Numerical methods in contextEdexcel A-Level Maths: Subtopic test

1. 1
   A manufacturer finds that the break-even selling price, £$x$, of a product satisfies $f(x)=x^3-4x-7=0$.

   (a)

   Which statement correctly justifies that $f(x)=0$ has a root between $x=2$ and $x=3$?

   [1 mark]

   • A$f(2)=7>0$ and $f(3)=-8<0$; $f$ is continuous, so it changes sign and has a root in $[2,3]$
   • B$f(2)=-7<0$ and $f(3)=8>0$; $f$ is continuous, so it changes sign and has a root in $[2,3]$
   • C$f(2)=-7$ and $f(3)=8$ are different, so the root is where $f$ is closest to zero, at $x=2$
   • D$f(2)=-7<0$ and $f(3)=8>0$, so $f$ is increasing and there is no root in $[2,3]$

   (b)

   Rearranging gives $x_{n+1}=\sqrt[3]{4x_n+7}$. Starting with $x_0=2$, find $x_1$ to 3 decimal places.

   [1 mark]

   • A$5.000$
   • B$0.250$
   • C$2.564$
   • D$2.466$

   (c)

   Show that $x^3-4x-7=0$ can be rearranged to give $x=\sqrt{4+\frac{7}{x}}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A scientist models the concentration of drug A as $5e^{-0.4t}$ mg per litre and of drug B as $t$ mg per litre, where $t$ is the time in hours since the start. The concentrations are equal when $t$ is a root of $f(t)=t-5e^{-0.4t}=0$.

   (a)

   Find $f(2)$ to 3 decimal places.

   [1 mark]

   • A$-0.247$
   • B$0.247$
   • C$-2.246$
   • D$-9.128$

   (b)

   Rearranging gives $t_{n+1}=5e^{-0.4t_n}$. Starting with $t_0=2$, find $t_2$ to 3 decimal places.

   [1 mark]

   • A$2.247$
   • B$2.215$
   • C$2.036$
   • D$11.128$

   (c)

   Show that the root of $f(t)=0$ lies between $t=2.1$ and $t=2.2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A decorative pond is in the shape of a segment of a circle of radius $10$ m. The chord of the segment subtends an angle of $\theta$ radians at the centre of the circle, and the area of the pond is $30$ m$^2$.

   (a)

   Show that $\theta-\sin\theta=0.6$.

   [3 marks]

   (b)

   Use the iteration $\theta_{n+1}=0.6+\sin\theta_n$ with $\theta_0=1.5$ to find $\theta_1$, $\theta_2$ and $\theta_3$ to 4 decimal places, and write down the value of $\theta$ to 2 decimal places. (Work in radians.)

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An open box is made from a rectangular sheet of card $30$ cm by $20$ cm by cutting a square of side $x$ cm from each corner and folding up the sides. The volume of the box is $900$ cm$^3$.

   (a)

   (i) Show that $x^3-25x^2+150x-225=0$.
   (ii) Show that this equation has a root between $x=2$ and $x=3$.

   [6 marks]

   (b)

   (i) Show that the equation can be rearranged to $x=\frac{225}{x^2-25x+150}$.
   (ii) Use the iteration $x_{n+1}=\frac{225}{x_n^2-25x_n+150}$ with $x_0=2$ to find $x_1$ and $x_2$ to 3 decimal places.
   (iii) By considering the sign of $f(x)=x^3-25x^2+150x-225$ at $x=2.295$ and $x=2.305$, show that the root is $2.30$ to 2 decimal places.

   [6 marks]

   Total for question 4: 12 marks

End of questions