Numerical solution of equations: past paper questions

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• Question 6May/June 2026, paper 31Structured10 marks

  A curve has the equation $y={\mathrm{e}}^{(x^2+4x)}{\cos }^{2}x$, where $\cos x\mathrm{\ne }0$. Show that the $x$-coordinates of the stationary points of this curve satisfy the equation $\tan x=x+2$.

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• Question 5Oct/Nov 2024, paper 31Structured7 marks

  By sketching a suitable pair of graphs, show that the equation $2+{\mathrm{e}}^{-0.2x}=\ln (1+x)$ has only one root.

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• Question 8Oct/Nov 2023, paper 31Structured8 marks

  By sketching a suitable pair of graphs, show that the equation $$\sqrt{x}={\mathrm{e}}^x-3$$ has only one root.

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• Question 8May/June 2026, paper 33Structured8 marks

  By sketching a suitable pair of graphs, show that the equation $x^3=4-2x$ has exactly one real root.

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• Question 8Oct/Nov 2025, paper 33Structured10 marks

  The curve with equation $y={\mathrm{e}}^{-5x}\ln 5x$ has a stationary point at $x=p$. Show that $p$ satisfies the equation $\ln 5p=\frac{1}{5p}$.

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• Question 2Oct/Nov 2024, paper 33Structured5 marks

  Let $\mathrm{f}(x)=2x^3-5x^2+4$. Show that if a sequence of values given by the iterative formula $$x_{n+1}=\sqrt{\frac{4}{5-2x_n}}$$ converges, then it converges to a root of the equation $\mathrm{f}(x)=0$.

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• Question 9May/June 2023, paper 31Structured10 marks

  The constant $a$ is such that ${\int }_0^{a}x{\mathrm{e}}^{-2x}\mathrm{d}x=\frac{1}{8}$. Show that $a=\frac{1}{2}\ln (4a+2)$.

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• Question 9May/June 2025, paper 31Structured10 marks

  The constant $a$ is such that ${\int }_1^{a}6x\ln x\mathrm{d}x=4$. Show that $a=\exp \left(\frac{1}{6}(\frac{5}{a^2}+3)\right)$, where $\exp (x)$ denotes ${\mathrm{e}}^x$.

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• Question 6May/June 2024, paper 31Structured9 marks

  By sketching a suitable pair of graphs, show that the equation $\mathrm{cosec}\frac{1}{2}x={\mathrm{e}}^x-3$ has exactly one root, denoted by $\alpha$, in the interval $0<x<\pi$.

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• Question 6May/June 2025, paper 32Structured7 marks

  By sketching a suitable pair of graphs, show that the equation $$\mathrm{\mid }x-2\mathrm{\mid }=2\sin \frac{1}{2}x$$ has only one root in the interval $0<x<\pi$.

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• Question 5May/June 2024, paper 32Structured6 marks

  It is given that the equation ${\mathrm{e}}^{2x}=5+\cos 3x$ has only one root. Show by calculation that this root lies in the interval $0.7<x<0.8$.

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