Numerical Solutions of Equations (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

The equation has a root in the interval

Use linear interpolation once on the interval to determine an approximation to , giving your answer to 3 decimal places.

The equation has another root in the interval

Determine

Hence, using as a first approximation to , apply the Newton–Raphson process once to to determine a second approximation to , giving your answer to 3 decimal places.

(a) Show that the equation has a root in the interval

(b) Use linear interpolation once on the interval to find an approximation to , giving your answer to 2 decimal places.

(a) Determine .

The equation has a root in the interval

(b) Using as a first approximation to , apply the Newton–Raphson procedure once to to find a second approximation to , giving your answer to 3 decimal places.

Show that the equation has a root, , in the interval

Starting with the interval , use interval bisection twice to show that lies in the interval

(i) Determine.

[1]

(ii) Using as a first approximation for , apply the Newton-Raphson process once to to determine a second approximation for , giving your answer to 3 decimal places.

[3]

Use linear interpolation once on the interval to obtain a different approximation for , giving your answer to 3 decimal places.

The equation has a single root, , that lies in the interval

(i) Determine

[2]

(ii) Explain why 0.25 cannot be used as an initial approximation for in the Newton-Raphson process.

[1]

(iii) Taking 0.15 as a first approximation to apply the Newton-Raphson process once to to obtain a second approximation to
Give your answer to 3 decimal places.

[2]

Use linear interpolation once on the interval to find another approximation to
Give your answer to 3 decimal places.

Show that the equation has a root in the interval [0.4, 0.5]

Determine .

Using as a first approximation to , apply the Newton-Raphson procedure once to to find a second approximation to , giving your answer to 3 decimal places.

The equation has another root in the interval [4.8, 4.9]

Use linear interpolation once on the interval [4.8, 4.9] to find an approximation to , giving your answer to 3 decimal places.

The table below shows values of for some values of , with values of given to 4 decimal places where appropriate.

Complete the table giving the values to 4 decimal places.

The equation has exactly one positive root,

determine an interval of width one that contains

Hence use interval bisection twice to obtain an interval of width 0.25 that contains

Given also that the equation has a negative root, , in the interval

use linear interpolation once on this interval to find an approximation for

Give your answer to 3 significant figures.

Show that the equation has a root, , in the interval

(i) Find

(ii) Hence, using as a first approximation to , apply the Newton–Raphson procedure once to to calculate a second approximation to , giving your answer to 3 decimal places.

Use linear interpolation once on the interval to find another approximation to . Give your answer to 3 decimal places.

(a) Show that the equation has a root in the interval

[2]

(b) Taking as a first approximation, apply the Newton–Raphson process twice to to obtain an approximate value of .Give your answer to 3 decimal places. Show your working clearly.

[4]

where is measured in radians.

The equation has a single root in the interval .

Use linear interpolation on the values at the end points of this interval to obtain an approximation to . Give your answer to 3 decimal places.

Show that the equation , where is in radians, has a root in the interval

Starting with the interval ,use interval bisection twice to find an interval of width in which lies.

Show that the equation has a root in the interval

Determine .

Using as a first approximation to , apply the Newton-Raphson procedure once to to calculate a second approximation to , giving your answer to 3 decimal places.