, 2
1 The table shows some values of x and the associated values of f(x).
x 1.9 2.0 2.1
f(x) 0.216 92 0.2 0.184 84
(a) Determine an estimate of f l(2.0) using the forward difference method. [2]
(b) Determine an estimate of f l(2.0) using the central difference method. [2]
2 The numbers p and q are approximated by
P = 323 and Q = 162.
P has been found by rounding p to the nearest whole number.
Q has been found by chopping q to the nearest whole number.
(a) (i) Find the maximum possible relative error in using P to approximate p. [1]
(ii) Find the maximum possible relative error in using Q to approximate q. [1]
200
(b) Determine the range of possible values of R = p -2q
. [3]
(c) Explain why your answer to part (b) is so large. [1]
y0.5
1.3
3 Approximations to 1 + x3 dx using the midpoint rule, the trapezium rule and Simpson’s rule
with n = 1 and n = 2 are shown in the table. The table is incomplete.
n Mn Tn S2n
1 1.081 111
2 1.074 256
(a) Complete the copy of the table in the Printed Answer Booklet. Give your answers to
6 decimal places. [4]
(b) Without doing any further calculations, state the value of y 1.3 1 + x3 dx as accurately as
0.5
possible. You must justify the precision quoted. [1]
© OCR 2025 Y434/01 Jun25
, 3
4 The Newton-Raphson method is to be used to find the positive root of the equation
tanh x - x2 + 4 = 0 .
The diagram shows part of the graph of y = tanh x - x2 + 4.
y
5
x
–3 –2 –1 0 1 2 3 4
–5
–10
(a) On the copy of the diagram in the Printed Answer Booklet, show how the Newton-Raphson
method works to find x1 using the starting value x0 = 1. [1]
(b) Use the Newton-Raphson method using the starting value x0 = 1 to determine the values of
x1 and x2 correct to 8 decimal places. [4]
(c) Continue the iteration to determine the value of the positive root of the equation
tanh x - x2 + 4 = 0 correct to 7 decimal places. [2]
© OCR 2025 Y434/01 Jun25 Turn over
, 4
5 The diagram shows part of the graph of y = sinh x - 3x + 1.
y
10
5
x
–1 0 1 2 3 4
–5
The largest positive root, a, of the equation sinh x - 3x + 1 = 0 is to be found using the secant
method with x0 = 2 and x1 = 3.
Table 5.1 shows the associated spreadsheet output.
Table 5.1
◢ A B C D E
2 r xr f(xr) xr +1 f(xr +1)
3 0 2 –1.37314 3 2.01787
4 1 3 2.017875 2.404935 –0.7211
5 2 2.404935 –0.72109 2.561598 –0.2451
6 3 2.561598 –0.24513 2.642285 0.06018
7 4 2.642285 0.060175 2.626382 –0.0035
8 5 2.626382 –0.00348 2.627252 –5E-05
9 6 2.627252 –4.5E-05 2.627264 3.5E-08
(a) Write down a suitable formula for cell D4. [2]
(b) State the most accurate approximation to a in the spreadsheet output. [1]
(c) Determine whether this approximation is accurate to 6 decimal places. [2]
(d) Explain what would happen if the initial entry in cell D3 were to be changed from 3 to 1. [1]
© OCR 2025 Y434/01 Jun25
1 The table shows some values of x and the associated values of f(x).
x 1.9 2.0 2.1
f(x) 0.216 92 0.2 0.184 84
(a) Determine an estimate of f l(2.0) using the forward difference method. [2]
(b) Determine an estimate of f l(2.0) using the central difference method. [2]
2 The numbers p and q are approximated by
P = 323 and Q = 162.
P has been found by rounding p to the nearest whole number.
Q has been found by chopping q to the nearest whole number.
(a) (i) Find the maximum possible relative error in using P to approximate p. [1]
(ii) Find the maximum possible relative error in using Q to approximate q. [1]
200
(b) Determine the range of possible values of R = p -2q
. [3]
(c) Explain why your answer to part (b) is so large. [1]
y0.5
1.3
3 Approximations to 1 + x3 dx using the midpoint rule, the trapezium rule and Simpson’s rule
with n = 1 and n = 2 are shown in the table. The table is incomplete.
n Mn Tn S2n
1 1.081 111
2 1.074 256
(a) Complete the copy of the table in the Printed Answer Booklet. Give your answers to
6 decimal places. [4]
(b) Without doing any further calculations, state the value of y 1.3 1 + x3 dx as accurately as
0.5
possible. You must justify the precision quoted. [1]
© OCR 2025 Y434/01 Jun25
, 3
4 The Newton-Raphson method is to be used to find the positive root of the equation
tanh x - x2 + 4 = 0 .
The diagram shows part of the graph of y = tanh x - x2 + 4.
y
5
x
–3 –2 –1 0 1 2 3 4
–5
–10
(a) On the copy of the diagram in the Printed Answer Booklet, show how the Newton-Raphson
method works to find x1 using the starting value x0 = 1. [1]
(b) Use the Newton-Raphson method using the starting value x0 = 1 to determine the values of
x1 and x2 correct to 8 decimal places. [4]
(c) Continue the iteration to determine the value of the positive root of the equation
tanh x - x2 + 4 = 0 correct to 7 decimal places. [2]
© OCR 2025 Y434/01 Jun25 Turn over
, 4
5 The diagram shows part of the graph of y = sinh x - 3x + 1.
y
10
5
x
–1 0 1 2 3 4
–5
The largest positive root, a, of the equation sinh x - 3x + 1 = 0 is to be found using the secant
method with x0 = 2 and x1 = 3.
Table 5.1 shows the associated spreadsheet output.
Table 5.1
◢ A B C D E
2 r xr f(xr) xr +1 f(xr +1)
3 0 2 –1.37314 3 2.01787
4 1 3 2.017875 2.404935 –0.7211
5 2 2.404935 –0.72109 2.561598 –0.2451
6 3 2.561598 –0.24513 2.642285 0.06018
7 4 2.642285 0.060175 2.626382 –0.0035
8 5 2.626382 –0.00348 2.627252 –5E-05
9 6 2.627252 –4.5E-05 2.627264 3.5E-08
(a) Write down a suitable formula for cell D4. [2]
(b) State the most accurate approximation to a in the spreadsheet output. [1]
(c) Determine whether this approximation is accurate to 6 decimal places. [2]
(d) Explain what would happen if the initial entry in cell D3 were to be changed from 3 to 1. [1]
© OCR 2025 Y434/01 Jun25
