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

1a

3 marks

q26-paper2-spec2025-cie-igcse-further-maths

$f (x) = x (x + 2) (x - 3)$

On the diagram, sketch the graph of $y = f (x) for - 3 ⩽ x ⩽ 4$ .
Show the values of the intersections with the axes.

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1b

3 marks

Expand and simplify.

$x (x + 2) (x - 3)$

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1

4 marks

On the diagram, sketch the graph of $y = (x + 1) {(x - 3)}^{2}$ .
Label the values where the graph meets the $x$ -axis and the $y$ -axis.

cie-igcse-2020-oct-nov-p4-tz3-q7b

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1a

2 marks

$f (x) = \frac{20}{x} + x, x \neq 0$

Complete the table.

$x$	-10	-8	-5	-2	-1.6		1.6	2	5	8	10
$f (x)$	-12	-10.5	-9	-12	-14.1		14.1	12			12

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1b

5 marks

On the grid, draw the graph of $y = f (x)$ for $- 10 ⩽ x ⩽ - 1.6$ and $1.6 ⩽ x ⩽ 10$ .

cie-igcse-spec-p4-q3b-graph

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1c

2 marks

Using your graph, solve the equation $f (x) = 11$ .

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

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1d

2 marks

$k$ is a prime number and $f (x) = k$ has no solutions.

Find the possible values of $k$ .

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1e

1 mark

The gradient of the graph of $y = f (x)$ at the point $(2, 12)$ is $- 4$ .

Write down the coordinates of the other point on the graph of $y = f (x)$ where the gradient is $- 4$ .

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

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1f

6 marks

i)

The equation

f (x) = x^{2}

can be written as

x^{3} + p x^{2} + q = 0 .

Show that

p = - 1

and

q = - 20

.

[2]

ii)

On the grid, draw the graph of

y = x^{2}

for

- 4 ⩽ x ⩽ 4

.

[2]

iii)

Using your graphs, solve the equation

x^{3} - x^{2} - 20 = 0 .

x

= .............................................. [1]

iv)

The diagram shows a sketch of the graph of

y = x^{3} - x^{2} - 20 .

P

is the point

(n, 0)

.

Write down the value of

n

.

n =

.............................................. [1]

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$x$	-1	-0.5	0	0.5	1	1.5	2
$y$	-3	1.75				0.75	3

$x$	0.5	1	2	3	4	5	6
$f (x)$	-7.9	-3.8		0.9		5.5	8.3

$x$	-4	-3	-2	-1		1	2	3	4
$g (x)$	-2.3		-4.5	-9		9	4.5		2.3

$x$	$0$	$1$	$2$	$3$	$4$
$y$		$2$	$4$	$8$

Function	$y = x + 1$	$y = 1 - \frac{x}{3}$	$y = 2 x^{2}$	$y = - \frac{4}{x}$
Diagram	E

$x$	-3.5	-3	-2.5	-2	-1.5	-1	-0.5	0	0.5	1	1.5
$y$	-4.1		5.1	6	5.4	4	2.6		2.9		12.1

$x$	-3	-2.5	-2	-1.5	-1	0	1	1.5	2	2.5	3
$y$	-19	-9.1		0.1	1	-1	-3	-2.1	1	7.1

Further Graphs & Tangents (Cambridge O Level Maths)

Topic Questions