2024_OCR: A Level Further Mathematics B (MEI) Y434/01 Numerical Methods
(Merged Question Paper & Marking Scheme)
Key areas to revise:
1. Root-Finding Methods:
o Study methods for finding approximate solutions to equations, including the bisection method,
Newton-Raphson method, and iteration methods. Understand the concepts of convergence
and error estimation, and how to apply these methods to solve equations that cannot be solved
algebraically.
2. Solving Systems of Equations:
o Learn methods to solve systems of linear equations numerically, such as Gaussian elimination,
LU decomposition, and iterative methods like Jacobi and Gauss-Seidel. Understand how to
apply these methods to solve large systems efficiently.
3. Numerical Integration:
o Focus on methods for approximating definite integrals, such as trapezium rule, Simpson’s
rule, and midpoint rule. Understand how to derive and apply these methods for finding the area
under curves when an exact integral cannot be computed.
4. Numerical Differentiation:
o Study methods for approximating derivatives, including forward, backward, and central
difference formulas. Learn about their accuracy and how to apply these methods to estimate the
rate of change of functions numerically.
5. Euler’s Method and Runge-Kutta Methods:
o Understand how Euler’s method and Runge-Kutta methods are used to solve ordinary
differential equations (ODEs) numerically. Study the error estimation and the differences
between these methods in terms of efficiency and accuracy.
6. Interpolation and Approximation:
o Study the methods of linear interpolation and polynomial interpolation (e.g., Lagrange
interpolation). Learn how to approximate unknown values between data points and understand
the importance of minimizing error in approximations.
7. Error Analysis and Estimation:
o Learn how to estimate the errors in numerical methods and the impact of truncation and
rounding errors. Understand the concept of relative error and absolute error, and how to use
them to assess the reliability of numerical solutions.
8. Numerical Methods for Curve Fitting:
o Focus on the use of least squares fitting for approximating data with a curve, including linear
regression and polynomial fitting. Understand how numerical methods can be used to find the
best-fit line or curve for a given set of data points.
9. Applications of Numerical Methods:
o Explore real-world applications where numerical methods are essential, such as solving
problems in physics, engineering, economics, and biology, where analytical solutions may not be
possible.
, Oxford Cambridge and RSA
Wednesday 19 June 2024 – Afternoon
A Level Further Mathematics B (MEI)
Y434/01 Numerical Methods
Time allowed: 1 hour 15 minutes
*1417498445*
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined page at the end of the Printed Answer
Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2024 [A/508/5598] OCR is an exempt Charity
DC (DE/SG) 333178/4 Turn over
, 2
1 The table shows some values of x, together with the associated values of a function, f(x).
x 1.9 2 2.1
f(x) 0.5842 0.6309 0.6753
(a) Use the information in the table to calculate the most accurate estimate of f '(2) possible. [2]
(b) Calculate an estimate of the error when f(2) is used as an estimate of f(2.05). [2]
2 You are given that a = tanh(1) and b = tanh(2).
A is the approximation to a formed by rounding tanh(1) to 1 decimal place.
B is the approximation to b formed by rounding tanh(2) to 1 decimal place.
(a) Calculate the following.
• The relative error RA when A is used to approximate a.
•
The relative error RB when B is used to approximate b. [3]
A a
(b) Calculate the relative error RC when C = is used to approximate c = . [2]
B b
(c) Comment on the relationship between RA , RB and RC . [1]
© OCR 2024 Y434/01 Jun24
, 3
3 The equation x2 - cosh (x - 2) = 0 has two roots, a and b, such that a 1 b .
(a) Use the iterative formula
xn +1 = g(xn) where g(xn) = ,
starting with x0 = 1, to find a correct to 3 decimal places. [2]
The diagram shows the part of the graphs of y = x and y = g(x) for 0 G x G 7.
y
10
y = g(x)
8
y=x
6
4
2
0
0 2 4 6 x
(b) Explain why the iterative formula used to find a cannot successfully be used to find b, even
if x0 is very close to b. [1]
(c) Use the relaxed iteration
xn +1 = (1 - m) xn + mg(xn),
with m =-0.21 and x0 = 6.4, to find b correct to 3 decimal places. [2]
In part (c) the method of relaxation was used to convert a divergent sequence of approximations
into a convergent sequence.
(d) State one other application of the method of relaxation. [1]
© OCR 2024 Y434/01 Jun24 Turn over
(Merged Question Paper & Marking Scheme)
Key areas to revise:
1. Root-Finding Methods:
o Study methods for finding approximate solutions to equations, including the bisection method,
Newton-Raphson method, and iteration methods. Understand the concepts of convergence
and error estimation, and how to apply these methods to solve equations that cannot be solved
algebraically.
2. Solving Systems of Equations:
o Learn methods to solve systems of linear equations numerically, such as Gaussian elimination,
LU decomposition, and iterative methods like Jacobi and Gauss-Seidel. Understand how to
apply these methods to solve large systems efficiently.
3. Numerical Integration:
o Focus on methods for approximating definite integrals, such as trapezium rule, Simpson’s
rule, and midpoint rule. Understand how to derive and apply these methods for finding the area
under curves when an exact integral cannot be computed.
4. Numerical Differentiation:
o Study methods for approximating derivatives, including forward, backward, and central
difference formulas. Learn about their accuracy and how to apply these methods to estimate the
rate of change of functions numerically.
5. Euler’s Method and Runge-Kutta Methods:
o Understand how Euler’s method and Runge-Kutta methods are used to solve ordinary
differential equations (ODEs) numerically. Study the error estimation and the differences
between these methods in terms of efficiency and accuracy.
6. Interpolation and Approximation:
o Study the methods of linear interpolation and polynomial interpolation (e.g., Lagrange
interpolation). Learn how to approximate unknown values between data points and understand
the importance of minimizing error in approximations.
7. Error Analysis and Estimation:
o Learn how to estimate the errors in numerical methods and the impact of truncation and
rounding errors. Understand the concept of relative error and absolute error, and how to use
them to assess the reliability of numerical solutions.
8. Numerical Methods for Curve Fitting:
o Focus on the use of least squares fitting for approximating data with a curve, including linear
regression and polynomial fitting. Understand how numerical methods can be used to find the
best-fit line or curve for a given set of data points.
9. Applications of Numerical Methods:
o Explore real-world applications where numerical methods are essential, such as solving
problems in physics, engineering, economics, and biology, where analytical solutions may not be
possible.
, Oxford Cambridge and RSA
Wednesday 19 June 2024 – Afternoon
A Level Further Mathematics B (MEI)
Y434/01 Numerical Methods
Time allowed: 1 hour 15 minutes
*1417498445*
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined page at the end of the Printed Answer
Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2024 [A/508/5598] OCR is an exempt Charity
DC (DE/SG) 333178/4 Turn over
, 2
1 The table shows some values of x, together with the associated values of a function, f(x).
x 1.9 2 2.1
f(x) 0.5842 0.6309 0.6753
(a) Use the information in the table to calculate the most accurate estimate of f '(2) possible. [2]
(b) Calculate an estimate of the error when f(2) is used as an estimate of f(2.05). [2]
2 You are given that a = tanh(1) and b = tanh(2).
A is the approximation to a formed by rounding tanh(1) to 1 decimal place.
B is the approximation to b formed by rounding tanh(2) to 1 decimal place.
(a) Calculate the following.
• The relative error RA when A is used to approximate a.
•
The relative error RB when B is used to approximate b. [3]
A a
(b) Calculate the relative error RC when C = is used to approximate c = . [2]
B b
(c) Comment on the relationship between RA , RB and RC . [1]
© OCR 2024 Y434/01 Jun24
, 3
3 The equation x2 - cosh (x - 2) = 0 has two roots, a and b, such that a 1 b .
(a) Use the iterative formula
xn +1 = g(xn) where g(xn) = ,
starting with x0 = 1, to find a correct to 3 decimal places. [2]
The diagram shows the part of the graphs of y = x and y = g(x) for 0 G x G 7.
y
10
y = g(x)
8
y=x
6
4
2
0
0 2 4 6 x
(b) Explain why the iterative formula used to find a cannot successfully be used to find b, even
if x0 is very close to b. [1]
(c) Use the relaxed iteration
xn +1 = (1 - m) xn + mg(xn),
with m =-0.21 and x0 = 6.4, to find b correct to 3 decimal places. [2]
In part (c) the method of relaxation was used to convert a divergent sequence of approximations
into a convergent sequence.
(d) State one other application of the method of relaxation. [1]
© OCR 2024 Y434/01 Jun24 Turn over
