OCR GCE AS Level Further Maths B (MEI) Numerical Methods (optional - choose any two of Y411/Y412/Y413/Y414) June 2025 question paper Y414

The method of interval bisection is used to find a sequence of approximations to one of the roots of the equation e^x - x² - 3x = 0.

The table shows the associated spreadsheet output.

	A	B	C	D	E	F
1	a	f(a)	b	f(b)	c	f(c)
2	2	-2.61094	3	2.085537	2.5	-1.56751
3	2.5	-1.56751	3	2.085537	2.75	-0.16987
4	2.75	-0.16987	3	2.085537	2.875	0.834799
5	2.75	-0.16987	2.875	0.834799	2.8125	0.303839
6

Write down a suitable formula for cell E2.

The formula in cell A3 is = IF (F2 < 0,E2,A2).

Write down a similar formula for cell C3.

Complete row 6 of the table in the Printed Answer Booklet.

Without doing any more calculations, write down the value of the root correct to the maximum

number of decimal places which seems justified. You must explain the precision quoted.

The table gives three values of x and the associated values of y.

Use Lagrange’s form of the interpolating polynomial to construct a polynomial of degree 2 for the values in the table. Give your answer in the form

y = ax² + bx + c,

where a, b and c are constants to be determined.

Find the relative error when π is chopped to 3 decimal places.

Find the relative error when π is rounded to 3 decimal places.

You are given that y = π² - 5 and z = (π - 5)².

You are also given the following information.

• Y is an approximation to y.

• Z is an approximation to z.

• Y and Z are found by using π = 3.14.

A student states that the relative error in using Y to approximate y is exactly the same as the relative error in using Z to approximate z, because in each case the calculation involves squaring and subtracting 5.

Without doing any calculations, explain whether the student is correct.

A student is using a spreadsheet to investigate the curve y = f (x) at the point where x = 2. Some of the spreadsheet output is shown below.

Table 4.1 shows some values of x and the associated values of f (x).

Table 4.2 shows two approximations to f ′ (2) found using the same method with h = 0.001 and h = 0.01.

The formula in cell G10 is = (H5 - G5) /G9.

The formula in cell H10 is = (I 5 - G5)/ H 9.

State the method being used to find these approximations to f ′ (2).

• Explain why your answer is not the same as the value displayed in cell G10.

State the value of f ′ (2) as accurately as you can. You must justify the precision quoted.

Hence determine an estimate for the error when f (2) is used as an approximation to f (2.1).

Explain whether your answer to part (d) is likely to be an over-estimate or an under-estimate.

Simpson’s rule.

	F	G	H	I	J
21	n	Tn	Sn	difference	ratio
22	1	0.84557616
23	2	0.88144672	0.89340357
24	4	0.89066651	0.89373977	0.00034
25	8	0.89299453	0.89377054	3.1E-05	0.092
26	16	0.89357822	0.89377279	2.2E-06	0.073
27	32	0.89372426	0.89377294	1.5E-07	0.066
28	64	0.89376077	0.89377295	9.4E-09	0.063
29	128	0.8937699	0.89377295	5.9E-10	0.063

Write down the value displayed in cell I25 using standard mathematical notation.

Write down a suitable formula for cell H23.

Explain what the values in column J tell you about the order of Simpson’s rule in this

case.

Explain whether this is usual.

The equation 0.5 ln x + x² - x - 2 = 0 has a root α, such that 1 < α < 2.

A student attempts to find α using the iterative formula

x_n+1 = g(x_n) with x₀ = 1, where g(x) = 0.5 ln x + x² - 2.

The associated spreadsheet output is shown in the table.

	K	L
1	n	xn
2	0	1
3	1	-1
4	2	#NUM!

Explain why #NUM! is displayed in cell L4.

Explain why the iterative formula x_n+1 = g(x_n) may not be used to successfully find α,

whatever the value of x₀.

Use the relaxed iteration

x_n+1 = (1 - λ)x_n + λg(x_n),

with λ =-0.328 and x₀ = 1, to determine the value of α correct to 6 decimal places.

A student makes a cup of tea and investigates how long it takes to reach room temperature. The student places the cup of tea on a table in the kitchen and records the temperature T °C of the tea t minutes after it was made, at 5 minute intervals starting with t = 5. The data collected are shown in the table.

Show that a quadratic polynomial would be a reasonable model for these data.

Use Newton’s forward difference interpolation method to determine a suitable quadratic

polynomial for these data.

The temperature in the kitchen is 19°C. It takes one hour for the tea to cool to room temperature.

Use the model to find an estimate of the temperature of the tea after 1 hour.

Use the model to find an estimate of the temperature of the tea after an hour and a half.

Hence identify a limitation of the model.

Table 8.1 shows some values of x and the associated values of f (x).

Table 8.1

Use the central difference method to determine an approximation to f ′ (0.9).

Table 8.2 shows a sequence of approximations to f ′ (0.9) for different values of the step length, h together with some further analysis.

Table 8.2

Use extrapolation to determine the value of f ′ (0.9) as accurately as you can. You must justify

the precision quoted.

In this question you must show detailed reasoning

The equation 0.2x⁴ + 0.8x - 0.2 = 0 has a root α such that 0 < α < 1.

Use the Newton-Raphson method with x₀ = 1 to determine the value of α correct to

6 decimal places.

The equation 0.2x⁴ + 0.8x - 0.2 = 0 has another root β such that -2 < β <-1.

The Newton-Raphson method is used to find a sequence of approximations to β. The associated spreadsheet output is shown in the table below, together with some further analysis.

Explain what the values in column D suggest about the order of convergence of this

sequence of approximations.

Explain whether this is usual.