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

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 location 2.6.2—Iterative approximations 2.6.3—Iteration formulae

[Maximum number: 3]

Figure for Question 2.6 Numerical solution of equations question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\frac{2 \ln x}{3 x+1}$ . The curve crosses the x-axis at the point A and has a maximum point B. The shaded region is bounded by the curve and the lines x=3 and y=0.

Show by calculation that the x-coordinate of B lies between 3.0 and 3.1.

Equate numerator of first derivative to zero

M1

May be implied.

Consider sign of $\frac{2}{x}(3 x+1)-6 \ln x$ for 3.0 and 3.1

M1

OE

Obtain $0.074 \ldots$ and $-0.14 \ldots$ or equivalents and justify conclusion

A1

AG - necessary detail needed.
0.00075 and -0.001275 .

[Maximum number: 5]

(a)

It is given that the equation $x=\sqrt{\frac{9}{x+2}}$ has a single root.

Show by calculation that this root lies between 1.5 and 2.0.

[ 2 ]

Consider sign of $x-\sqrt{\frac{9}{x+2}}$ or equivalent for 1.5 and 2
M1
Obtain $-0.1 \ldots$ and 0.5 or equivalents and justify conclusion
A1
AG

(b)

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

[ 3 ]

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Covered syllabus points

2.6 Numerical solution of equations question 1

2.6 Numerical solution of equations question 2

Question (a)

Question (b)