CAIE A-Level Mathematics 2.6 Numerical Solution of Equations Question Bank

CAIE A-Level Mathematics 2.6 Numerical Solution of Equations Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise locating roots by sign changes or graph intersections, rearranging equations for fixed-point iteration and showing enough rounded iterates to justify the requested…

Exam points

  • evaluate f at both interval endpoints and use opposite signs to establish a root
  • rearrange the equation as x = F(x) without changing the required fixed point
  • iterate xₙ₊₁ = F(xₙ), record values at the stated precision and justify final rounding

Covered syllabus points

2.6.1—Root location2.6.2—Iterative approximations2.6.3—Iteration formulae

Question 3(c)

[Maximum number: 3]

（c）Use an iterative formula，based on the equation in part（b），to find the x－coordinate of the point of intersection correct to 3 decimal places．Give the result of each iteration to 5 decimal places．

Question 6

Question 6(a)

(a)

By sketching a suitable pair of graphs, show that the equation

∣x−2∣=2sin⁡12x|x-2|=2 \sin \frac{1}{2} x

has only one root in the interval 0<x<π0<x<\pi.

[ 2 ]

Question 6(b)

(b)

Show by calculation that this root lies between 1 and 1.5.

[ 2 ]

Question 6(c)

(c)

Use the iterative formula xn+1=2−2sin⁡12xnx_{n+1}=2-2 \sin \frac{1}{2} x_{n} with an initial value of 1.03 to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]