OCR GCE AS Level Further Maths B (MEI) Numerical Methods (optional - choose any two of Y411/Y412/Y413/Y414) June 2024 question paper Y414

The difference table shows some values of x and the associated values of y.

Complete the copy of the difference table in the Printed Answer Booklet.

Explain whether a cubic polynomial would be suitable to model the relationship between

y and x.

A spreadsheet is used to find the approximate value of π + e. The output is shown in the table.

and the formula in cell B3 is = BI + B2.

Show that the approximations to π and e displayed in cells B1 and B2 respectively have both

been rounded.

Find the relative error when 3.141 593 is used to approximate π.

Explain why the value displayed in cell B3 is not equal to 2.718 282 + 3.141593.

Following an injury a professional footballer has to undergo a rehabilitation programme before she can resume playing. As part of the programme she is required to take weekly fitness tests to assess her progress. When she achieves a score of 95% or over she is deemed fit enough to be part of the match day squad.

The footballer was ill at the end of week 3 and so did not take the fitness test. The percentage score she achieved, P, at the end of each week, w, is shown for w = 1, 2 and 4 in the table.

The medical staff at the footballer’s club believe the data may be modelled by a polynomial.

Use Lagrange’s form of the interpolating polynomial to construct a polynomial of degree 2 to

model the relationship between P and w.

The club are due to play a match at the end of week 6 and another match at the end of week 7.

Determine whether, according to the model, the footballer will be available to play in either

of these matches.

After returning to the match day squad, the medical staff continue to give the footballer a fitness test at the end of each week.

Explain why the model is no longer viable from the end of week 9.

The graph shows part of the graph of y = f (x).

The equation f (x) = 0 has two positive roots, α and β, where α > β.

The method of False Position is used to find α. A spreadsheet is used to find a sequence of approximations to α. The output is shown in Table 4.1.

Table 4.1

Write down the equation which is being solved.

The formula in cell V5 is = IF (AA 4 < 0, Z 4, V 4).

Write down a similar formula for cell X5.

By considering the output in Table 4.1, state the value of α as accurately as possible. You

must justify the precision quoted.

The Newton-Raphson method is used with x₀ = 2 to find a sequence of approximations to β. The associated spreadsheet output is shown in Table 4.2.

Table 4.2

Use the information in Table 4.2 to state the value of β as accurately as possible. You must

justify the precision quoted.

Explain what the ratios of differences tell you about the order of convergence of the sequence

of approximations to β.

State two advantages of using the Newton-Raphson method instead of the method of False

Position for finding the roots of an equation.

Table 5.1 shows some values of x and the associated values of f (x).

Table 5.1

Use the forward difference method to obtain four approximations to f ′ (5). Write your

answers in the copy of Table 5.2 in the Printed Answer Booklet.

State the value of f ′ (5) as accurately as you can. You must justify the precision quoted.

Ali uses a smaller value of h to find an approximation to f ′ (5). Ali’s spreadsheet output is shown in Table 5.3.

Table 5.3

Explain whether Ali’s approximation to f ′ (5) is likely to be more accurate or less accurate

than your answer to part (b).

Determine an approximation of the error in using f (5) to approximate f (5.02).

The equation 5 log₃x - x = 0 has two roots, α and β.

You are given that 1 < α < 2.

Use the iterative formula x_n+1 = g(x_n), where g(x_n) = and x₀ = 2, to determine the value

of α correct to 6 decimal places.

You are given that β ≈ 10.9.

The table shows some approximations to g ′ (10.9) found using the central difference method.

Explain why it is preferable to use the central difference method instead of the forward

difference method where possible.

Show how the approximation with h = 0.1 is calculated.

Write down the value of g ′ (10.9) as accurately as you can. You must justify the precision

quoted.

Explain why your answer to part b(iii) suggests that the iterative formula

may not be used to successfully find β, however close the starting value is to β.

Use the relaxed iteration

x_n+1 = (1 - λ)x_n + λg(x_n) with λ =-0.269 and x₀ = 10.9

to determine the value of β correct to 6 decimal places.

A student uses the midpoint rule, the trapezium rule and Simpson’s Rule to find a series of

Write down a suitable spreadsheet formula for cell I2.

Write down a suitable spreadsheet formula for cell K4.

By considering the values in column H, explain what may be inferred about the shape of the

curve y = f (x).

State the order of convergence of the sequence of values in column K.

Explain what the values of the ratios of differences tell you about the order of Simpson’s

Rule in this case.

In this question you must show detailed reasoning.