Further Graphs & Tangents (Cambridge (CIE) O Level Maths): Exam Questions

$y=x^2+\frac{1}{x},x\ne 0$

Complete the table.

On the grid, draw the graph of $y=x^2+\frac{1}{x}$ for $0.2\le x\le 2.5.$ The graph of $y=x^2+\frac{1}{x}$ for $-2.5\le x\le -0.2$ has been drawn for you.

By drawing suitable straight lines on the grid, solve the following equations.

$$i)$\frac{}{}$$$ $x^2+\frac{1}{x}=-2$

$x=$ ................................................ [1]

$$ii)$\frac{}{}$$$ $x^2+\frac{1}{x}+x-1=0$

$x=$ ................................................ [2]

$k$ is an integer and the equation $x^2+\frac{1}{x}=k$ has three solutions. Write down a possible value of $k$.

$k=$ ................................................

The table shows some values of  $y=\frac{x^2}{2}+\frac{1}{x^2}-\frac{2}{x}$, $x\ne 0$.

    Complete the table.

On the grid, draw the graph of  $y=\frac{x^2}{2}+\frac{1}{x^2}-\frac{2}{x}$  for  $-3\le x\le -0.3$  and  $0.2\le x\le 3$.

Use your graph to solve  $\frac{x^2}{2}+\frac{1}{x^2}-\frac{2}{x}\le 0$.

..................... $\le x\le$ ....................

Find the smallest positive integer value of $k$ for which  $\frac{x^2}{2}+\frac{1}{x^2}-\frac{2}{x}=k$ has two solutions for  $-3\le x\le -0.3$  and  $0.2\le x\le 3$.

By drawing a suitable straight line, solve  $\frac{x^2}{2}+\frac{1}{x^2}-\frac{2}{x}=3x+1$  for  $-3\le x\le -0.3$  and  $0.2\le x\le 3$.

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

The table shows some values for $y=x^3+x^2-5x$.

 
Complete the table.

On the grid, draw the graph of  $y=x^3+x^2-5x$  for  $-3\le x\le 3$.

Use your graph to solve the equation  $x^3+x^2-5x=0$.

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

By drawing a suitable tangent, find an estimate of the gradient of the curve at $x=2$.

Write down the largest value of the integer, $k$, so that the equation  $x^3+x^2-5x=k$  has three solutions for  $-3\le x\le 3$.

$k$ = ..................................................

The table shows some values for $y=\frac{3}{10}{x}^3-2x$ for $-3\le x\le 3$.

Complete the table.

On the grid, draw the graph of $y=\frac{3}{10}{x}^3-2x$ for $-3\le x\le 3$.

On the grid, draw a suitable straight line to solve the equation $\frac{3}{10}{x}^3-2x=\frac{1}{2}\left(1-x\right)$ for $-3\le x\le 3.$  

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

For $-3\le x\le 3$, the equation $\frac{3}{10}{x}^3-2x=1$ has $n$ solutions.   Write down the value of $n$.  

$n=$..................................................

The table shows some values of  $y=x^3-3x^2+x$.

Complete the table.

On the grid, draw the graph of  $y=x^3-3x^2+x$  for  $-0.75\le x\le 2.75$.

Use your graph to complete the inequalities in $x$ for which $y>-1$.  

.................... $<x<$ .................... and  $x>$ ....................

The equation  $x^3-3x^2+2x-1=0$ can be solved by drawing a straight line on the grid.

i) Write down the equation of this line.

[2]

ii) On the grid, draw this line and use it to solve the equation  $x^3-3x^2+2x-1=0$. 

$x$ = ............................................... [3]

By drawing a suitable tangent, find an estimate for the gradient of the graph of  $y=x^3-3x^2+x$  at  $x=-0.25$.

Complete the table of values for $y=\frac{x^3}{3}-\frac{1}{2x^2},x\ne 0$.

On the grid, draw the graph of $y=\frac{x^3}{3}-\frac{1}{2x^2}$ for $-3\le x\le -0.3$ and $0.3\le x\le 3$.

i) By drawing a suitable tangent, find an estimate of the gradient of the curve at $x=-2$.

[3]

ii) Write down the equation of the tangent to the curve at $x=-2$. Give your answer in the form $y=mx+c$.

$y=$ ................................................ [2]

Use your graph to solve the equations.

i) $\frac{x^3}{3}-\frac{1}{2x^2}=0$

$x=$ ................................................ [1]

ii) $\frac{x^3}{3}-\frac{1}{2x^2}+4=0$

$x=$ .................... or $x=$ .................... or $x=$ .................... [3]

The table shows some values for $y=\frac{x^3}{3}–2x+5$correct to 1 decimal place.

Complete the table.

Draw the graph of $y=\frac{x^3}{3}–2x+5\mathrm{for}–3⩽x⩽3.$

By drawing a suitable line on the grid, find the roots of the equation $\frac{x^3}{3}–1.5x+1=0.$

The grid shows the graph of$y=x^3-2x^2-5x+6$.

i) $x^3-2x^2-5x+6=k$ has exactly 2 solutions.
Use the graph to find the possible values of $k$.

[2]

ii) By drawing a suitable line on the grid, find the solutions of  $x^3-2x^2-7x+5=0$

x = .................. , x = .................. , x = .................. [4]

The table shows some values for $y=-\frac{x^3}{3}+3x+2$ correct to one decimal place.

Complete the table.

Draw the graph of $y=-\frac{x^3}{3}+3x+2$ for $-3\le x\le 3$.

By drawing a suitable line on the grid, find the roots of the equation$-\frac{x^3}{3}+1.5x+0.4=0$

The table shows some values of $y=\frac{1}{2x}-\frac{x}{4}$ for $0.15\le x\le 3.5.$

Complete the table. 

On the grid, draw the graph of $y=\frac{1}{2x}-\frac{x}{4}$ for $0.15\le x\le 3.5.$
The last two points have been plotted for you.

Use your graph to solve the equation $\frac{1}{2x}-\frac{x}{4}=\frac{1}{2}$ for $0.15\le x\le 3.5.$   

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

i) On the grid, draw the line $y=2-x.$   

[2]

ii) Write down the x co-ordinates of the points where the line $y=2-x$ crosses the graph of $y=\frac{1}{2x}-\frac{x}{4}$ for $0.15\le x\le 3.5$.  

$x$ = .................... and $x$ = .................... [2]

Show that the graph of $y=\frac{1}{2x}-\frac{x}{4}$ can be used to find the value of $\sqrt{2}$ for $0.15\le x\le 3.5.$

The diagram shows the graph of $y=f\left(x\right)$ where $f\left(x\right)=x^2-\frac{2}{x}-2,x\ne 0$.

On the grid, draw a suitable straight line to solve the equation $x^2-\frac{2}{x}-7=-3x$ for $-3\le x\le 3$.  

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

By drawing a suitable tangent, find an estimate of the gradient of the curve at $x=-2$.

i) Complete the table for $y=\text{g}\left(x\right)$ where $\text{g}\left(x\right)=2^{-x}$ for $-3\le x\le 3$.

[3]

ii) On the grid, draw the graph of $y=\text{g}\left(x\right)$.  

[3]

iii) Use your graph to find the positive solution to the equation $\text{f}\left(x\right)=\text{g}\left(x\right)$.  

$x$ = .............................................. [1]

The graph of $y=10-8x^2$ for  $-1.5\le x\le 1.5$  is drawn on the grid.

And the table shows some values for $y=x^3+3x+4$.

i) Complete the table.

[3]

ii) On the grid, draw the graph of $y=x^3+3x+4$ for $-1.5\le x\le x1.5$.

[4]

Show that the values of $x$ where the two curves intersect are the solutions to the equation $x^3+8x^2+3x-6=0$.

By drawing a suitable straight line, solve the equation $x^3+5x+2=0$  for  $-1.5\le x\le 1.5$.

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

The table shows some values for $y=2x+\frac{1}{x}-3$ for  $0.125\le x\le 3.$

Complete the table.

On the grid, draw the graph of $y=\frac{1}{2}x+\frac{1}{x}-3$ for $0.125\le x\le 3.$

Use your graph to solve $2x+\frac{1}{x}-3\ge 2.$  

.................................................   ................................................. 

The equation $\frac{1}{x}=7-3x$ can be solved using your graph in part (b) and a straight line.

i) Write down the equation of this straight line.  

[2]

ii) Draw this straight line and solve the equation $\frac{1}{x}=7-3x.$  

$x=$.................... or $x=$.................... [3]

The table shows some values of $x$ and the corresponding values of $y$ for $y=2x^3–3x^2+5.$

i) Complete the table. 

[1]

ii) Using a scale of $4$ cm to represent $1$ unit, draw a horizontal $x$-axis for $–1.5\le x\le 2$.
Using a scale of $2$ cm to represent $5$ units, draw a vertical $y$-axis for $–10\le y\le 10$.
Draw the graph of $y=2x^3–3x^2+5$ for $–1.5\le x\le 2.$

[3]

iii) Use your graph to estimate the gradient of the curve when $x=1.5$.

[2]

iv) By drawing a suitable line on your graph, find the solution of the equation $2x^3–3x^2+4=0$.

$x$ = ...................[2]

The graph shows a sketch of the curve $y=\frac{p}{x}$
Two points on the curve are $(3,0.4)$ and $(q,2.4)$

i) Find $p$ and $q$.

$p$ = ...................
$q$ = ...............[2]

ii) Calculate the gradient of the straight line joining the points $(3,0.4)$ and $(q,2.4)$.

[2]