Free Numerical solution of equations questions for A Level Mathematics (9709), part of Pure Mathematics 3. Each question has the answer and the marking points an examiner would look for.
Generate a full Numerical Solution of Equations worksheet →All Pure Mathematics 3 questionsThe equation ln x = 5 − x has one root, α.
Show by calculation that α lies between 3 and 4.
Use the iterative formula xn+1 = 5 − ln xn to find α correct to 3 decimal places.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | sign change f(x) = ln x + x − 5 at 3 and 4 M1 sign change shown A1 | 2 |
| (b) | 3.693 correct iteration M1 iterates to 5 d.p. A1 3.693 A1 | 3 |
A chord AB of a circle with centre O and radius r divides the circle into two regions whose areas are in the ratio 1 : 3. Angle AOB = θ radians.
Show that θ = sin θ + π2.
Show by calculation that the equation has a root between 2.3 and 2.4.
Use the iterative formula θn+1 = ½(θn + sin θn + π2) with θ1 = 2 to find θ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | ½r2(θ − sin θ) = 14πr2 segment area = ½r2θ − ½r2 sin θ M1 equated to πr24 M1 rearranged to the given form A1 | 3 |
| (b) | sign change of θ − sin θ − π2 between 2.3 and 2.4 evaluate θ − sin θ − π2 at both ends: −0.017 and 0.154 M1 sign change and conclusion A1 | 2 |
| (c) | 2.310 (2.24005, 2.29757, 2.30784, 2.30955, 2.30983, …) uses the iteration correctly at least once M1 obtains final answer after sufficient iterations to justify accuracy to 3 d.p. M1 2.310 A1 | 3 |
The equation x3 − 4x − 9 = 0 has a root α. Use the iterative formula xn+1 = ∛(4xn + 9) with x1 = 2 to find α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
| Part | Answer and marking points | Marks |
|---|---|---|
| α = 2.707 (iterates 2.57128, 2.68168, 2.70200, 2.70570…) Uses iteration correctly at least once M1 Iterates to sufficient accuracy M1 2.707 A1 | 3 |
The equation ln x = 4 − x has one root, α.
Show by calculation that α lies between 2 and 3.
Use the iterative formula xn+1 = 4 − ln xn to find α correct to 3 decimal places.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | sign change f(x) = ln x + x − 4 at 2 and 3 M1 sign change shown A1 | 2 |
| (b) | 2.926 correct iteration M1 iterates to 5 d.p. A1 2.926 A1 | 3 |
A chord AB of a circle with centre O and radius r divides the circle into two regions whose areas are in the ratio 1 : 5. Angle AOB = θ radians.
Show that θ = sin θ + π3.
Show by calculation that the equation has a root between 1.9 and 2.0.
Use the iterative formula θn+1 = ½(θn + sin θn + π3) with θ1 = 2 to find θ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | ½r2(θ − sin θ) = 16πr2 segment area = ½r2θ − ½r2 sin θ M1 equated to πr26 M1 rearranged to the given form A1 | 3 |
| (b) | sign change of θ − sin θ − π3 between 1.9 and 2.0 evaluate θ − sin θ − π3 at both ends: −0.093 and 0.044 M1 sign change and conclusion A1 | 2 |
| (c) | 1.969 (1.97825, 1.97179, 1.96983, 1.96923, 1.96905, …) uses the iteration correctly at least once M1 obtains final answer after sufficient iterations to justify accuracy to 3 d.p. M1 1.969 A1 | 3 |
The equation x3 − 4x − 1 = 0 has one positive root α.
Show by calculation that α lies between 2 and 3.
Use the iterative formula xn+1 = ∛(4xn + 1) to find α correct to 2 decimal places.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | sign change evaluate at both ends M1 sign change, so root A1 | 2 |
| (b) | 2.11 correct iteration M1 iterates to 4 d.p. A1 2.11 A1 | 3 |
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Meet Riyaz →Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in