Pure: Numerical methodsEdexcel A-Level Maths: Topic test

1. 1
   The function $f(x)=x^3+2x-9$ is defined for all real $x$, and the equation $f(x)=0$ is to be solved numerically.

   (a)

   Which statement about the real roots of $f(x)=0$ is correct?

   [1 mark]

   • AThere are three real roots, because $f$ is a cubic.
   • BThere is no real root, because $f(0)<0$ and $f(1)<0$.
   • CThere is exactly one real root, because $f'(x)=3x^2+2>0$ so $f$ is strictly increasing.
   • DThere are exactly two real roots, because $f(0)<0$ and $f(2)>0$.

   (b)

   Which rearrangement of $f(x)=0$ can be used as an iteration formula $x_{n+1}=g(x_n)$?

   [1 mark]

   • A$x_{n+1}=\dfrac{9-x_n^3}{2}$
   • B$x_{n+1}=\dfrac{x_n^3-9}{2}$
   • C$x_{n+1}=\sqrt[3]{9+2x_n}$
   • D$x_{n+1}=\dfrac{9}{x_n^2-2}$

   (c)

   Show that $f(x)=0$ has a root between $1.7$ and $1.8$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The equation $x^3-7x+3=0$ has a root $\alpha$ close to $0.4$. Let $f(x)=x^3-7x+3$. The Newton-Raphson method is used to find $\alpha$.

   (a)

   Which expression is the Newton-Raphson formula for this equation?

   [1 mark]

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

   (b)

   Starting with $x_0=0.4$, what is $x_1$ to $3$ decimal places?

   [1 mark]

   • A$0.360$
   • B$0.440$
   • C$0.664$
   • D$0.040$

   (c)

   A student starts instead with $x_0=1.5$ and finds that $x_1$ is far from every root. Find $x_1$ and explain why it is a poor estimate.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The depth of a river is measured at $1$ m intervals across its width of $4$ m. The depths, in metres, from one bank to the other are $0,\ 1.2,\ 1.9,\ 1.6,\ 0$. The depth, plotted against distance across the river, is a smooth concave curve (it curves downwards, like an arch).

   (a)

   Use the trapezium rule with $4$ strips to estimate the cross-sectional area of the river, in m$^2$.

   [3 marks]

   (b)

   The channel keeps the same cross-section for $250$ m. Estimate the volume of water in this stretch, and state, with a reason, whether your estimate is an over-estimate or an under-estimate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The equation $\ln x+x-3=0$ has a single root $\alpha$. Let $f(x)=\ln x+x-3$. Work in radians throughout and give decimal answers to $4$ decimal places.

   (a)

   Show that $\alpha$ lies between $2$ and $3$. The equation is rearranged as $x=3-\ln x$ and the iteration $x_{n+1}=3-\ln x_n$ is used with $x_0=2$. Find $x_1$, $x_2$ and $x_3$, and describe how the values behave in relation to $\alpha$.

   [6 marks]

   (b)

   Use the Newton-Raphson method with $x_0=2$ to find $x_1$ and $x_2$, and hence give the value of $\alpha$ to $3$ decimal places. Compare how quickly this converges with the iteration $x_{n+1}=3-\ln x_n$ in part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The eccentric anomaly $x$ (in radians) of a satellite satisfies Kepler's equation $x-1=0.5\sin x$. The iteration $x_{n+1}=1+0.5\sin x_n$ is used with $x_0=1$.

   (a)

   What is $x_1$ to $3$ decimal places?

   [1 mark]

   • A$1.842$
   • B$1.421$
   • C$1.009$
   • D$1.500$

   (b)

   Which statement correctly explains why this iteration converges to the root?

   [1 mark]

   • ABecause $x_0=1$ is a positive integer.
   • BBecause $\sin x$ always lies between $-1$ and $1$, so every $x_n$ lies between $0.5$ and $1.5$.
   • CBecause $x_{n+1}>x_n$ for every value of $n$, so the sequence must stop.
   • DBecause $|g'(x)|=|0.5\cos x|\leqslant0.5<1$ near the root, so successive errors shrink.

   (c)

   Show that $x-1=0.5\sin x$ has a root between $1.4$ and $1.6$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The integral $I=\int_0^{1.5}\mathrm{e}^{x}\,\mathrm{d}x$ is estimated using the trapezium rule with $3$ strips of equal width.

   (a)

   What is the estimate, to $3$ significant figures?

   [1 mark]

   • A$3.55$
   • B$3.48$
   • C$2.46$
   • D$14.2$

   (b)

   Which statement is correct?

   [1 mark]

   • AThe estimate is an under-estimate, because the curve is convex.
   • BThe estimate is an under-estimate, because the function is increasing.
   • CThe estimate is an over-estimate, because the curve is convex, so the chords lie above the curve.
   • DThe estimate is an over-estimate, because the function is increasing.

   (c)

   Find the exact value of $I$ and hence the percentage error in the estimate, to $3$ significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The cube root of $10$ is to be found as the positive root of $f(x)=x^3-10=0$ using the Newton-Raphson method.

   (a)

   Show that the method gives $x_{n+1}=\dfrac{2x_n^3+10}{3x_n^2}$, and find $x_1$ when $x_0=2$.

   [3 marks]

   (b)

   Find $x_2$ to $5$ decimal places, and explain why the method cannot be started with $x_0=0$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Water flows into a reservoir at a rate $R(t)=40+30\sin\!\left(\dfrac{\pi t}{12}\right)$ m$^3$ per hour, where $t$ is the time in hours after midnight and $0\leqslant t\leqslant12$. Work in radians.

   (a)

   Use the trapezium rule with $4$ strips to estimate the volume that flows in between $t=0$ and $t=12$. State, with a reason, whether this is an over-estimate or an under-estimate, and confirm your answer by integration.

   [6 marks]

   (b)

   The volume that has flowed in by time $t$ is $V(t)=40t+\dfrac{360}{\pi}\left(1-\cos\dfrac{\pi t}{12}\right)$ m$^3$. The time $T$ at which $300$ m$^3$ has flowed in satisfies $f(T)=0$, where $f(t)=V(t)-300$. Show that $5<T<6$, then use the Newton-Raphson method with $t_0=5$ to find $t_1$ and $t_2$ to $4$ decimal places, and state $T$ to $3$ significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions