Further Pure 1: Further numerical methodsEdexcel A-Level Further Maths: Topic test

1. 1
   The function $y$ satisfies $\frac{dy}{dx}=xy$, with $y=2$ when $x=1$. Values of $y$ are approximated using a step length of $h=0.1$.

   (a)

   Using the approximation $\left(\frac{dy}{dx}\right)_n\approx\frac{y_{n+1}-y_n}{h}$, what is the approximate value of $y$ when $x=1.1$?

   [1 mark]

   • A$0.2$
   • B$2.1$
   • C$2.2$
   • D$2.4$

   (b)

   Using the same approximation and the value of $y$ at $x=1.1$ found in part (a), what is the approximate value of $y$ when $x=1.2$?

   [1 mark]

   • A$2.442$
   • B$2.4$
   • C$2.42$
   • D$2.464$

   (c)

   Using the approximation $\left(\frac{dy}{dx}\right)_n\approx\frac{y_{n+1}-y_{n-1}}{2h}$, together with $y(1)=2$ and the value $y(1.1)=2.2$ from part (a), estimate $y(1.2)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $y$ satisfies $\frac{d^2y}{dx^2}=x-2y$. It is given that $y=1$ when $x=0$ and $y=1.2$ when $x=0.1$. The equation is solved numerically with step length $h=0.1$ using $\left(\frac{d^2y}{dx^2}\right)_n\approx\frac{y_{n+1}-2y_n+y_{n-1}}{h^2}$.

   (a)

   Which expression gives $y_{n+1}$?

   [1 mark]

   • A$2y_n-y_{n-1}+h(x_n-2y_n)$
   • B$2y_n+y_{n-1}+h^2(x_n-2y_n)$
   • C$y_n-y_{n-1}+h^2(x_n-2y_n)$
   • D$2y_n-y_{n-1}+h^2(x_n-2y_n)$

   (b)

   What is the approximate value of $y$ when $x=0.2$?

   [1 mark]

   • A$1.4$
   • B$1.377$
   • C$1.423$
   • D$1.17$

   (c)

   Given that $y(0.2)=1.377$, find an estimate for $y(0.3)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_{0}^{2}x^{4}\,dx$.

   (a)

   Find the exact value of $I$ and use Simpson's rule with 2 strips to estimate $I$.

   [3 marks]

   (b)

   Use Simpson's rule with 4 strips to estimate $I$, and find the percentage error of your estimate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $y$ satisfies $\frac{dy}{dx}=\sqrt{1+x^3}$, with $y=0$ when $x=0$.

   (a)

   Use the approximation $\left(\frac{dy}{dx}\right)_n\approx\frac{y_{n+1}-y_n}{h}$ with a step length of $0.5$ to estimate $y(2)$, giving your answer to 3 decimal places.

   [6 marks]

   (b)

   The exact value of $y(2)$ is $\int_0^2\sqrt{1+x^3}\,dx$. Use Simpson's rule with 4 strips to estimate this integral and compare your answer with the estimate from part (a). State, with a reason, which estimate is likely to be more accurate and why the estimate in part (a) is too small.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $y$ satisfies $\frac{dy}{dx}=2y-x$. It is given that $y=1$ when $x=0$ and $y=1.2$ when $x=0.1$. The equation is solved numerically with step length $h=0.1$ using $\left(\frac{dy}{dx}\right)_n\approx\frac{y_{n+1}-y_{n-1}}{2h}$.

   (a)

   Which expression gives $y_{n+1}$?

   [1 mark]

   • A$y_{n-1}+2h(2y_n-x_n)$
   • B$y_n+2h(2y_n-x_n)$
   • C$y_{n-1}+h(2y_n-x_n)$
   • D$y_{n-1}+2h(2y_{n+1}-x_{n+1})$

   (b)

   What is the approximate value of $y$ when $x=0.2$?

   [1 mark]

   • A$1.43$
   • B$1.4$
   • C$1.66$
   • D$1.46$

   (c)

   Given that $y(0.2)=1.46$, find an estimate for $y(0.3)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Simpson's rule with 4 strips is used to estimate $\int_{1}^{3}\ln\left(x^2+1\right)dx$.

   (a)

   What is the width of each strip?

   [1 mark]

   • A$0.25$
   • B$0.5$
   • C$2$
   • D$0.4$

   (b)

   What is the value of $\ln\left(x^2+1\right)$ at $x=1.5$, to 4 decimal places?

   [1 mark]

   • A$0.8109$
   • B$0.9163$
   • C$1.1787$
   • D$1.8109$

   (c)

   Use Simpson's rule with 4 strips to estimate the integral, giving your answer to 3 decimal places.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The population $P$ (in thousands) of a colony of bacteria at time $t$ hours satisfies $\frac{dP}{dt}=0.2P-0.004P^2$, with $P=10$ when $t=0$. A step length of $h=1$ hour is used.

   (a)

   Use the approximation $\left(\frac{dP}{dt}\right)_n\approx\frac{P_{n+1}-P_n}{h}$ to estimate $P$ when $t=1$ and when $t=2$.

   [3 marks]

   (b)

   Using $P(0)=10$ and $P(1)=11.6$, and the approximation $\left(\frac{dP}{dt}\right)_n\approx\frac{P_{n+1}-P_{n-1}}{2h}$, estimate $P(2)$ and $P(3)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The function $y$ satisfies $\frac{d^2y}{dx^2}=2x-y$, with $y=1$ and $\frac{dy}{dx}=0$ when $x=0$. The equation is solved numerically with step length $h=0.5$, using $\left(\frac{dy}{dx}\right)_n\approx\frac{y_{n+1}-y_n}{h}$ for the first step and $\left(\frac{d^2y}{dx^2}\right)_n\approx\frac{y_{n+1}-2y_n+y_{n-1}}{h^2}$ for later steps.

   (a)

   Find estimates of $y$ at $x=0.5$, $1$, $1.5$ and $2$.

   [6 marks]

   (b)

   Use Simpson's rule with the four strips given by your values in part (a) to estimate $\int_0^2y\,dx$. Hence find an estimate of the mean value of $y$ over the interval $0\leqslant x\leqslant2$, and state one way to improve the accuracy of these estimates.

   [6 marks]

   Total for question 8: 12 marks

End of questions