CAIE A-Level Mathematics 3.6 Numerical Solution of Equations Question Bank

CAIE A-Level Mathematics 3.6 Numerical Solution of Equations Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise using graph intersections or sign changes to locate roots and applying convergent fixed-point iterations with enough displayed values to justify the requested decimal…

Exam points

  • sketch a suitable pair of graphs or evaluate endpoint signs to isolate one root
  • verify that the fixed-point equation is algebraically equivalent to the original equation
  • iterate to the stated working precision until consecutive values guarantee the rounded root

Covered syllabus points

3.6.1—Root location3.6.2—Iterative approximations3.6.3—Iteration formulae

Question 7

[Maximum number: 5]

The polynomial p(x) is defined by

p(x)=2x4+kx3+kx2+17x+18,\mathrm{p}(x)=2 x^{4}+k x^{3}+k x^{2}+17 x+18,

where k is a constant. It is given that (x+2) is a factor of p(x).

Question 7(c)

(a)

Show by calculation that −1.4<β<−1.0-1.4<\beta<-1.0.

[ 2 ]

Question 7(d)

(b)

Use an iterative formula, based on the equation in part (b), to find the value of β\beta correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

[ 3 ]

Question 7

[Maximum number: 5]

The polynomial p(x) is defined by

p(x)=2x4+kx3+kx2+17x+18,\mathrm{p}(x)=2 x^{4}+k x^{3}+k x^{2}+17 x+18,

where k is a constant. It is given that (x+2) is a factor of p(x).

Question 7(c)

(a)

Show by calculation that −1.4<β<−1.0-1.4<\beta<-1.0.

[ 2 ]

Question 7(d)

(b)

Use an iterative formula, based on the equation in part (b), to find the value of β\beta correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

[ 3 ]