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

$\mathrm{f} ( x ) = 5 - 1.25 x \mathrm{g} ( x ) = 1.5 x - 1$

On the diagram, sketch the graph of $y = \mathrm{f} ( x )$ for $0\leq x\leq4.$

Extract:

g(x) = 1.5 x - 1

On the diagram, sketch the graph of $y = \mathrm{g} ( x )$ for $0\leq x\leq4.$

Extract:

straight f open parentheses x close parentheses equals 5 minus 1.25 x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight g open parentheses x close parentheses space equals 1.5 x minus 1

Find $x$ when $f(x)=g(x).$

$2^{3x+1}=8^{2-x}$

Find the value of $x$

$x=$.......................

$f(x)=x^{3}-5x+3$ for $-3\leq x\leq3$

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

Extract:

f(x) = x^3 - 5x + 3 for -3 ≤ x ≤ 3

Find the coordinates of the local maximum.

( ...................... , ......................)

Extract:

f(x) = x^3 - 5x + 3 for -3 ≤ x ≤ 3

Find the zeros of the graph of $y=f(x).$

$f(x)=x^{3}-2x$ $g(x)=1$

On the diagram, sketch the graphs of $y=f(x)$ and $y=g(x)$ for $-2\leq x\leq2.$

Extract:

straight f left parenthesis x right parenthesis equals x cubed minus 2 x straight g left parenthesis x right parenthesis equals 1 On the diagram, sketch the graphs of y = f(x) and y = g(x) for -2 ≤ x ≤ 2.

Use your graphs to solve $x^{3}-2x-1=0$.

Extract:

straight f left parenthesis x right parenthesis equals x cubed minus 2 x straight g left parenthesis x right parenthesis equals 1 On the diagram, sketch the graphs of y = f(x) and y = g(x) for negative 2 less or equal than x less or equal than 2.

Use your graphs to solve $x^{3}-2x>1$.

Use a graphical method to solve the inequality.

Show a sketch of the graph.

$2^{x}+x>5$

$\mathrm{f} ( x ) = \frac{1}{\sin x^{\circ}}$ for $0\leq x\leq360$

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

Extract:

f(x) = 1/sin(x°) for 0 ≤ x ≤ 360

Find the coordinates of the local minimum point.

Extract:

f(x) = 1 / sin x for 0 ≤ x ≤ 360

Write down the equations of the three asymptotes of the graph of $y=f(x)$.

Extract:

f(x) = 1/sin(x°) for 0 ≤ x ≤ 360

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

Write down the range of values of $k$.

Extract:

Graph with x-axis labeled from 0 to 360 and y-axis labeled from -10 to 10. Both axes have arrows indicating positive direction. y = f(x) = 1/sin(x) for 0 ≤ x ≤ 360 On the diagram, sketch the graph of y = f(x).

By sketching another graph on the diagram, solve the equation $\frac{1}{\sin x^{\circ}} = 5 \sin ( \frac{x}{2} )^{\circ}$ for $0\leq x\leq360^{\circ}$.

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 $x$.

Extract:

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

V = (600 - x)/4 + sqrt(70000 + x^2)

Find, correct to the nearest metre,

(i) the minimum visibility,

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

Extract:

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

V = (600 - x)/4 + sqrt(70000 + x^2)

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

$\mathrm{f} ( x ) = ( x - 2 )^{3} - 5 x + 12$ for $-1\leq x\leq4.7$

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

Write down the coordinates of the local maximum.

Extract:

f(x) = (x - 2)^3 - 5x + 12

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

Find the values of $k$.

$\mathrm{g} ( x ) = - ( x - 1 )^{2}$ for $-1\leq x\leq4.7$

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

Extract:

f(x) = (x - 2)^3 - 5x + 12 for -1 ≤ x ≤ 4.7 g(x) = -(x - 1)^2 for -1 ≤ x ≤ 4.7

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

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 $\begin{gathered} y = \frac{1}{x^{2} + 1} \end{gathered}$ and $\begin{gathered} y = x^{3} - 2 x \end{gathered}$.

Solve.

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

Solve.

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

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.

$x = log_{p} ( x + q )$

Find $p$ and $q$.

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

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

Extract:

f(x) = (x^2 + 3x)/[(x-2)(x+1)]

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

Extract:

f(x) = (x^2 + 3x)/[(x - 2)(x + 1)]

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

Extract:

f(x) = (x^2 + 3x)/[(x - 2)(x + 1)] On the diagram sketch the graph of y = f(x) for values of x between -6 and 6.

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

(i) On the diagram, sketch the graph of $y = \mathrm{g} ( x )$ for $-6\leq x\leq6.$

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

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

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

On the diagram, sketch the graph of $y = \mathrm{f} ( x )$ for $0\leq x\leq360.$

Extract:

f(x) = (cos x°)^2 g(x) = 0.5 - 0.001x Graph with a horizontal x-axis from 0 to 360 and a vertical y-axis from -1 to 1. Both axes are labelled x and y respectively.

On the same diagram, sketch the graph of $y=g(x)$ for $0\leq x\leq360.$

Extract:

f(x) = (cos x°)^2 g(x) = 0.5 - 0.001x

Solve $\mathrm{f} ( x ) = \mathrm{g} ( x )$ for $0\leq x\leq360.$

Extract:

f(x) = (cos x°)^2 g(x) = 0.5 - 0.001x for 0 ≤ x ≤ 360

Solve $\mathrm{f} ( x ) < \mathrm{g} ( x )$ for $0\leq x\leq360.$

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

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

Extract:

f(x) = x + 5/[(x-2)(x+3)]

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

Extract:

f(x) = x + \frac{5}{(x-2)(x+3)}

(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 $\mathrm{f} ( x ) = k$ has exactly one solution.

Extract:

f(x) = x + 5/[(x-2)(x+3)] g(x) = -4 - x

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

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

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

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.