the interval [0, 1].
Apply two iterations of the false position method to obtain an estimate of this root. Give your
answer to 3 decimal places.
Determine whether or not your answer is correct to 3 decimal places.
[8]
2
3
Û
The integral I = Ù 1 + 2 x dx is to be found numerically.
ı0
Obtain the estimates given by the mid-point rule and the trapezium rule with h 2. Use these
values to obtain a Simpson’s rule estimate of I.
Given that the mid-point rule estimate with h 1 is 3.510 411, obtain as efficiently as possible a
second trapezium rule estimate and a second Simpson’s rule estimate.
Give the value of I to the accuracy that appears justified.
4776 June 2006
[8]
3
4
Given the data in the table below, find three estimates of f ( 2 ) .
h
0
0.1
0.01
0.001
f(2 + h)
1.4427
1.3478
1.4324
1.4416
Discuss briefly the likely accuracy of these estimates.
5
[6]
Show, by means of a difference table, that the function g ( x ) tabulated below is approximately but
not exactly quadratic.
x
1
2
3
4
5
6
g(x)
3.2
12.8
28.4
50.2
77.9
111.6
Use Newton’s forward difference formula to estimate the value of g ( 1.5 ) .
[7]
Section B (36 marks)
6
(i) Show that the equation
x2 tan x
(*)
(where x is in radians) has a root in the interval [4.6, 4.7].
Use the bisection method with starting values 4.6 and 4.7 to find this root with maximum
possible error 0.0125.
[9]
(ii) You are now given that equation (*) also has a root in the interval [7.7, 7.9]. Show that 7.7
and 7.9 are not suitable starting points for the bisection method. Explain with the aid of a
sketch graph how this situation arises.
[5]
(iii) Using only the fact that equation (*) has a root in the interval [7.7, 7.9], write down the best
possible estimate of the root. Determine whether or not this estimate is correct to 1 decimal
place.
[4]
4776 June 2006
[Turn over
4
7
The following values of the function f ( x ) are known.
x
1
2
4
f(x)
–3
8
36
It is required to estimate D f ( 2 ) and I f (x) dx.
4
1
(i) Use the forward difference method to estimate D.
Use the trapezium rule to obtain the best possible estimate of I.
[4]
(ii) Use Lagrange’s method to find the quadratic that passes through the given points.
Hence find new estimates of D and I.
[11]
(iii) Comment on the extent to which the estimates in part (i) agree with those in part (ii).
4776 June 2006
[3]
4776/01
ADVANCED SUBSIDIARY GCE UNIT
MATHEMATICS (MEI)
Numerical Methods
THURSDAY 25 JANUARY 2007
Morning
Time: 1 hour 30 minutes
Additional materials:
Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer booklet.
•
Answer all the questions.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
•
Read each question carefully and make sure you know what you have to do before starting your
answer.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
HN/3
© OCR 2007 [M/102/2666]
OCR is an exempt Charity
[Turn over
2
Section A (36 marks)
1
A calculator gives the answer to a calculation as 1.711 224 5 10 9 8, correct to 8 significant
figures. Find the largest possible absolute error and the largest possible relative error in this value.
Though the calculator displays numbers such as 1.711 224 5 to 8 digit accuracy, it stores them
internally to 11 digit accuracy. Explain briefly why this is done.
[5]
2
The approximation
tan x x 13 x 3
is valid for small values of x in radians.
(i) Find the absolute and relative errors in the approximation for x 0.2.
[4]
A much more accurate approximation is given by
tan x x 13 x 3 kx5,
where k is a constant.
(ii) Use the first result in part (i) to estimate k, giving your answer to 2 significant figures.
3
[3]
An equation is being solved numerically using a fixed-point iteration of the form
xr1 g ( xr ) .
The iteration has been used to obtain the values shown in the following table.
r
0
1
2
3
xr
0.35
0.354767
0.356462
0.357067
Differences
Ratio of
differences
Copy and complete the table to show the differences in successive values of xr and the ratios of
those differences. Use extrapolation to estimate the root to which this iteration is converging,
giving your answer to the accuracy that appears justified.
[8]
© OCR 2007
4776/01 Jan 07
3
4
Show, graphically or otherwise, that the equation x2 cos x where x is in radians has exactly one
root for x 0. Show further that the root lies in the interval ( 0.7, 0.9 ) .
Use the secant method to find the root correct to 3 decimal places.
5
[8]
The function f ( x ) has the values shown in the table.
x
0
0.25
0.5
f(x)
1.1105
1.2446
1.4065
(i) Use the forward difference formula with h 0.5 and h 0.25 to obtain two estimates of
f ( 0 ) . Comment on the likely accuracy of these results and on the number of decimal places
that it would be safe to quote.
[4]
(ii) Obtain the best estimate you can of the value of f ( 0.25 ) . Comment on the likely accuracy of
this result in relation to those in part (i). To how many decimal places would you quote the
answer?
[4]
Section B (36 marks)
6
The following values of x and y were obtained in an experiment. The values of x are exact; the
values of y are correct to 2 decimal places. It is required to estimate a , the value of x for which y 0.
x
0.9
1.1
1.2
1.4
1.5
y
–0.43
–0.09
0.15
0.78
1.15
(i) Use Lagrange’s method to find the equation of the straight line joining the data points for
x 1.1 and x 1.2. Hence estimate a .
By considering the maximum possible errors in the values of y obtain a range of possible
values of a . Hence give the value of a to the accuracy that is justified.
[10]
(ii) Obtain a further estimate of a by fitting a quadratic to the data points for x 1.1, 1.2 and 1.4.
[8]
© OCR 2007
4776/01 Jan 07
[Turn over
4
7
This question concerns the function f ( x ) x x. (This can also be written as f ( x ) 1
.) The table
xx
below shows some values of the function.
x
1
1.5
2
f(x)
1
0.544331 0.25
2
Û
(i) Use the values in the table to find the Simpson’s rule estimate of Ù f ( x ) dx with h 0.5.
ı1
Find the Simpson’s rule estimate with h 0.25.
[7]
You are now given that the Simpson’s rule estimate with h 0.125 is 0.572 344 to 6 dp. Let the
three Simpson’s rule estimates with h 0.5, 0.25, 0.125 be denoted by a, b and c respectively.
(ii) Find the value of the ratio of differences
comment.
cb
. State the theoretical value of this ratio and
ba
[5]
(iii) Extrapolate from b and c to obtain a further estimate of the integral.
Give the value of the integral to the accuracy that appears to be justified, explaining your
reasoning.
[6]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate
(UCLES), which is itself a department of the University of Cambridge.
© OCR 2007
4776/01 Jan 07
4776/01
ADVANCED SUBSIDIARY GCE UNIT
MATHEMATICS (MEI)
Numerical Methods
WEDNESDAY 20 JUNE 2007
Afternoon
Time: 1 hour 30 minutes
Additional materials:
Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer booklet.
•
Answer all the questions.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
•
Read each question carefully and make sure you know what you have to do before starting your
answer.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
HN/4
© OCR 2007 [M/102/2666]
OCR is an exempt Charity
[Turn over
2
Section A (36 marks)
1
2
Show that the equation x + 1 + x = 3 has a root in the interval ( 1, 1.4 ) .
Use the bisection method to obtain an estimate of the root with maximum possible error 0.025.
Determine how many additional iterations of the bisection process would be required to reduce the
maximum possible error to less than 0.005.
[8]
0.5
2
1
Û
dx, find the values given by the trapezium rule and the mid-point rule,
For the integral Ù
ı0 1 + x 4
taking h 0.5 in each case.
Hence show that the Simpson’s rule estimate with h 0.25 is 0.493 801.
You are now given that the Simpson’s rule estimate with h 0.125 is 0.493 952. Use extrapolation
to determine the value of the integral as accurately as you can.
[8]
3
A triangle has sides a, 3 and 4. The angle opposite side a is ( 90 e ) °, where e is small. See Fig. 3.
a
3
(90 + e)∞
4
Fig. 3
Use the cosine rule to calculate a when e 5.
The approximation
cos ( 90 e ) ° pe
180
with e 5 is now used in the cosine rule to find an approximate value for a.
Find the absolute and relative errors in this approximate value of a.
© OCR 2007
4776/01 June 07
[5]
3
4
The number x is represented in a computer program by the approximation X. You are given that
X x ( 1 r ) where r is small.
(i) State what r represents.
[1]
(ii) Use the first two terms in a binomial expansion to show that the relative error in X n as an
approximation to xn is approximately nr.
[2]
22
(iii) A lazy programmer has approximated p by 7 . Find the relative error in this approximation.
Use the result in part (ii) to write down the approximate relative errors in the values of p 2 and
p when p is taken as 22
[5]
7.
5
The function f ( x ) has the values shown in the table.
x
–1
0
4
f(x)
3
2
9
Use Lagrange’s interpolation method to obtain the quadratic function that fits the three data points.
Hence estimate the value of x for which f ( x ) takes its minimum value.
[7]
Section B (36 marks)
6
(i) Explain, with the aid of a sketch, the principle underlying the Newton-Raphson method for
[3]
the solution of the equation f ( x ) 0.
(ii) Draw a sketch of the function f ( x ) tan x 2x for 0 x 12 p ( x in radians ) . Mark on
your sketch the non-zero root, a , of the equation tan x 2x 0. Show by means of your
sketch that, for some starting values, the Newton-Raphson method will fail to converge to a .
Identify two distinct cases that can arise.
[6]
(iii) Given that the derivative of tan x is 1 tan 2 x, show that the Newton-Raphson iteration for the
solution of the equation tan x 2x 0 is
xr1 xr ( tan xr 2xr )
( tan2 xr 1 )
.
Use this iteration with x0 1.2 to determine a correct to 4 decimal places.
Show carefully that this iteration is faster than first order.
[Question 7 is printed overleaf.]
© OCR 2007
4776/01 June 07
[9]
4
7
The function g ( x ) has the values shown in the table.
x
g(x)
1
2.87
2
4.73
3
6.23
4
7.36
5
8.05
(i) Draw up a difference table for g ( x ) as far as second differences. State with a reason whether
or not g ( x ) is quadratic.
[5]
(ii) Draw up another difference table, based this time on x 1, 3, 5. Use Newton’s forward
difference formula to find the quadratic approximation to g ( x ) based on these three points.
Simplify the coefficients of this quadratic.
[8]
(iii) Find the absolute and relative errors when this quadratic is used to estimate g ( 2 ) and g ( 4 ) .
[5]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate
(UCLES), which is itself a department of the University of Cambridge.
© OCR 2007
4776/01 June 07
4776/01
ADVANCED SUBSIDIARY GCE
MATHEMATICS (MEI)
Numerical Methods
THURSDAY 24 JANUARY 2008
Morning
Time: 1 hour 30 minutes
*CUP/T38505*
Additional materials: Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
•
Read each question carefully and make sure you know what you have to do before starting
your answer.
•
Answer all the questions.
•
•
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
The number of marks for each question is given in brackets [ ] at the end of each question or
part question.
•
The total number of marks for this paper is 72.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
SP (KN) T38505/4
© OCR 2008 [M/102/2666]
OCR is an exempt Charity
[Turn over
2
Section A (36 marks)
1
The equation f(x) = 0 is known to have a single root. Given that f(2) = 0.24 and f(3) = 0.03, use the
secant method to obtain an estimate of the root. Show, by means of a sketch, that this estimate could be
very inaccurate.
[5]
2
For the integral I =
2 – x dx, find the values given by
冕 –––––
2+ x
1
0
(A) the trapezium rule with h = 1,
(B) the mid-point rule with h = 1.
Use these two values to obtain a further trapezium rule estimate and a Simpsonʼs rule estimate of the
integral.
[8]
3
The function f(x) has the values shown in the table.
x
0
1
3
f(x)
2.00
2.57
3.85
Use Lagrangeʼs method to find the estimate of f(2) given by fitting a quadratic function to the data.
[7]
4
Show that the equation x3(2 – x) = 1 has a root in the interval (1.5, 2). Use the bisection method to find
the root with maximum possible error 0.0625.
[6]
Determine how many further iterations of the bisection process would be required to reduce the
maximum possible error to less than 0.005.
[2]
5
A numerical derivative is being found using the forward difference approximation. Show, by means of
a sketch, that a large value of h may lead to a large error.
[3]
The function g(x) has the values shown in the table correct to 3 decimal places.
x
2
2.001
2.01
2.1
g(x)
3.610
3.612
3.633
3.849
Obtain three estimates of the derivative of the function at x = 2. Use your answers to show that, in
numerical differentiation, a smaller value of h may not always lead to greater accuracy.
[5]
© OCR 2008
4776/01 Jan08
3
Section B (36 marks)
6
The function f(x) has the values shown in the table.
x
3
4
5
6
f(x)
1
3
–1
–10
(i) Use Newtonʼs forward difference interpolation formula to fit a quadratic to the points at x = 3, 4, 5.
Use this quadratic to estimate
(A) the value of x at which f(x) takes its maximum value,
(B) the value of x in the interval (4, 5) for which f(x) = 0.
Show that the quadratic does not pass through the fourth data point.
[12]
(ii) Use Newtonʼs forward difference interpolation formula to estimate f(4.5) using a cubic. (Note that
you are not required to find the cubic in terms of x.)
冕 f(x) dx.
6
Hence obtain a Simpsonʼs rule estimate of
3
[6]
[Question 7 is printed overleaf.]
© OCR 2008
4776/01 Jan08
[Turn over
4
7
(i) The number 2.506 628 is known to be correct to 6 decimal places. Write down the maximum
possible error and calculate the maximum possible relative error.
[3]
(ii) A computer adds up 1000 numbers each of which has been rounded to 6 decimal places. Calculate
the maximum possible error in the sum. Explain why an error of this magnitude is unlikely to arise
in practice.
[3]
(iii) A computer adds up 1000 numbers each of which has been chopped to 6 decimal places. Calculate
the maximum possible error in the sum. What is the most likely error in practice? Explain your
answer.
[5]
(iv) A computer program in which numbers are rounded to 7 significant figures is used to sum the
following numbers. All intermediate answers used in calculations are rounded to 7 significant
figures.
1,
0.000 000 1,
0.000 000 2,
0.000 000 3,
0.000 000 4.
Find the answers the program will give if the numbers are summed
(A) from left to right,
(B) from right to left.
Explain the difference in the two answers.
[3]
(v) A simple computer program is written to find the following sum.
1 + ––
1 + … + –––––
1 .
1 + ––
––
3
3
3
2
3
10003
1
The answer obtained is 1.202 051. When the terms are summed in reverse order the answer is
1.202 056. State, with an explanation, which of these is likely to be more accurate.
When the same two calculations are performed on a spreadsheet the two answers that are displayed
are identical. What two features of a spreadsheet explain why this happens?
[4]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),
which is itself a department of the University of Cambridge.
© OCR 2008
4776/01 Jan08
4776/01
ADVANCED SUBSIDIARY GCE
MATHEMATICS (MEI)
Numerical Methods
MONDAY 16 JUNE 2008
Afternoon
Time: 1 hour 30 minutes
*CUP/T38891*
Additional materials: Answer Booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
•
Read each question carefully and make sure you know what you have to do before starting
your answer.
•
Answer all the questions.
•
•
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
The number of marks for each question is given in brackets [ ] at the end of each question or
part question.
•
The total number of marks for this paper is 72.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
SP (KN) T38891/4
© OCR 2008 [M/102/2666]
OCR is an exempt Charity
[Turn over
2
Section A (36 marks)
1
The equation f(x) = 0 is known to have a single root in the interval (3, 3.5). Given that f(3) = 0.5 and
f(3.5) = –0.8, estimate the root using linear interpolation.
State the maximum possible error in this estimate.
2
[6]
The function f(x) has the values shown in the table. The value of k is to be determined.
x
1
3
5
7
9
f(x)
2
1
5
k
2
Use a difference table to obtain the value of k, assuming that f(x) is a cubic.
3
[6]
The function f(x) = 1 + 3x is to be differentiated numerically.
Use the central difference method with h = 0.2 to estimate the derivative at x = 2. Obtain further
estimates with h = 0.1 and h = 0.05.
By considering the differences between successive estimates, find the value of the derivative to an
accuracy of 3 decimal places.
[8]
4
Show that a Newton-Raphson iteration to find the cube root of 25 is
xr3 – 25
.
xr + 1 = xr – –––––––
3xr2
Perform three steps of this iteration, beginning with x0 = 4. Show, by considering the differences
between successive iterates, that the convergence is faster than first order.
[8]
5
(i) Find sin 86° – sin 85° to the accuracy given by your calculator.
[1]
(ii) A simple spreadsheet works to an accuracy of 6 significant figures. All intermediate answers used
in calculations are rounded to 6 significant figures.
Write down the values of sin 86° and sin 85° as given by this spreadsheet. Hence find the value the
spreadsheet gives for sin 86° – sin 85°.
[3]
(iii) You are now given that sin 86° – sin 85° = 2 cos 85.5° sin 0.5°. Find the value the spreadsheet
gives for this expression.
[2]
(iv) Use your working from parts (ii) and (iii) to explain how two expressions that are mathematically
identical can nevertheless evaluate differently.
[2]
© OCR 2008
4776/01 Jun08
3
6
The integral
冕
Section B (36 marks)
3
1
1 + sin x dx, where x is in radians, is to be evaluated numerically.
(i) Copy and complete the following table.
[7]
h
Mid-point rule estimate
Trapezium rule estimate
2
M1 = 2.763 547
T1 =
1
M2 =
T2 =
0.5
M4 =
T4 =
(ii) Show that the differences between successive mid-point rule estimates reduce by a factor of
about 4.
State a result about the differences between successive trapezium rule estimates.
[4]
1
(iii) Now let S1 = – (2M1 + T1), with S2 and S4 defined similarly.
3
Calculate S1, S2, S4 and the differences S2 – S1, S4 – S2. By considering these differences, give the
value of the integral to the accuracy that appears justified.
[7]
7
The equation x2 = 4 + 1–x has three roots.
(i) Show graphically that the equation has exactly one root for x > 0. Find the integer a such that this
positive root lies in the interval (a, a + 1).
Use the fixed-point iteration
xr + 1 =
√(4 + ––x1r )
(*)
to determine the positive root correct to 4 decimal places.
[7]
(ii) The equation also has two negative roots. Without doing any calculations, explain why the iteration
(*) cannot be used to find these negative roots.
Use the fixed-point iteration
xr + 1 = –
√(4 + ––x1r )
(**)
to find a negative root near to x = –2 correct to 4 decimal places.
[5]
(iii) The third root of the equation lies in the interval (–1, 0). Show that the iteration (**) used in part
(ii) will not converge to this third root. Use another fixed point iteration to find the third root
correct to 4 decimal places.
[6]
© OCR 2008
4776/01 Jun08
ADVANCED SUBSIDIARY GCE
4776
MATHEMATICS (MEI)
Numerical Methods
*OCE/V02215*
Candidates answer on the Answer Booklet
OCR Supplied Materials:
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Morning
Duration: 1 hour 30 minutes
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Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces
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Read each question carefully and make sure that you know what you have to do before starting your answer.
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INFORMATION FOR CANDIDATES
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The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to
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The total number of marks for this paper is 72.
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Turn over
2
Section A (36 marks)
1
(i) Show by means of a difference table that a quadratic function fits the following data points.
x
–3
–1
1
3
y
–16
–2
4
2
[3]
(ii) Obtain the equation of the quadratic function, expressing your answer in its simplest form.
2
(i) Use the formula for the difference of two squares to show that
(
(ii) A spreadsheet shows
x +1 − x
)(
Use the spreadsheet figures to obtain values of
(A) by subtraction,
(B) by using (*)
(*) [2]
x + 1 + x = 1.
50001 as 223.6090 and
)
50000 as 223.6068.
50001 –
50000
Comment on your results.
3
[5]
[5]
(i) For the integral
I=

0.8
0
1 − x 5 dx
find the trapezium rule and mid-point rule estimates with h = 0.8 in each case. Use these estimates
to obtain a Simpson’s rule estimate.
[4]
(ii) Given that the mid-point rule estimate with h = 0.4 is 0.784 069 to 6 significant figures, obtain a
second Simpson’s rule estimate. Without doing any further calculations, give a value for I to the
accuracy that is justified.
[4]
4
(i) An approximation to cos x, where x is small and in radians, is given by
cos x ≈ 1 – 0.5 x2.
Find the absolute and relative errors in this approximation when x = 0.3.
[4]
(ii) The formula
cos x ≈ 1 – 0.5 x2 + k x4
gives a better approximation if k is suitably chosen. By considering x = 0.3 again, estimate k. [2]
© OCR 2009
4776 Jan09
3
5
A student is investigating the iteration
xr + 1 = xr2 – 3xr + 3
for different starting values x0.
Determine the values of x1 and x2 in each of the cases x0 = 3, x0 = 2.99, x0 = 3.01.
Evaluate the derivative of x2 – 3x + 3 at x = 3.
Comment on your results.
[7]
Section B (36 marks)
6
(i) Show that the equation
sin x + cos x = 1.5, (*)
where x is in radians, has a root in the interval (0.2, 0.3).
Perform two iterations of the bisection method and give the interval within which the root lies, the
best estimate of the root, and the maximum possible error in that estimate.
[6]
(ii) Now perform two iterations of the secant method, starting with x0 = 0.2 and x1 = 0.3. Give an
estimate of the root to an appropriate number of significant figures.
Comment on the relative rate of convergence of the bisection method and the secant method. [6]
(iii) You are given that equation (*) also has a root a which is 1.298 504 to 6 decimal places. An
iteration to find this root produces the following sequence of values.
r
0
1
2
3
4
xr
1.4
1.314 351
1.298 887
1.298 504
1.298 504
By considering the values of xr – a, show that this iteration displays second order convergence
making it clear what that means.
[6]
[Question 7 is printed overleaf.]
© OCR 2009
4776 Jan09
Turn over
4
7
A function f(x) has values, correct to 6 significant figures, as given in the table.
x
–0.4
–0.2
–0.1
0
0.1
0.2
0.4
f(x)
0.601 201
0.711 982
0.765 298
0.816 603
0.865 314
0.911 308
0.994 506
(i) Obtain three estimates of f9(0) using the forward difference method with h equal to 0.4, 0.2, 0.1.
Show that the differences between these estimates are approximately halved as h is halved. [4]
(ii) Obtain three estimates of f9(0) using the central difference method. Show, by considering the
differences between these estimates, that the central difference method converges more rapidly
than the forward difference method.
[4]
(iii) D1 and D2 are two estimates of a quantity d.
(A) Suppose that the error in D2 is approximately half of the error in D1. Write down
expressions for the errors in D1 and D2 and hence show that d ≈ 2D2 – D1.
(B) Now suppose that the error in D2 is approximately a quarter of the error in D1. Show that
4D2 – D1
.
[5]
d≈
3
(iv) Use the results in part (iii)(A) and part (iii)(B) to obtain two further estimates of f9(0). Give an
estimate of f9(0) to the accuracy that you consider justified.
[5]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
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© OCR 2009
4776 Jan09
ADVANCED SUBSIDIARY GCE
4776/01
MATHEMATICS (MEI)
Numerical Methods
*OCE/T61324*
Candidates answer on the Answer Booklet
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•
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•
Graph paper
Wednesday 20 May 2009
Afternoon
Duration: 1 hour 30 minutes
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*
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6
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INSTRUCTIONS TO CANDIDATES
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Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer all the questions.
Do not write in the bar codes.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
© OCR 2009 [M/102/2666]
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OCR is an exempt Charity
Turn over
2
Section A (36 marks)
1
A quadratic function, f(x), is to be fitted to the data shown in the table.
x
0
0.4
1
y
1.6
2.4
1.8
(i) Use Lagrange’s method to find f(x), simplifying the coefficients.
[6]
(ii) Explain why Newton’s forward difference interpolation formula would not have been useful for
this purpose.
[1]
2
Show that the equation
x2 + 1 = 3
x
has a root in the interval (1, 2).
Use the Newton-Raphson method to find this root, giving it correct to 6 significant figures.
3
[7]
The numbers X and Y shown below are known to be correct to 3 decimal places.
X = 2.718
Y = 3.142
(i) State the maximum possible errors in X, X + Y, X – Y, 10X + 20Y.
[4]
(ii) Find the maximum possible relative errors in X and Y. Hence state approximately the maximum
[4]
possible relative errors in XY and X.
Y
4
You are given that, for A and B in radians and A ≈ B,
sin A – sin B
A+B
.
≈ cos
A–B
2
(*)
A computer program calculates values of sine and cosine correct to 6 decimal places.
(i) In the case A = 1.01, B = 1, find the values of the left and right sides of (*) as calculated by this
program.
[2]
(ii) Identify two distinct reasons for the difference in these two values.
[2]
(iii) Explain briefly why the right side of (*) is likely to be evaluated more accurately than the left as A
gets progressively closer to B.
[2]
5
Sketch, on the same axes, the graphs y = x and y = 1 – x4 for 0 ⭐ x ⭐ 1. You should use the same scale
on each axis.
Show numerically that the iteration xr+1 = 1 – xr4, starting with x0 = 0.6, diverges.
Illustrate this divergence on your sketch, showing x0, x1, x2, x3.
© OCR 2009
4776/01 Jun09
[8]
3
Section B (36 marks)
6
The integral
冕
0
0.8
3 + x − x 2 dx is to be evaluated numerically.
(i) Find, as efficiently as possible, the mid-point rule estimates and the trapezium rule estimates for
h = 0.8 and 0.4.
[6]
(ii) Use the values in part (i) to show that the first Simpson’s rule estimate is 1.427 959 (correct to
6 decimal places), and to find a second Simpson’s rule estimate.
[3]
(iii) Given that, for h = 0.2, the mid-point rule estimate is 1.428 782 and the trapezium rule estimate is
1.426 497, calculate a third Simpson’s rule estimate.
[2]
(iv) Show that the differences between successive mid-point rule estimates reduce by a factor of about
0.25 as h is halved. Find the corresponding factor for the Simpson’s rule estimates. Hence give the
integral to the accuracy that appears justified.
[7]
7
(i) Use Newton’s forward difference interpolation formula to find the quadratic function that passes
through the following data points.
x
1
f(x) 0.6
1.2
1.4
–0.1
0.4
[8]
(ii) Use the quadratic function to estimate f⬘(1.2). Show that the central difference formula gives
exactly the same estimate. What does this suggest about the central difference formula?
[5]
(iii) Use the quadratic function to estimate f⬘(1). Show that the forward difference does not give the
same value. What does this show about the forward difference method? Which of these two
estimates is likely to be more accurate?
[5]
© OCR 2009
4776/01 Jun09
ADVANCED SUBSIDIARY GCE
4776/01
MATHEMATICS (MEI)
Numerical Methods
* O C E / 1 3 3 2 6 *
Candidates answer on the Answer Booklet
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•
Graph paper
•
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Afternoon
Duration: 1 hour 30 minutes
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None
*
4
7
7
6
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INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces
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Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer all the questions.
Do not write in the bar codes.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
© OCR 2010 [M/102/2666]
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OCR is an exempt Charity
Turn over
2
Section A (36 marks)
1
Show that the equation
2x +
冢 冣 =3
1
2
x
has a root between x = 1.3 and x = 1.5. Use the bisection method to find an estimate of this root with a
maximum possible error less than 0.02.
Determine how many further iterations would be required to reduce the maximum possible error to less
than 0.001.
[8]
2
An integral,
冕
b
f(x) dx, is being evaluated numerically. Some mid-point rule and trapezium rule
a
estimates are shown in the table.
h
Mid-point rule
Trapezium rule
1
2.579 768
2.447 490
0.5
2.547 350
Find the trapezium rule estimate for h = 0.5.
Find two Simpson’s rule estimates and hence state, with a reason, the value of the integral to the
accuracy that appears justified.
[7]
3
(i) Given that f(x) = x3 – x2 + 1, find f(0.5).
Use the formula f(x + h) ≈ f(x) + h fʹ(x) to show that
f(0.5 + h) ≈ 0.875 – 0.25 h.
[3]
(ii) Hence determine the approximate range of values of x for which f(x) = 0.875 correct to 3 decimal
places.
[4]
4
(i) Show algebraically that
(k + 1)2 + (k – 1)2 – 2k2 = 2
(*)
for all values of k.
[2]
(ii) Use your calculator to evaluate the left hand side of (*) for increasingly large values of k (e.g. 103,
106, 109, …). State briefly two important results in numerical methods that are illustrated by your
working.
[4]
© OCR 2010
4776/01 Jan10
3
5
A function f(x) has the following values correct to 3 decimal places.
x
f(x)
0
1
2
3
4
1.883
2.342
2.874
3.491
4.206
(i) Show, by means of a difference table, that a cubic polynomial fits these data points closely but not
exactly.
[4]
(ii) Use Newton’s forward difference formula to estimate the value of f(1.5).
[4]
Section B (36 marks)
6
(i) The derivative of a function is to be estimated numerically. Show, with the aid of a sketch, that the
central difference method will generally be more accurate than the forward difference method. [4]
(ii) The table shows two values of tan xº correct to 7 significant figures.
x
tan xº
60
62
1.732 051
1.880 726
Use these two values to estimate the derivative of tan xº at x = 60.
Use your calculator to find two further estimates of this derivative, using the forward difference
method and taking h = 1 and h = 0.5.
[4]
(iii) Use the central difference method with h = 2, h = 1 and h = 0.5 to obtain three estimates of the
derivative of tan xº at x = 60.
[4]
(iv) Show that the differences between the estimates in part (ii) reduce by a factor of about 0.5 as h is
halved.
By considering the differences between the estimates in part (iii) show that the central difference
method seems to converge more rapidly than the forward difference method.
[6]
[Question 7 is printed overleaf.]
© OCR 2010
4776/01 Jan10
Turn over
4
7
(i) Show, by means of a sketch or otherwise, that the equation
x = 3 sin x,
(*)
where x is in radians, has a root, α, in the interval ( 12 π, π). Determine how many other non-zero
roots, if any, the equation has.
[3]
(ii) Determine whether or not the iteration
xr+1 = 3 sin xr,
starting with x0 = 2, converges to α. Illustrate your answer with a staircase or cobweb diagram as
appropriate.
[7]
(iii) Show that equation (*) may be rearranged into the form
x = sin x + 23 x.
Show that the corresponding iteration, starting with x0 = 2, converges rapidly. State to 5 decimal
places the value to which the iteration converges. Verify that this value for α is correct to 5 decimal
places.
[8]
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MATHEMATICS (MEI)
Numerical Methods
Monday 24 May 2010
Afternoon
* O C E / 1 3 6 6 0 *
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Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespaces
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Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestion.
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Turn over
2
Section A (36 marks)
1
(i) Show that the equation
1 = 3 – x2
x
(*)
has a root, a, between x =1 and x = 2.
Show that the iteration
xr+1 =
1 ,
3 – x 2r
[5]
with x0 = 1.5, converges, but not to a.
(ii) By rearranging (*), find another iteration that does converge to a. You should demonstrate the
convergence by carrying out several steps of the iteration.
[3]
2
A function f(x) has the values shown in the table.
x
f(x)
2.8
3
3.2
0.9508
0.9854
0.9996
(i) Taking the values of f(x) to be exact, use the forward difference method and the central difference
method to find two estimates of f′(3). State which of these you would expect to be more accurate.
[5]
(ii) Now suppose that the values of f(x) have been rounded to the four significant figures shown. Find,
for each method used in part (i), the largest possible value it gives for the estimate of f′(3).
[2]
3
(i) X is an approximation to the number x such that X = x (1 + r). State what r represents.
[4]
Show that, provided r is small, X n ≈ x n (1 + nr).
(ii) The number G = 0.577 is an approximation to the number g. G is about 0.04% smaller than g.
State, in similar terms, relationships between
(A) G2 and g2,
(B)
4
G and g .
[3]
The expression, sin x + tan x, where x is in radians, can be approximated by 2x for values of x close to
zero.
(i) Find the absolute and relative errors in this approximation when x = 0.2 and x = 0.1.

3

(ii) A better approximation is sin x + tan x ≈ 2 x + x , where k is an integer.
k
Use your results from part (i) to estimate k.
©OCR2010
4776/01Jun10
[4]
[3]
3
5
A quadratic function, f(x), is to be determined from the values shown in the table.
x
f(x)
1
3
6
–10
–12
30
Explain why Newton’s forward difference formula would not be useful in this case.
Use Lagrange’s interpolation formula to find f(x) in the form ax2 + bx + c.
[7]
Section B (36 marks)
6
The integral

I=
1.8
1
x 3 + 1 dx
is to be estimated numerically. You are given that, correct to 6 decimal places, the mid-point rule
estimate with h = 0.8 is 1.547 953 and that the trapezium rule estimate with h = 0.8 is 1.611 209.
(i) Find the mid-point rule and trapezium rule estimates with h = 0.4 and h = 0.2.
Hence find three Simpson’s rule estimates of I.
(ii) Write down, with a reason, the value of I to the accuracy that appears to be justified.
[7]
[2]
(iii) Taking your answer in part (ii) to be exact, show in a table the errors in the mid-point rule and
trapezium rule estimates of I.
Explain what these errors show about
(A) the relative accuracy of the mid-point rule and the trapezium rule,
(B) the rates of convergence of the mid-point rule and the trapezium rule.
7
[8]
(i) Show that the equation
x5 – 8x + 5 = 0
(*)
has a root in the interval (0, 1).
Find this root, using the Newton-Raphson method, correct to 6 significant figures.
Show, by considering the differences between successive iterates, that the convergence of the
Newton-Raphson iteration is faster than first order.
[11]
(ii) You are now given that equation (*) has a root in the interval (1.4, 1.5). Find this root, correct to 3
significant figures, using the secant method. Determine whether or not the secant method is faster
than first order.
[8]
©OCR2010
4776/01Jun10
ADVANCED SUBSIDIARY GCE
4776/01
MATHEMATICS (MEI)
Numerical Methods
* O C E / 2 7 6 2 7 *
Friday 14 January 2011
Afternoon
Candidates answer on the answer booklet.
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8 page answer booklet
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MEI Examination Formulae and Tables (MF2)
Duration: 1 hour 30 minutes
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Scientific or graphical calculator
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4
7
7
6
0
1
*
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Write your name, centre number and candidate number in the spaces provided on the
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Use black ink. Pencil may be used for graphs and diagrams only.
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your answer.
Answer all the questions.
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Do not write in the bar codes.
INFORMATION FOR CANDIDATES
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The number of marks is given in brackets [ ] at the end of each question or part
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This document consists of 4 pages. Any blank pages are indicated.
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2
Section A (36 marks)
1
(i) Show that the equation 1 + x = tan x, where x is in radians, has a root in the interval [1, 1.2].
[2]
(ii) Show numerically that the iteration xr+1 = tan xr – 1 with x0 = 1.1 diverges.
[2]
(iii) Use another iteration to find the root correct to 3 decimal places.
[4]
兰 f(x) dx, using the mid-point rule (M) and the trapezium
4
2
The table shows some estimates of an integral,
2
rule (T ), for given values of h.
h
M
T
2
1.987 467
1.354 440
1
1.830 595
0.5
Copy the table and fill in the additional estimates that can be found.
Obtain the Simpson’s rule estimates that can be found.
Give the value of the integral to the accuracy that appears justified.
3
[8]
The table shows values of g(x) correct to 4 decimal places.
x
0
0.5
1
g(x)
1.4509
1.6799
2.0100
(i) Use the forward difference method to find two estimates of gʹ(0). State, with a reason, which of
these is likely to be more accurate.
[4]
(ii) Use the central difference method to find an estimate of gʹ(0.5). Comment on the likely accuracy
of this estimate compared to those in part (i).
[2]
© OCR 2011
4776/01 Jan11
3
4
A bank’s computer system calculates the interest payable on each savings account every day. A running
total is kept of the daily amounts of interest, and accounts are credited with this interest at the end of
each year. The bank used to round the daily amounts of interest payable to the nearest 0.01 of a penny,
but they decide to chop to the nearest 0.01 of a penny instead.
(i) Find the maximum possible loss in a year to a savings account because of the chopping, and
explain how this loss could occur. State, with a reason, what the average loss will be.
[4]
(ii) The bank calculates that chopping in this way will generate an additional profit of about £150 000
per year. Estimate the number of savings accounts the bank has.
[2]
5
The function P(x) is known to be a polynomial. Some values of P(x) are given in the table.
x
1
3
5
7
9
P(x)
–10
3
44
129
274
(i) Use a difference table to determine, with a reason, the least possible degree of polynomial that will
fit all the data points.
[4]
(ii) Assuming that P(x) is of this degree, extend your table to find the values of P(–1) and P(11).
[4]
Section B (36 marks)
6
In this question,
f(x) =
sin x
x
–
,
x
sin x
where x is in radians. For small non-zero values of x, f(x) may be approximated by g(x) or by h(x),
where
1
g(x) = 3 x2
and
h(x) =
2x2
.
6 – x2
(i) Find the absolute and relative errors in g(x) and h(x) as approximations to f(x) for
(A) x = 0.2,
(B) x = 0.1
[9]
4g(x) + h(x)
. Explain by reference to part (i) why this
5
would be expected to be a good approximation.
Find the absolute and relative errors when this third approximation is used to estimate f(0.2) and
f(0.1).
[6]
x
(iii) Use your calculator to evaluate
when x = 10–4.
sin x
(ii) A third approximation to f(x) is given by
When x = 10–4, a cheap calculator evaluates f(x) as zero. Use an approximate formula to find a
better value for f(10–4). Explain why the cheap calculator makes an error.
[3]
© OCR 2011
4776/01 Jan11
Turn over
4
7
(i) Show that the equation f(x) = 0, where
f(x) = x7 + x5 – 1,
(*)
has a root in the interval [0, 1].
By considering f ʹ(x) show that there are no other roots.
Sketch the graph of y = f(x) for x ⭓ 0.
[7]
(ii) Obtain the Newton-Raphson iteration based on (*). Starting with x0 = 0.6, find x1 and x2. Illustrate
this iteration on your sketch of y = f(x).
[7]
(iii) Use the Newton-Raphson iteration to find x1 and x2 in the cases
(A) x0 = 0.3,
(B) x0 = 0.9.
Comment on your results in each case.
[4]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
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opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2011
4776/01 Jan11
ADVANCED SUBSIDIARY GCE
4776/01
MATHEMATICS (MEI)
Numerical Methods
QUESTION PAPER
* 4 7 1 3 9 0 0 6 1 1 *
Candidates answer on the printed answer book.
OCR supplied materials:
• Printedanswerbook4776/01
• MEIExaminationFormulaeandTables(MF2)
Wednesday 18 May 2011
Morning
Duration:1hour30minutes
Other materials required:
• Scientificorgraphicalcalculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the printed answer book and the question paper.
•
•
•
•
•
•
•
•
•
Thequestionpaperwillbefoundinthecentreoftheprintedanswerbook.
Writeyourname,centrenumberandcandidatenumberinthespacesprovidedontheprinted
answerbook.Pleasewriteclearlyandincapitalletters.
Write your answer to each question in the space provided in the printed answer book.
Additionalpapermaybeusedifnecessarybutyoumustclearlyshowyourcandidatenumber,
centre number and question number(s).
Useblackink.Pencilmaybeusedforgraphsanddiagramsonly.
Readeachquestioncarefully.Makesureyouknowwhatyouhavetodobeforestartingyour
answer.
Answerall the questions.
Donot write in the bar codes.
Youarepermittedtouseagraphicalcalculatorinthispaper.
Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATION FOR CANDIDATES
Thisinformationisthesameontheprintedanswerbookandthequestionpaper.
•
•
•
•
Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestiononthe
question paper.
Youareadvisedthatananswermayreceiveno marksunlessyoushowsufficientdetailofthe
workingtoindicatethatacorrectmethodisbeingused.
Thetotalnumberofmarksforthispaperis72.
Theprintedanswerbookconsistsof12pages.Thequestionpaperconsistsof4pages.Anyblank
pagesareindicated.
INSTRUCTIONS TO EXAMS OFFICER/INVIGILATOR
•
Donotsendthisquestionpaperformarking;itshouldberetainedinthecentreordestroyed.
©OCR2011 [M/102/2666]
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2
Section A (36 marks)
1
The equation f(x) = 0, where f(x) is a continuous function, is known to have a single root in the interval
[0.4, 1.8].
(i) Suppose the root is to be found using the bisection method. State the best possible estimate of
the root at the start of the process. State also the maximum possible error associated with that
estimate.
Determine how many iterations of the bisection process would be required to reduce the maximum
possible error to less than 0.05.
[4]
(ii) Given now that f(0.4) = –0.2 and f(1.8) = 0.5, find an estimate of the root using the false position
method.
[3]
2
The function g(x) has the values shown in the table.
x
1.80
2.00
2.20
g(x)
2.66
2.85
3.02
(i) Takingthedatatobeexact,usethecentraldifferenceformulatoestimateg′(2).
[2]
(ii) Suppose instead that the x values are exact but the values of g(x) are rounded to 2 decimal places.
Findanappropriaterangeofestimatesofg′(2).
[3]
(iii) Now suppose that all the values in the table have been rounded to 2 decimal places. Find the
appropriaterangeofestimatesofg′(2)inthiscase.
[3]
3
The function Q(x) is known to be quadratic and it has the values shown in the table.
x
–1
1
5
Q(x)
–4
–12
20
(i) Write down the estimate of Q(0) obtained by linear interpolation.
[1]
(ii) Use Lagrange’s method to write down an expression for Q(x). [You are not required to simplify
this expression.]
[5]
(iii) Find the exact value of Q(0).
©OCR2011
[2]
4776/01Jun11
3
4
(i) Show that the equation x = 1 – x4 has a root in the interval [0.7, 0.8].
[2]
(ii) Show, by considering the derivative of 1 – x4, that the iteration xr +1 = 1 – xr4, with a starting value
in the interval [0.7, 0.8], will diverge.
[4]
5
(i) Find the absolute error and the relative error when X = 3.162 is used as an approximation to
x=√10.
[3]
(ii) Find the relative error if X 4 is used as an approximation to x4.
[3]
(iii) State, in terms of k, the approximate relative error if X k is used as an approximation to x k.
[1]
Section B (36 marks)
6
The integral I = 
2
2.8
1+ x 3 dx is to be determined numerically. You should give all your answers to
7 decimal places unless instructed otherwise.
(i) Find mid-point rule and trapezium rule estimates of I, taking h = 0.8.
Use these two estimates to find a second trapezium rule estimate and a Simpson’s rule estimate
of I.
[8]
(ii) Find the mid-point rule estimate with h = 0.4, and hence obtain a second Simpson’s rule estimate
of I.
[3]
(iii) You are now given that the mid-point rule estimate of I with h = 0.2 is 3.091 429 8, correct to
7 decimal places.
Find a third Simpson’s rule estimate. Show by considering ratios of differences that Simpson’s
rule is of order h4.
Give the value of I to the accuracy that appears justified.
[7]
[Question 7 is printed overleaf.]
©OCR2011
4776/01Jun11
Turn over
4
7
The function f(x) has the exact values shown in the table.
x
1
3
5
f(x)
4
–2
10
(i) Use Newton’s forward difference interpolation formula to find the quadratic function that fits the
data. (There is no need to simplify your answer.)
[6]
(ii) Hence estimate the values of f(2) and f(6). State, with a reason, which of these estimates is likely
to be more accurate.
[3]
(iii) Now suppose that f(7) = 11. Find the cubic function that fits all the data. Use this cubic to estimate
f(2) and f(6).
[7]
(iv) Comment on (A) the absolute changes and (B) the relative changes in the estimates of f(2) and f(6)
from part (ii) to part (iii).
[2]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
departmentoftheUniversityofCambridge.
©OCR2011
4776/01Jun11
ADVANCED SUBSIDIARY GCE
4776/01
MATHEMATICS (MEI)
Numerical Methods
QUESTION PAPER
* 4 7 1 3 9 0 0 6 1 1 *
Candidates answer on the printed answer book.
OCR supplied materials:
• Printedanswerbook4776/01
• MEIExaminationFormulaeandTables(MF2)
Wednesday 18 May 2011
Morning
Duration:1hour30minutes
Other materials required:
• Scientificorgraphicalcalculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the printed answer book and the question paper.
•
•
•
•
•
•
•
•
•
Thequestionpaperwillbefoundinthecentreoftheprintedanswerbook.
Writeyourname,centrenumberandcandidatenumberinthespacesprovidedontheprinted
answerbook.Pleasewriteclearlyandincapitalletters.
Write your answer to each question in the space provided in the printed answer book.
Additionalpapermaybeusedifnecessarybutyoumustclearlyshowyourcandidatenumber,
centre number and question number(s).
Useblackink.Pencilmaybeusedforgraphsanddiagramsonly.
Readeachquestioncarefully.Makesureyouknowwhatyouhavetodobeforestartingyour
answer.
Answerall the questions.
Donot write in the bar codes.
Youarepermittedtouseagraphicalcalculatorinthispaper.
Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATION FOR CANDIDATES
Thisinformationisthesameontheprintedanswerbookandthequestionpaper.
•
•
•
•
Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestiononthe
question paper.
Youareadvisedthatananswermayreceiveno marksunlessyoushowsufficientdetailofthe
workingtoindicatethatacorrectmethodisbeingused.
Thetotalnumberofmarksforthispaperis72.
Theprintedanswerbookconsistsof12pages.Thequestionpaperconsistsof4pages.Anyblank
pagesareindicated.
INSTRUCTIONS TO EXAMS OFFICER/INVIGILATOR
•
Donotsendthisquestionpaperformarking;itshouldberetainedinthecentreordestroyed.
©OCR2011 [M/102/2666]
DC(LEO)27629/4
OCRisanexemptCharity
Turn over
2
Section A (36 marks)
1
The equation f(x) = 0, where f(x) is a continuous function, is known to have a single root in the interval
[0.4, 1.8].
(i) Suppose the root is to be found using the bisection method. State the best possible estimate of
the root at the start of the process. State also the maximum possible error associated with that
estimate.
Determine how many iterations of the bisection process would be required to reduce the maximum
possible error to less than 0.05.
[4]
(ii) Given now that f(0.4) = –0.2 and f(1.8) = 0.5, find an estimate of the root using the false position
method.
[3]
2
The function g(x) has the values shown in the table.
x
1.80
2.00
2.20
g(x)
2.66
2.85
3.02
(i) Takingthedatatobeexact,usethecentraldifferenceformulatoestimateg′(2).
[2]
(ii) Suppose instead that the x values are exact but the values of g(x) are rounded to 2 decimal places.
Findanappropriaterangeofestimatesofg′(2).
[3]
(iii) Now suppose that all the values in the table have been rounded to 2 decimal places. Find the
appropriaterangeofestimatesofg′(2)inthiscase.
[3]
3
The function Q(x) is known to be quadratic and it has the values shown in the table.
x
–1
1
5
Q(x)
–4
–12
20
(i) Write down the estimate of Q(0) obtained by linear interpolation.
[1]
(ii) Use Lagrange’s method to write down an expression for Q(x). [You are not required to simplify
this expression.]
[5]
(iii) Find the exact value of Q(0).
©OCR2011
[2]
4776/01Jun11
3
4
(i) Show that the equation x = 1 – x4 has a root in the interval [0.7, 0.8].
[2]
(ii) Show, by considering the derivative of 1 – x4, that the iteration xr +1 = 1 – xr4, with a starting value
in the interval [0.7, 0.8], will diverge.
[4]
5
(i) Find the absolute error and the relative error when X = 3.162 is used as an approximation to
x=√10.
[3]
(ii) Find the relative error if X 4 is used as an approximation to x4.
[3]
(iii) State, in terms of k, the approximate relative error if X k is used as an approximation to x k.
[1]
Section B (36 marks)
6
The integral I = 
2
2.8
1+ x 3 dx is to be determined numerically. You should give all your answers to
7 decimal places unless instructed otherwise.
(i) Find mid-point rule and trapezium rule estimates of I, taking h = 0.8.
Use these two estimates to find a second trapezium rule estimate and a Simpson’s rule estimate
of I.
[8]
(ii) Find the mid-point rule estimate with h = 0.4, and hence obtain a second Simpson’s rule estimate
of I.
[3]
(iii) You are now given that the mid-point rule estimate of I with h = 0.2 is 3.091 429 8, correct to
7 decimal places.
Find a third Simpson’s rule estimate. Show by considering ratios of differences that Simpson’s
rule is of order h4.
Give the value of I to the accuracy that appears justified.
[7]
[Question 7 is printed overleaf.]
©OCR2011
4776/01Jun11
Turn over
4
7
The function f(x) has the exact values shown in the table.
x
1
3
5
f(x)
4
–2
10
(i) Use Newton’s forward difference interpolation formula to find the quadratic function that fits the
data. (There is no need to simplify your answer.)
[6]
(ii) Hence estimate the values of f(2) and f(6). State, with a reason, which of these estimates is likely
to be more accurate.
[3]
(iii) Now suppose that f(7) = 11. Find the cubic function that fits all the data. Use this cubic to estimate
f(2) and f(6).
[7]
(iv) Comment on (A) the absolute changes and (B) the relative changes in the estimates of f(2) and f(6)
from part (ii) to part (iii).
[2]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
departmentoftheUniversityofCambridge.
©OCR2011
4776/01Jun11