OCR A Level Further Maths B (MEI) Numerical Methods (minor option - choose one with a major, or three alone) June 2022 question paper Y434

C = 3.7622 and S = 3.6269 are used to approximate cosh2 and sinh2 respectively.

Determine whether these approximations are the result of chopping or rounding the values of

cosh2 and sinh2.

Calculate the relative error when C² - S² is used to approximate cosh²2 - sinh²2, giving your

answer correct to 3 significant figures.

Without doing any further calculations, explain whether the same value for the relative error

is obtained when (C - S)² is used to approximate (cosh 2 - sinh 2)².

The table shows some values of x and the associated values of y = f (x).

Explain why your answer to part (b) is likely to be closer than your answer to part (a) to the

When x = 5 it is given that y = 1.4645 and = 0.1820, correct to 4 decimal places.

Determine an estimate of the error when f (5) is used to estimate f (5.024).

The equation f(x) = sin^-1(x) - x + 0.1 = 0 has a root α such that -1 < α < 0.

Alex uses an iterative method to find a sequence of approximations to α. Some of the associated spreadsheet output is shown in the table.

	C	D	E
4	r	xr	f(xr)
5	0	-1	-0.4707963
6	1	-0.8	-0.0272952
7	2	-0.787691	-0.0193610
8	3	-0.7576546	-0.0020574
9	4	-0.7540834	-0.0001740
10	5
11	6

The formula in cell D7 is

and equivalent formulae are in cells D8 and D9.

State the method being used.

Use the values in the spreadsheet to calculate x₅ and x₆, giving your answers correct to

7 decimal places.

State the value of α as accurately as you can, justifying the precision quoted.

Alex uses a calculator to check the value in cell D9, his result is - 0.7540832686.

Explain why this is different to the value displayed in cell D9.

The value displayed in cell E11 in Alex’s spreadsheet is - 1.4629 E - 09.

Write this value in standard mathematical notation.

Fig. 4.1 shows part of the graph of y = e^x - x² - x - 1.1.

The equation e^x - x² - x - 1.1 = 0 has a root α such that 1 < α < 2.

Ali is considering using the Newton-Raphson method to find α. Ali could use a starting value of x₀ = 1 or a starting value of x₀ = 2.

State what is being calculated in the following columns of the spreadsheet.

Column N

Column O

Kai uses the midpoint rule, trapezium rule and Simpson’s rule to find approximations to

	F	G	H	I
3	n	Mn	Tn	S2n
4	1	0.2436699	0.1479020
5	2	0.2306967

Write down a suitable spreadsheet formula for cell H5.

Complete the copy of the table in the Printed Answer Booklet, giving the values correct to

7 decimal places.

Use the information in Fig. 6.1 to find the value of the root as accurately as you can,

justifying the precision quoted.

The relaxed iteration x_n+1 = (1 - λ)x_n + λg(x_n), with λ = 0.51 and x₀ = 0, is to be used to find the root of the equation sin^-1(x) - x² + 1 = 0.

Complete the copy of Fig. 6.2 in the Printed Answer Booklet, giving the values of x_r correct to 7 decimal places and the values in the difference column and ratio column correct to

Write down the value of the root correct to 7 decimal places.

Explain why extrapolation could not be used in this case to find an improved approximation

using this sequence of iterates.

In this case the method of relaxation has been used to speed up the convergence of an iterative scheme.

Name another application of the method of relaxation.

Sam decided to go on a high-protein diet. Sam’s mass in kg, M, after t days of following the diet is recorded in Fig. 7.1.

A difference table for the data is shown in Fig. 7.2.

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

Sam’s doctor uses these data to construct a cubic interpolating polynomial to model Sam’s mass at time t days after starting the diet.

Find the model in the form M = at³ + bt² + ct + d, where a, b, c and d are constants to be

determined.

Subsequently it is found that when t = 40, M = 78.7 and when t = 50, M = 80.05.

Determine whether the model is a good fit for these data.

By completing the extended copy of Fig. 7.2 in the Printed Answer Booklet, explain why a

quartic model may be more appropriate for the data.

Refine the doctor’s model to include a quartic term.

Explain whether the new model for Sam’s mass is likely to be appropriate over a longer

period of time.