Iteration (OCR GCSE Maths: Higher): Exam Questions

The equation $x^3+2x-5=0$ has a solution between $x=1$ and $x=2$.

Find this solution correct to 1 decimal place.
Show your working.  

x = ......................................................

The equation $x^3+x^2-1=0$ has a solution between $x=0$ and $x=1$.

Find this solution correct to 1 decimal place.
Show your working.  

x = .......................................................

It is known that the equation  $x^3-4x=12$  has a solution that lies between $x=2$ and $x=3$.

Use a suitable method to find the solution correct to 1 decimal place.
You must show your working clearly.  

x = .......................................................

The equation $t^4-t-3=0$ has a negative solution and a positive solution.
The positive solution lies between $t=1$ and $t=2$.

Use a suitable method to find the positive solution correct to 1 decimal place.
Show your working.  

t = .......................................................

Show that the equation $x^4-6=x^3$ has a solution between $x=-2$ and $x=-1$.

Show that the equation $x^4-x^2-9=0$ has a solution between $x=1$ and $x=2$.

Find this solution correct to 1 decimal place.

Show your working.

x = .......................................................

The equation $x^3+5x-7=0$ has a solution between $x=1$ and $x=2$.

Use a suitable method to find this solution correct to two decimal places.

A value of $k$ between 2 and 3 satisfies the equation $k^3+2=\frac{60}{k^2}$

By writing the equation in the form $k^5+p{k}^2+q=0$ where $p$ and $q$ are integers you should find, determine the value of $k$ correct to one decimal place.

Student A decides to solve the equation $x^2-3x+1=0$ using the quadratic formula and gets the two solutions  

$\frac{3\pm \sqrt{5}}{2}$  

Student B decides to solve the equation using a sign-change method between $x=a$ and $x=a+1$, where $a$ is an integer.
They want to find the larger solution to 1 decimal place.

Show Student B's working, stating clearly the value of $a$.

The volume of a sphere of radius $r$ cm, where $2<r<3$, is 5 cm³ more than the volume of a square-based cuboid with height 10 cm, width $r$ cm and length $r$ cm.

Use a suitable method to find $r$ to 1 decimal place.

[In this question, you may use that the volume of a sphere of radius $r$ is $\frac{4}{3}\mathrm{\pi }{r}^3$]

The graph of $y=2x^2+2x-1$ is shown below.

The roots of the equation $2x^2+2x-1=0$ are at $p$ and $q$.

Calculate $y$ when $x=1$.

Without solving the equation, explain why $q$ must lie between $0$ and $1$.

An iteration formula for solving the equation $2x^2+2x-1=0$ is

$x_{n+1}=\frac{1-2x_n^2}{2}$

Starting with $x_0=0.4$, find $x_1$.

Give your answer correct to $2$ decimal places.

The exact value of $q$ is

$\frac{-2+\sqrt{12}}{4}$

Write $\frac{-2+\sqrt{12}}{4}$ in the form $\frac{a+\sqrt{b}}{c}$ where $a$, $b$ and $c$ are integers.