Numerical solution of equations: past paper questions

Where the questions come from

• Question 5May/June 2026, paper 21Structured10 marksHas a diagram

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=2x^4+3x^3+4x^2-8x\mathrm{.}$$ Find the quotient when $\mathrm{p}(x)$ is divided by $(2x+3)$, and show that the remainder is 21.

  Try this question
• Question 7Oct/Nov 2025, paper 21Structured11 marks

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=2x^4+k{x}^3+k{x}^2+17x+18,$$ where $k$ is a constant. It is given that $(x+2)$ is a factor of $\mathrm{p}(x)$. Find the value of $k$.

  Try this question
• Question 5Oct/Nov 2024, paper 21Structured8 marks

  It is given that ${\int }_a^{a^3}\frac{10}{2x+1}\mathrm{d}x=7$, where $a$ is a constant greater than 1. Show that $a=\sqrt[3]{0.5{\mathrm{e}}^{1.4}(2a+1)-0.5}$.

  Try this question
• Question 7Oct/Nov 2023, paper 21Structured11 marks

  The curve with equation ${\mathrm{e}}^{2x}-18x+y^3+y=11$ has a stationary point at $(p,q)$. Find the exact value of $p$.

  Try this question
• Question 6May/June 2026, paper 22Structured8 marksHas a diagram

  The diagram shows the curve with equation $y=\mathrm{f}(x)$, where $$\mathrm{f}(x)=a{x}^4+4x^3-3a{x}^2-21x-18$$ and $a$ is a constant. It is given that $(x+2)$ is a factor of $\mathrm{f}(x)$. Find the value of $a$.

  Try this question
• Question 4Oct/Nov 2024, paper 22Structured7 marks

  Sketch the graphs of $y=1+{\mathrm{e}}^{2x}$ and $y=\mathrm{\mid }x-4\mathrm{\mid }$ on the same diagram.

  Try this question
• Question 6May/June 2026, paper 23Structured8 marksHas a diagram

  The diagram shows the curve with equation $y=\mathrm{f}(x)$, where $$\mathrm{f}(x)=a{x}^4+4x^3-3a{x}^2-21x-18$$ and $a$ is a constant. It is given that $(x+2)$ is a factor of $\mathrm{f}(x)$. Find the value of $a$.

  Try this question
• Question 3May/June 2023, paper 21Structured7 marks

  It is given that ${\int }_0^a(3{\mathrm{e}}^{2x}-1)\mathrm{d}x=12$, where $a$ is a positive constant. Show that $a=\frac{1}{2}\ln (9+\frac{2}{3}a)$.

  Try this question
• Question 4May/June 2025, paper 22Structured8 marksHas a diagram

  The diagram shows parts of the curves with equations $y=4{\mathrm{e}}^{-2x}$ and $y=1+0.5\sin 3x$. Point $P$ is a point of intersection of the curves, and the shaded region is bounded by the two curves and the $y$-axis.

  Try this question
• Question 6May/June 2024, paper 22Structured9 marksHas a diagram

  The diagram shows the curve with equation $y=\frac{\ln (2x+1)}{x+3}$. The curve has a maximum point $M$. Find an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$.

  Try this question
• Question 4May/June 2023, paper 22Structured7 marksHas a diagram

  The diagram shows the graph of $y=3-{\mathrm{e}}^{-\frac{1}{2}x}$. On the diagram, sketch the graph of $y=\mathrm{\mid }5x-4\mathrm{\mid }$, and show that the equation $3-{\mathrm{e}}^{-\frac{1}{2}x}=\mathrm{\mid }5x-4\mathrm{\mid }$ has exactly two real roots.

  Try this question