FP1: Numerical solution of equationsEdexcel International A Level Further Maths: Topic test

1. 1
   Let $\mathrm{f}(x)=x^3+4x-3$. The equation $\mathrm{f}(x)=0$ has a single real root $\alpha$, and $0<\alpha<1$.

   (a)

   Find $\mathrm{f}(0.5)$.

   [1 mark]

   • A$0.875$
   • B$-0.875$
   • C$-0.5$
   • D$-4.875$

   (b)

   Interval bisection is applied to $[0,1]$, using $\mathrm{f}(0.5)=-0.875$ and $\mathrm{f}(0.75)=0.421875$. Which interval of width $0.25$ contains $\alpha$?

   [1 mark]

   • A$[0,0.25]$
   • B$[0.75,1]$
   • C$[0.5,0.75]$
   • D$[0.25,0.5]$

   (c)

   Use linear interpolation on the interval $[0,1]$ to find an estimate of $\alpha$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\mathrm{f}(x)=x^4+2x-7$. The equation $\mathrm{f}(x)=0$ has a root $\alpha$ with $1<\alpha<2$.

   (a)

   Which formula gives the Newton-Raphson iteration for $\alpha$?

   [1 mark]

   • A$x_{n+1}=x_n-\dfrac{x_n^4+2x_n-7}{4x_n^3+2}$
   • B$x_{n+1}=x_n+\dfrac{x_n^4+2x_n-7}{4x_n^3+2}$
   • C$x_{n+1}=x_n-\dfrac{4x_n^3+2}{x_n^4+2x_n-7}$
   • D$x_{n+1}=x_n-\dfrac{x_n^4+2x_n-7}{4x_n^3}$

   (b)

   Use the Newton-Raphson method once with $x_0=1.5$ to find $x_1$ to 4 decimal places.

   [1 mark]

   • A$1.5685$
   • B$1.4213$
   • C$0.4375$
   • D$1.4315$

   (c)

   Show that $\alpha$ lies between $1.4$ and $1.5$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\mathrm{f}(x)=\mathrm{e}^x+x-4$. The equation $\mathrm{f}(x)=0$ has a root $\alpha$ with $1<\alpha<2$.

   (a)

   Show that $\mathrm{f}(1)<0<\mathrm{f}(2)$, and use linear interpolation on $[1,2]$ to find an estimate of $\alpha$ to 3 decimal places.

   [3 marks]

   (b)

   Use the Newton-Raphson method twice, starting with $x_0=1$, to find $x_1$ and $x_2$. Give each to 4 decimal places.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{f}(x)=x^3-2x^2-4$. The equation $\mathrm{f}(x)=0$ has a single real root $\alpha$.

   (a)

   (i) Show that $2<\alpha<3$.
   (ii) Use interval bisection twice, starting with the interval $[2,3]$, to find an interval of width $0.25$ that contains $\alpha$.
   (iii) Use linear interpolation on that interval to estimate $\alpha$ to 3 decimal places.

   [6 marks]

   (b)

   (i) Find $\mathrm{f}'(x)$.
   (ii) Use the Newton-Raphson method once, with $x_0=2.6$, to find $x_1$ to 4 decimal places.
   (iii) Explain why the Newton-Raphson method cannot be used with $x_0=0$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $\mathrm{f}(x)=x^3+6x-2$. The equation $\mathrm{f}(x)=0$ has a single real root $\alpha$, and $0.3<\alpha<0.4$.

   (a)

   Find $\mathrm{f}(0.3)$.

   [1 mark]

   • A$0.173$
   • B$0.7$
   • C$-1.773$
   • D$-0.173$

   (b)

   Given that $\mathrm{f}(0.3)=-0.173$ and $\mathrm{f}(0.4)=0.464$, use linear interpolation to estimate $\alpha$ to 3 decimal places.

   [1 mark]

   • A$0.327$
   • B$0.373$
   • C$0.350$
   • D$0.273$

   (c)

   Use the Newton-Raphson method once, with $x_0=0.3$, to find $x_1$ to 4 decimal places.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The equation $x\ln x=2$ has a single root $\alpha$. Let $\mathrm{f}(x)=x\ln x-2$ for $x>0$.

   (a)

   Given that $\mathrm{f}(2.2)=-0.265$, $\mathrm{f}(2.3)=-0.084$, $\mathrm{f}(2.4)=0.101$ and $\mathrm{f}(2.5)=0.291$, which interval contains $\alpha$?

   [1 mark]

   • A$[2.2,2.3]$
   • B$[2.3,2.4]$
   • C$[2.4,2.5]$
   • D$[2.0,2.2]$

   (b)

   Find $\mathrm{f}'(x)$.

   [1 mark]

   • A$\ln x$
   • B$\dfrac1x$
   • C$\ln x+1$
   • D$1+\dfrac1x$

   (c)

   Use interval bisection once on $[2.3,2.4]$ to find a smaller interval containing $\alpha$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $\mathrm{f}(x)=\sqrt{x}+x-4$ for $x>0$. The equation $\mathrm{f}(x)=0$ has a root $\alpha$ with $2<\alpha<3$.

   (a)

   Find $\mathrm{f}'(x)$ and use the Newton-Raphson method once, with $x_0=2$, to find $x_1$ to 4 decimal places.

   [3 marks]

   (b)

   Show that $2.4<\alpha<2.5$ and use linear interpolation on $[2.4,2.5]$ to estimate $\alpha$ to 4 decimal places.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $\mathrm{f}(x)=x^3+x^2-10$. The equation $\mathrm{f}(x)=0$ has a single real root $\alpha$.

   (a)

   (i) Show that $1.8<\alpha<1.9$.
   (ii) Use linear interpolation on $[1.8,1.9]$ to estimate $\alpha$ to 3 decimal places.
   (iii) Use the Newton-Raphson method once, with $x_0=1.9$, to find $x_1$ to 3 decimal places.

   [6 marks]

   (b)

   (i) Evaluate $\mathrm{f}(1.865)$ and $\mathrm{f}(1.875)$, and hence show that $\alpha=1.87$ to 2 decimal places.
   (ii) Starting with $x_1=1.868$, use the Newton-Raphson method once more to find $x_2$ to 4 decimal places.

   [6 marks]

   Total for question 8: 12 marks

End of questions