Using Calculators for Graphs, Equations & Inequalities (Cambridge (CIE) IGCSE International Maths: Extended): Exam Questions

$\mathrm{f}\left(x\right)=\frac{1}{\sin x°}$ for $0\le x\le 360$

On the diagram, sketch the graph of $y=\mathrm{f}\left(x\right)$.

Find the coordinates of the local minimum point.

Write down the equations of the three asymptotes of the graph of $y=\mathrm{f}\left(x\right)$.

The equation $\mathrm{f}\left(x\right)=k$ has no solutions.

Write down the range of values of $k$.

By sketching another graph on the diagram, solve the equation $\frac{1}{\sin x°}=5\sin \left(\frac{x}{2}\right)°$ for $0\le x\le 360°$.

A medicine delivery company uses drones to deliver medications to remote mountain villages.

A radar system is used to control the drones, as long as the controller is within a certain distance from the drone, called the radar visibility, $V$ metres,

A model for the radar visibility, $V$ metres, at different altitudes, $x$ metres up a particular mountain is

$V=\frac{600-x}{4}+\sqrt{70000+x^2}$

On the diagram, sketch the graph of $V$ against $x$.

Find, correct to the nearest metre,

(i) the minimum visibility,

[1]

(i) the altitude for which the visibility is at its minimum.

[1]

Write down a reason why the visibility given by the model might be different to the actual visibility.

$\mathrm{f}\left(x\right)={\left(x-2\right)}^3-5x+12$ for $-1\le x\le 4.7$

On the diagram, sketch the graph of $y=\mathrm{f}\left(x\right)$.

Write down the coordinates of the local maximum.

The equation ${\left(x-2\right)}^3-5x+12=k$ has exactly 2 solutions.

Find the values of $k$.

$\mathrm{g}\left(x\right)=-{\left(x-1\right)}^2$ for $-1\le x\le 4.7$

On the diagram, sketch the graph of $y=\mathrm{g}\left(x\right)$.

Solve $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$.

The curve  $y=x^2-2x+1$  is drawn on a grid.

A line is drawn on the same grid.

The points of intersection of the line and the curve are used to solve the equation  $x^2-7x+5=0.$

Find the equation of the line in the form  $y=mx+c.$

$y=$ ..................................................

On the diagram, sketch the graphs of $\begin{array}{rcl}y& =& \frac{1}{x^2+1}\end{array}$ and $\begin{array}{rcl}y& =& x^3-2x\end{array}$.

Solve.

$\begin{array}{rcl}\frac{1}{x^2+1}& =& x^3-2x\end{array}$

Solve.

$\begin{array}{rcl}\frac{1}{x^2+1}& <& x^3-2x\end{array}$

Find the coordinates of the minimum point of $y=x^3-2x$ where $x>0$.

The graph of $y=1+2^x$ has one asymptote. The equation of the asymptote is $y=1$.

Draw the graph of $y=1+2^x$ on your calculator. Use this to sketch the graph of $y=1+2^x$ on the axes provided.

Label clearly the $y$-intercept and the asymptote.

Solve the following equation.

$1+2^x=x+3$

Give your answers to 2 decimal places where appropriate.

Solve the following inequality.

$1+2^x>x+3$

The solutions in part (b) are also solutions to the equation below, where $p$ and $q$ are positive integers.

$x={\log }_p\left(x+q\right)$

Find $p$ and $q$.

$\mathrm{f}\left(x\right)=\frac{x^2+3x}{\left(x-2\right)\left(x+1\right)}$

On the diagram sketch the graph of $y=\mathrm{f}\left(x\right)$ for values of $x$ between $-6$ and $6.$

Write down the equations of the asymptotes parallel to the $y$‑axis.

Find the zeros of the graph of $y=\mathrm{f}\left(x\right)$.

$\mathrm{g}\left(x\right)=x-3$

(i) On the diagram, sketch the graph of $y=\mathrm{g}\left(x\right)$ for $-6\le x\le 6.$

[1]

(ii) Use your graphs to solve $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$.

[3]

(iii) Solve $\mathrm{g}\left(x\right)>\mathrm{f}\left(x\right)$.

[3]

$\mathrm{f}\left(x\right)={\left(\cos x°\right)}^2$ $\mathrm{g}\left(x\right)=0.5-0.001x$

On the diagram, sketch the graph of $y=\mathrm{f}\left(x\right)$ for $0\le x\le 360.$

On the same diagram, sketch the graph of $y=\mathrm{g}\left(x\right)$ for $0\le x\le 360.$

Solve $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$ for $0\le x\le 360.$

Solve $\mathrm{f}\left(x\right)<\mathrm{g}\left(x\right)$ for $0\le x\le 360.$

$\mathrm{f}\left(x\right)=x+\frac{5}{\left(x-2\right)\left(x+3\right)}$

Sketch the graph of $y=\mathrm{f}\left(x\right)$ for values of $x$ between $-5$ and $5$.

Write down the equations of the asymptotes parallel to the $y$-axis.

(i) Find the coordinates of the local maximum.

(............ , ............) [2]

(ii) Find the coordinates of the local minimum.

(............ , ............) [2]

(iii) Write down the range of values of $k$ for which $\mathrm{f}\left(x\right)=k$ has exactly one solution.

[2]

$\mathrm{g}\left(x\right)=-4-x$

(i) Solve the equation $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right).$

[3]

(ii) Find the solutions to the inequality $\mathrm{f}\left(x\right)>\mathrm{g}\left(x\right).$

[3]

The graph of $y=\frac{1}{2x}$ is shown below.

Write down the equations of any asymptotes.

On the same diagram, sketch the graph of $y=4x^2-x^4$.

Label clearly the coordinates of any turning points and any intercepts with the coordinate axes.

If $x>0$, solve the following inequality.

$4x^2-x^4>\frac{1}{2x}$

Give your answer to 2 decimal places.