Further Graphs & Tangents | Cambridge O Level Maths Topic Questions 2025

1a

2 marks

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

Complete the table.

$x$	0.2	0.3	0.5	1	1.5	2	2.5
$y$	5.0	3.4	2.3		2.9		6.7

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1b

4 marks

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

cie-igcse-2020-oct-nov-p4-tz2-q7b

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1c

3 marks

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]

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1d

1 mark

$k$ is an integer and the equation $x^{2} + \frac{1}{x} = k$ has three solutions.

Write down a possible value of $k$ .

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

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1

3 marks

cie-igcse-2020-may-jun-p4-tz3-q10b

The diagram shows the graph of another function.

By drawing a suitable tangent, find an estimate for the gradient of the function at the point P.

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1a

3 marks

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

$x$	0.15	0.2	0.5	1	1.5	2	2.5	3	3.5
$y$	3.30		0.88		-0.04		-0.43	-0.58	-0.73

Complete the table.

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1b

4 marks

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

cie-igcse-2019-may-jun-p4-tz2-q5b

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1c

1 mark

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

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

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1d

4 marks

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}{2 x} - \frac{x}{4}

for

0.15 \leq x \leq 3.5

.

x

= .................... and

x

= .................... [2]

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1e

2 marks

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

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$x$	$- 3$	$- 2$	$- 1$	$- 0.5$	$- 0.3$		$0.2$	$0.3$	$0.5$	$1$	$2$	$3$
$y$	$5.3$	$3.3$		$8.1$	$17.8$			$4.5$	$0.1$	$- 0.5$	$1.3$

$x$	$- 3$	$- 2$	$- 1.5$	$- 1$	$0$	$1$	$1.5$	$2$	$2.5$	$3$
$y$	$- 3$	$6$	$6.4$		$0$		$- 1.9$	$2$	$9.4$

$x$	-3	-2	-1.5	-1	0	1	1.5	2	3
$y$			2.0	1.7	0		-2.0	-1.6

$x$	-0.75	-0.5	-0.25	0	0.5	1	1.5	2	2.5	2.75
$y$	-2.9	-1.4	-0.5		-0.1	-1	-1.9		-0.6

$x$	0.125	0.25	0.375	0.5	0.75	1	1.5	2	2.5	3
$y$	5.25	1.5	0.42			0	0.67	1.5		3.33

Further Graphs & Tangents (Cambridge O Level Maths)

Topic Questions