Figure — understand the maths, don't just answer it

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Surds and indices

39 questions

LessonNot started

\frac{7 - \sqrt[3]{x^4}}{x^3}

in the form

7x^p - x^q

, where

p

and

q

are constants.

(2 marks)

●●●●●Level 22 marksStart

Given that

y = \dfrac{(2x + 3)^2}{\sqrt{x}}, \qquad x > 0

find

\dfrac{\mathrm{d}y}{\mathrm{d}x}

, giving each term in its simplest form. (3 marks)

●●●●●Level 23 marksStart

Solve the equation

8^{x+1} = 4^{2x-1}

(4 marks)

●●●●●Level 24 marksStart

Given that

p = 5\sqrt[3]{x} \qquad\qquad q = \frac{1}{5x^{3}}

express each of the following in the form

kx^{n}

where

k

and

n

are fully simplified constants.
(a)

q

(1 mark)
(b)

\dfrac{p^{2}}{q}

(2 marks)
(c)

\left(p^{3}q\right)^{-\frac{1}{2}}

(2 marks)

●●●●●Level 25 marksStart

Find

\int \left(x^2 - 3\right)\left(2 + \frac{5}{x^2}\right)\mathrm{d}x

writing your answer in simplest form.

(4 marks)

●●●●●Level 24 marksStart

Quadratic equations

110 questions

LessonNot started

The displacement,

r

metres, of a particle from a fixed point

O

at time

t

seconds is

r = 7.2t - 0.8t^2

for

t \ge 0

(a) Find the value of

r

when

t = 2

(1 mark)
(b) Determine the range of values of

t

for which the displacement is positive.

(2 marks)

●●●●●Level 23 marksStart

(a) Express

3x^2 - 18x + 31

in the form

3(x - a)^2 + b

, where

a

and

b

are constants.

(2 marks)
(b) Hence write down the minimum value of

3x^2 - 18x + 31

.

(1 mark)

●●●●●Level 23 marksStart

Simultaneous equations and inequalities

32 questions

LessonNot started

Solve the inequality

5 - 2x < 11

. (2 marks)

●●●●●Level 22 marksStart

Inequalities on graphs and regions

25 questions

LessonNot started

The inequality

y > 3x - 2

defines a region of the plane.
(a) Determine whether the point

(2, 5)

lies in the region. (2 marks)
(b) Determine whether the point

(3, 5)

lies in the region. (1 mark)

●●●●●Level 23 marksStart

A student is asked to shade the region satisfying

y > 2x + 1

.
(a) State whether the boundary line should be drawn solid or dashed, and why. (2 marks)
(b) Describe how to decide which side of the line to shade. (1 mark)

●●●●●Level 23 marksStart

The diagram shows the curve

y = 9 - x^2

and the line

y = x + 3

. The region

R

, shaded in the diagram, is bounded by the curve, the line and the

x

-axis.Write down three inequalities that define the region

R

.

(3 marks)

●●●●●Level 23 marksStart

Sketching curves

42 questions

LessonNot started

It is given that

\mathrm{f}(x) = x(x + 4)(b - x)

where

b

is a positive constant.

Sketch the graph of

y = \mathrm{f}(x)

(3 marks)

●●●●●Level 23 marksStart

The diagram shows part of the graph of

y = \text{f}(x)

, where

\text{f}(x)

is a cubic polynomial in

x

.Explain why one of the roots of the equation

\text{f}(x) = 0

cannot be found by the sign change method.

(2 marks)

●●●●●Level 22 marksStart

Transformations of graphs

36 questions

LessonNot started

The graph of

y = f(x)

is translated

2

units in the positive

x

-direction. The resulting graph is then stretched parallel to the

x

-axis with scale factor

\dfrac{1}{3}

.

The equation of the final graph is

Circle your answer.

A $y = f(3x - 2)$
B $y = f(3x - 6)$
C $y = f\left(\dfrac{x}{3} - 2\right)$
D $y = f\left(\dfrac{x - 2}{3}\right)$

●●●●●Level 21 markStart

The point

P(5,\ -2)

lies on the curve with equation

y = \text{f}(x)

,

x \in \mathbb{R}

Find the coordinates of the point to which

P

is mapped when the curve with equation

y = \text{f}(x)

is transformed to the curve with equation
(a)

y = \text{f}(x) - 5

(1 mark)
(b)

y = |\text{f}(x)|

(1 mark)
(c)

y = 3\text{f}(x + 1) - 4

(2 marks)

●●●●●Level 24 marksStart

The point

Q(-8,\ 6)

lies on the curve with equation

y = \text{f}(x)

,

x \in \mathbb{R}

Find the coordinates of the point to which

Q

is mapped when the curve with equation

y = \text{f}(x)

is transformed to the curve with equation
(a)

y = \text{f}(x + 4)

(1 mark)
(b)

y = \text{f}(4x)

(1 mark)
(c)

y = 5\text{f}(-x) - 3

(2 marks)

●●●●●Level 24 marksStart

The point

R(-3,\ 4)

lies on the curve with equation

y = \text{f}(x)

,

x \in \mathbb{R}

Find the coordinates of the point to which

R

is mapped when the curve with equation

y = \text{f}(x)

is transformed to the curve with equation
(a)

y = \text{f}(x) + 7

(1 mark)
(b)

y = \text{f}(x - 6)

(1 mark)
(c)

y = 4\text{f}(-x + 1) - 9

(2 marks)

●●●●●Level 24 marksStart

A sequence of transformations maps the curve

y = \sqrt{x}

to the curve

y = 2\sqrt{x + 5}

.

Give details of these transformations.

(3 marks)

●●●●●Level 23 marksStart

Polynomials and the factor theorem

50 questions

LessonNot started

It is given that

\mathrm{f}(x) = x(x + 4)(b - x)

where

b

is a positive constant.

Sketch the graph of

y = \mathrm{f}(x)

(3 marks)

●●●●●Level 23 marksStart

The polynomial

\text{P}(x)

is given by

\text{P}(x) = 6x^3 - x^2 - 14x + 8

Use the factor theorem to show that

(3x - 2)

is a factor of

\text{P}(x)

(2 marks)

●●●●●Level 22 marksStart

\text{f}(x) = (2x - 1)(x^2 + 6x + k) - 18 \qquad \text{where } k \text{ is a constant}

Given that

(x + 4)

is a factor of

\text{f}(x)

, find the value of

k

.

(3 marks)

●●●●●Level 23 marksStart

\text{f}(x) = (3x + 5)(x^2 - 6x + k) - 20 \qquad \text{where } k \text{ is a constant}

Given that

(x + 1)

is a factor of

\text{f}(x)

, find the value of

k

.

(3 marks)

●●●●●Level 23 marksStart

The cubic polynomial

f(x)

is defined by

f(x) = 2x^3 - x^2 - 13x - 6

.
(a) Use the factor theorem to show that

(2x + 1)

is a factor of

f(x)

.

(1 mark)
(b) Express

f(x)

in fully factorised form.

(3 marks)

●●●●●Level 24 marksStart

Algebraic fractions

30 questions

LessonNot started

(a) Simplify fully

\dfrac{2x^2 - x - 6}{x^2 - 4}

. (3 marks)
(b) Hence solve the equation

\dfrac{2x^2 - x - 6}{x^2 - 4} = 3

. (2 marks)

●●●●●Level 25 marksStart

Simplify

\dfrac{x^{2} + 5x + 6}{x^{2} - 4}

. (2 marks)

●●●●●Level 22 marksStart

The function f is defined by

\text{f}(x) = \dfrac{2x^2 + 7x + 3}{x^2 - x - 12} - \dfrac{6}{x - 4} \qquad x > 4

(a) Show that

\text{f}(x) = \dfrac{2x - 5}{x - 4} \qquad x > 4

(3 marks)
(b) Hence find

\text{f}'(x)

giving your answer in simplest form.

(2 marks)

●●●●●Level 25 marksStart

Partial fractions

46 questions

LessonNot started

Express

\dfrac{9x + 17}{(2x + 1)(x + 3)}

in partial fractions.

(3 marks)

●●●●●Level 23 marksStart

The modulus function

40 questions

LessonNot started

Solve

|x + 3| = 7

. (2 marks)

●●●●●Level 22 marksStart

Functions, domain and range

15 questions

LessonNot started

The function

f

is defined by

f(x) = x^2 - 1

. Find

f(-3)

.

●●●●●Level 21 markStart

The function

f

is defined by

f(x) = 2x + 3

. Find

f(4)

.

●●●●●Level 21 markStart

The function

f

is defined by

f(x) = \frac{3x + 1}{x - 4} \qquad x \in \mathbb{R},\ x > 4

(a) State the range of

f

.

(1 mark)
(b) Find

f^{-1}(x)

, stating the domain of

f^{-1}

.

(3 marks)

●●●●●Level 24 marksStart

Composite and inverse functions

45 questions

LessonNot started

SECTION A: PURE MATHEMATICS

The function

\mathrm{f}

is defined by

\mathrm{f}(x) = \dfrac{2x+1}{x-3}

,

\quad x \in \mathbb{R}

,

x \neq 3

.
(a) Find

\mathrm{f}^{-1}(x)

, stating the value of

x

for which

\mathrm{f}^{-1}

is not defined. (3 marks)
(b) Find

\mathrm{ff}(5)

. (2 marks)

●●●●●Level 25 marksStart

The function

\text{f}

is defined by

\text{f}(x) = x^2 - 6x + 14 \quad \text{for } x \in \mathbb{R}

The function

\text{g}

is defined by

\text{g}(x) = \ln(10 - 2x)

where

\text{g}

has its greatest possible domain.
(a) Using set notation, state the range of

\text{f}

(2 marks)
(b) State the domain of

\text{g}

(1 mark)

●●●●●Level 23 marksStart

The function

f

is defined by

f(x) = \frac{3x + 1}{x - 4} \qquad x \in \mathbb{R},\ x > 4

(a) State the range of

f

.

(1 mark)
(b) Find

f^{-1}(x)

, stating the domain of

f^{-1}

.

(3 marks)

●●●●●Level 24 marksStart