Numerical solution of equations: AS Level Mathematics questions with answers
Free Numerical solution of equations questions for AS Level Mathematics (9709), part of Pure Mathematics 2. Each question has the answer and the marking points an examiner would look for.
Numerical solution of equations practice questions
Question 1Structured · 8 marksA* challenge
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.
(a)
Show that θ = sin θ + π2.
[3]
(b)
Show by calculation that the equation has a root between 2.3 and 2.4.
[2]
(c)
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.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
½r2(θ − sin θ) = 14πr2
segment area = ½r2θ − ½r2 sin θ M1 equated to πr24M1 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
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
Question 2Structured · 5 marksA* challenge
The equation ln x = 5 − x has one root, α.
(a)
Show by calculation that α lies between 3 and 4.
[2]
(b)
Use the iterative formula xn+1 = 5 − ln xn to find α correct to 3 decimal places.
[3]
Show answers and mark scheme
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
Question 3Structured · 3 marks
The equation x3 − 8x − 7 = 0 has a root α. Use the iterative formula xn+1 = ∛(8xn + 7) with x1 = 2 to find α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
Uses iteration correctly at least once M1 Iterates to sufficient accuracy M1 3.193 A1
3
Question 4Structured · 8 marksA* challenge
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.
(a)
Show that θ = sin θ + π3.
[3]
(b)
Show by calculation that the equation has a root between 1.9 and 2.0.
[2]
(c)
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.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
½r2(θ − sin θ) = 16πr2
segment area = ½r2θ − ½r2 sin θ M1 equated to πr26M1 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
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
Question 5Structured · 5 marksA* challenge
The equation ln x = 4 − x has one root, α.
(a)
Show by calculation that α lies between 2 and 3.
[2]
(b)
Use the iterative formula xn+1 = 4 − ln xn to find α correct to 3 decimal places.
[3]
Show answers and mark scheme
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
Question 6Structured · 3 marks
The equation x3 − 5x − 5 = 0 has a root α. Use the iterative formula xn+1 = ∛(5xn + 5) with x1 = 2 to find α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
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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