Browse Algebra & functions

$OQR$ is a sector of a circle with centre $O$
$OPS$ is a sector of a smaller circle, also with centre $O$ , where $P$ lies on $OQ$ and $S$ lies on $OR$
angle $POS = 1.1$ radians
the area of sector $OPS$ is $4.95$ m $^2$

(a) show that

OP = 3

m

(2 marks)

Given also that

the area of the gravel path $PQRS$ is $40$ m $^2$
$PQ = x$ metres

(b) show that

11x^2 + 66x - 800 = 0

(3 marks)
(c) Solve the equation in part (b) to find the value of

x

, giving your answer to 3 significant figures.

(Solutions relying entirely on calculator technology are not acceptable.)

(2 marks)
(d) Hence find the perimeter of the gravel path, giving your answer to the nearest metre.

(3 marks)

●●●●●Level 510 marksStart

Figure 1Figure 1 shows a sketch of part of the graph with equation

y = \text{f}(x)

, where

\text{f}(x) = 3k - |kx - 6| \qquad x \in \mathbb{R}

and

k

is a constant with

k > 2

The graph

cuts the $y$ -axis at the point $A$
has a vertex at the point $P$

(a) Find, in simplest form in terms of

k

,

(i) the

y

coordinate of

A

(ii) the coordinates of

P

(3 marks)
(b) Find, in terms of

k

, the range of values of

x

which satisfy

3k - |kx - 6| \geqslant 2k

(3 marks)

Given that the line with equation

y = 2x + 5

intersects the graph of

y = \text{f}(x)

at 2 distinct points,
(c) find the range of values of

k

.

(4 marks)

●●●●●Level 510 marksStart

The continuous random variable

X

has cumulative distribution function given by

\mathrm{F}(x) = \begin{cases} 0 & x < 0 \\ ax^2 + bx^3 & 0 \leqslant x \leqslant k \\ 1 & x > k \end{cases}

where

a

b

and

k

are constants.
(a) Show that

ak^2 = 1 - bk^3

(1 mark)

Given that

\mathrm{E}(X) = \dfrac{7}{6}

(b) show that

bk^4 = 14 - 8k

(6 marks)

Given also that

\mathrm{Var}(X) = \dfrac{43}{180}

(c) show that

5k^2 + bk^5 = 16

(5 marks)

Given further that

k < 2.5

(d) find the value of

k

(4 marks)
(e) Find the value of

a

and the value of

b

(2 marks)

●●●●●Level 518 marksStart

Simultaneous equations and inequalities

32 questions

LessonNot started

(a) Solve the inequality

\dfrac{x + 4}{x - 2} \leqslant 3

(5 marks)
(b) Hence, or otherwise, solve the inequality

\dfrac{|x| + 4}{|x| - 2} \leqslant 3

(3 marks)

●●●●●Level 58 marksStart

The functions f and g are defined for all real values of

x

f(x) = 3x^2 - 6x \quad

and

\quad g(x) = 6x - 9

.
(a) Find the range of f.

(3 marks)
(b) Give a reason why f has no inverse.

(1 mark)
(c) Given that

fg(1) = g^{-1}(a)

, where

a

is a constant, determine the value of

a

.

(4 marks)
(d) Determine the set of values of

x

for which

f(x) > g(x)

. Give your answer in set notation.

(3 marks)

●●●●●Level 511 marksStart

The equation

(k - 1)x^2 - 2kx + (k + 3) = 0

where

k

is a constant, has at least one real root.

Find the complete set of possible values of

k

, fully justifying your answer. (7 marks)

●●●●●Level 57 marksStart

The diagram shows the graph of

y = |3x - 6|

.(a) State the coordinates of the points where the graph meets the axes.

(2 marks)
(b) Given that the graphs of

y = |3x - 6|

and

y = ax + 3

have two distinct points of intersection, determine

(i) the set of possible values of

a

,

(ii) the

x

-coordinates of the points of intersection of these graphs, giving your answers in terms of

a

.

(7 marks)

●●●●●Level 59 marksStart

Inequalities on graphs and regions

25 questions

LessonNot started

The line

y = mx + 4

and the curve

y = x^{2} - 2x + 8

are drawn on the same axes, where

m

is a constant.
(a) Find the two values of

m

for which the line is a tangent to the curve. (4 marks)
(b) For each of these values of

m

, find the coordinates of the point where the line touches the curve. (3 marks)
(c) Find the set of values of

m

for which the line meets the curve at two distinct points. (2 marks)

●●●●●Level 59 marksStart

The number of passengers carried each day by two bus routes is being monitored.

The number of passengers,

N_R

, in hundreds, carried each day by route R is modelled by the equation

N_R = |t - 7| + 6 \qquad t \geqslant 0

where

t

is the time in years since the two routes opened.

The number of passengers,

N_S

, in hundreds, carried each day by route S is modelled by the equation

N_S = 18 - |2t - 17| \qquad t \geqslant 0

where

t

is the time in years since the two routes opened.

Over a 16-year period the graph of

N_R

is a V-shape with a single minimum point, and the graph of

N_S

is an inverted V-shape with a single maximum point.

Use the equations of the models to answer parts (a), (b),
(c) and (d).
(a) Find the initial difference between the number of passengers carried each day by route R and the number carried each day by route S.

(2 marks)

When

t = T

a retail park opened close to route R and the number of passengers carried each day by route R began to increase.
(b) Suggest a value for

T

, giving a reason for your answer.

(2 marks)
(c) Find the range of values of

t

for which

N_R > N_S

giving your answer in set notation.

(5 marks)
(d) State a limitation of the model used for route S.

(1 mark)

●●●●●Level 510 marksStart

Figure 1

Figure 1 shows part of the curve

C

with equation

y = \text{f}(x)

where

\text{f}(x) = 3x^2 - 18x + 32

(a) Write

3x^2 - 18x + 32

in the form

a(x + b)^2 + c

where

a

b

and

c

are constants to be found.

(3 marks)

Given that

C

has a minimum at the point

P

(b) state the coordinates of

P

(1 mark)

The point

Q

lies on

C

and has

x

coordinate

1

The line

l

passes through

Q

and

P

, as shown in Figure 1.
(c) Show that an equation of

l

y = 23 - 6x

(3 marks)

The finite region

R

, shown shaded in Figure 1, is bounded by the

x

-axis, the

y

-axis,

l

and

C

.
(d) Use inequalities to define the region

R

.

(3 marks)

●●●●●Level 510 marksStart

Figure 1

Figure 1 shows a sketch of the curve

C

with equation

y = 2x^2 - 12x + 22

(a) Write

2x^2 - 12x + 22

in the form

a(x + b)^2 + c

where

a

b

and

c

are constants to be found.

(3 marks)

The point

M

is the minimum turning point of

C

, as shown in Figure 1.
(b) Deduce the coordinates of

M

(2 marks)

The line

l

is the normal to

C

at the point

P

, as shown in Figure 1.

Given that

l

has equation

x - 4y + k = 0

, where

k

is a constant,
(c) (i) find the coordinates of

P

(ii) show that

k = 22

(6 marks)

The finite region

R

, shown shaded in Figure 1, is bounded by

C

l

and the

y

-axis.
(d) Use inequalities to define the region

R

.

(3 marks)

●●●●●Level 514 marksStart

Figure 1Figure 1 shows a sketch of part of the graph with equation

y = \text{f}(x)

, where

\text{f}(x) = 3k - |kx - 6| \qquad x \in \mathbb{R}

and

k

is a constant with

k > 2

The graph

cuts the $y$ -axis at the point $A$
has a vertex at the point $P$

(a) Find, in simplest form in terms of

k

,

(i) the

y

coordinate of

A

(ii) the coordinates of

P

(3 marks)
(b) Find, in terms of

k

, the range of values of

x

which satisfy

3k - |kx - 6| \geqslant 2k

(3 marks)

Given that the line with equation

y = 2x + 5

intersects the graph of

y = \text{f}(x)

at 2 distinct points,
(c) find the range of values of

k

.

(4 marks)

●●●●●Level 510 marksStart

Sketching curves

42 questions

LessonNot started

Figure 2 shows the curve

C

with equation

y = \dfrac{8}{x} + 2x, \qquad x > 0

together with the line

l

with equation

y = 10

. The region

R

, shown shaded, is bounded by

C

and

l

.Figure 2
(a) Find the coordinates of the stationary point of

C

, and determine its nature. (5 marks)
(b) Find the

x

-coordinates of the two points where

l

meets

C

. (3 marks)
(c) Find the exact area of

R

. (5 marks)
(d) Write down an equation of each of the two asymptotes of

C

. (2 marks)

●●●●●Level 515 marksStart

Figure 1

Figure 1 shows a sketch of part of the curve

C_1

Given that

C_1

has equation $y = \mathrm{f}(x)$ where $\mathrm{f}(x)$ is a cubic function
cuts the $x$ -axis at $x = -2$ and touches the $x$ -axis at $x = 4$
passes through the point $(2, 24)$

(a) find

\mathrm{f}(x)

(3 marks)

The curve

C_2

has equation

y = \tfrac32(x + 2)(10 - x)

A copy of Figure 1, labelled Diagram 1, is shown below.Diagram 1
(b) On Diagram 1 sketch a graph of the curve

C_2

(2 marks)
(c) Use algebra to find the coordinates of the points where

C_1

and

C_2

intersect.

Show each stage of your working.

(5 marks)

●●●●●Level 510 marksStart

The polynomial

\text{p}(x)

is given by

\text{p}(x) = 27x^3 + 3(b - 6)x^2 - 2(b - 6)x - 8

where

b

is a constant.
(a) Use the factor theorem to prove that

(3x - 2)

is a factor of

\text{p}(x)

for all values of

b

(3 marks)
(b) The graph of

y = \text{p}(x)

meets the

x

-axis at exactly two points.
(b) (i) Sketch a possible graph of

y = \text{p}(x)

(ii) Given

\text{p}(x)

can be written as

\text{p}(x) = (3x - 2)(9x^2 + bx + 4)

find the value of

b

Fully justify your answer.

(7 marks)

●●●●●Level 510 marksStart

The curve

C

has equation

y = \dfrac{\ln x}{x}, \qquad x > 0.

(a) Determine the exact coordinates of the stationary point of

C

, and show that it is a maximum point. (5 marks)
(b) The equation

\ln x = kx,

where

k

is a constant, has exactly two distinct real roots. Determine the set of possible values of

k

, justifying your answer fully. (4 marks)

●●●●●Level 59 marksStart

(a) Sketch the graph of

y = x^2(3x + a)

where

a

is a positive constant.

(3 marks)
(b) The polynomial

\text{p}(x)

is given by

\text{p}(x) = x^2(3x + a) + 8

(b) (i) It is given that

(x + 2)

is a factor of

\text{p}(x)

Use the factor theorem to show that

a = 4

(ii) State the transformation which maps the curve with equation

y = x^2(3x + 4)

onto the curve with equation

y = x^2(3x + 4) + 8

(iii) The polynomial

\text{p}(x)

can be written as

(x + 2)(3x^2 + bx + c)

Without finding the values of

b

and

c

, use your answers to parts (a) and (b)(ii) to explain why

b^2 < 12c

(6 marks)

●●●●●Level 59 marksStart

The number of passengers carried each day by two bus routes is being monitored.

The number of passengers,

N_R

, in hundreds, carried each day by route R is modelled by the equation

N_R = |t - 7| + 6 \qquad t \geqslant 0

where

t

is the time in years since the two routes opened.

The number of passengers,

N_S

, in hundreds, carried each day by route S is modelled by the equation

N_S = 18 - |2t - 17| \qquad t \geqslant 0

where

t

is the time in years since the two routes opened.

Over a 16-year period the graph of

N_R

is a V-shape with a single minimum point, and the graph of

N_S

t = T

a retail park opened close to route R and the number of passengers carried each day by route R began to increase.
(b) Suggest a value for

T

, giving a reason for your answer.

(2 marks)
(c) Find the range of values of

t

for which

N_R > N_S

giving your answer in set notation.

(5 marks)
(d) State a limitation of the model used for route S.

(1 mark)

●●●●●Level 510 marksStart

Figure 1Figure 1 shows a sketch of the graph of

C_1

with equation

y = 8 - |2x - 6|

(a) Write down the coordinates of

(i) the vertex of

C_1

(ii) the intersection of

C_1

with the

y

-axis.

(2 marks)
(b) Find the

x

coordinates of the intersections of

C_1

with the

x

-axis.

(2 marks)

Diagram 1 is a copy of Figure 1.
(c) On Diagram 1, sketch the curve

C_2

with equation

y = \dfrac{1}{2}x^2 - 2

Identify clearly the coordinates of any points of intersection of

C_2

with the coordinate axes.

(3 marks)
(d) Using algebra, find the coordinates of the points of intersection of

C_1

and

C_2

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.

A curve has equation

y = \dfrac{x - 1}{x^2 + 3}

.
(a) State the coordinates of the points where the curve crosses the axes.

(2 marks)

The point

A

is where the curve crosses the

y

-axis. The point

B

on the curve has

x

-coordinate

-3

.
(b) Show that the line

AB

is parallel to the

x

-axis.

(2 marks)
(c) Determine the coordinates of the turning points of the curve.

(4 marks)
(d) Determine whether each turning point is a maximum or a minimum.

(5 marks)

●●●●●Level 513 marksStart

The diagram shows the graph of

y = |3x - 6|

.(a) State the coordinates of the points where the graph meets the axes.

(2 marks)
(b) Given that the graphs of

y = |3x - 6|

and

y = ax + 3

have two distinct points of intersection, determine

(i) the set of possible values of

a

,

(ii) the

x

-coordinates of the points of intersection of these graphs, giving your answers in terms of

a

.

(7 marks)

●●●●●Level 59 marksStart

(a) Sketch the curve with equation

y = -\dfrac{12}{x} \qquad x \neq 0

(2 marks)

The curve with equation

y = -\dfrac{12}{x}

x \neq 0

, is transformed onto the curve

C

with equation

y = 3 - \dfrac{12}{x} \qquad x \neq 0

(b) Fully describe this transformation.

(2 marks)

The curve

C

and the line with equation

y = kx + 9

where

k

is a constant, intersect at exactly two points,

P

and

Q

.

Given that the

x

coordinate of

P

-3

(c) show that

k = \dfrac{2}{3}

(2 marks)
(d) Find, using algebra, the coordinates of

Q

.

(Solutions relying entirely on calculator technology are not acceptable.)

(4 marks)

●●●●●Level 510 marksStart

Figure 1

Figure 1 shows a sketch of part of the curve

C

with equation

y = \mathrm{f}(x)

, where

\mathrm{f}(x) = (x + 3)(2x - 1)(x - 4)

(a) Use the given information to state the values of

x

for which

\mathrm{f}(x) > 0

(2 marks)
(b) Expand

(x + 3)(2x - 1)(x - 4)

, writing your answer as a polynomial in simplest form.

(3 marks)

The straight line

l

is the tangent to

C

at the point where

C

cuts the

y

-axis.

Given that

l

cuts

C

at the point

P

, as shown in Figure 1,
(c) find, using algebra, the

x

coordinate of

P

(Solutions relying on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

Figure 1 shows a sketch of the curve

C

with equation

y = \mathrm{f}(x)

, where

\mathrm{f}(x) = (2x + 3)(x - 4)^2

(a) Deduce the range of values of

x

for which

\mathrm{f}(x) \geqslant 0

(1 mark)
(b) Find

\mathrm{f}'(x)

, giving your answer in simplest form.

(3 marks)

The point

R(-1,\ 25)

lies on

C

.

Given that the tangent to

C

at the point

P

is parallel to the tangent to

C

at the point

R

(c) (i) show that the

x

coordinate of

P

satisfies the equation

3x^2 - 13x - 16 = 0

(ii) Hence find the

x

coordinate of

P

.

(4 marks)
(d) Find the point to which

R

is transformed when the curve with equation

y = \mathrm{f}(x)

is transformed to the curve with equation

(i)

y = \mathrm{f}(x + 2)

(ii)

y = \mathrm{f}\left(\tfrac12 x\right)

(2 marks)

●●●●●Level 510 marksStart

Figure 1Figure 1 shows a sketch of part of the graph with equation

y = \text{f}(x)

, where

\text{f}(x) = 3k - |kx - 6| \qquad x \in \mathbb{R}

and

k

is a constant with

k > 2

The graph

cuts the $y$ -axis at the point $A$
has a vertex at the point $P$

(a) Find, in simplest form in terms of

k

,

(i) the

y

coordinate of

A

(ii) the coordinates of

P

(3 marks)
(b) Find, in terms of

k

, the range of values of

x

which satisfy

3k - |kx - 6| \geqslant 2k

(3 marks)

Given that the line with equation

y = 2x + 5

intersects the graph of

y = \text{f}(x)

at 2 distinct points,
(c) find the range of values of

k

.

(4 marks)

●●●●●Level 510 marksStart

Transformations of graphs

36 questions

LessonNot started

(a) Given that

7\cos\theta - 24\sin\theta \equiv K\cos(\theta + \alpha)

where

K > 0

and

0 < \alpha < 90^\circ

, find the value of

K

and the value of

\alpha

, giving

\alpha

to 2 decimal places.

(3 marks)

An ecologist records the number of mice and the number of owls living on a farm over one year.

The number of mice,

M

, is modelled by

M = A + 7\cos(30t)^\circ - 24\sin(30t)^\circ

where

t

is the number of months after the start of the year and

A

is a constant.

According to the model, the greatest number of mice on the farm during the year is

145

(b) (i) Find a complete equation for the model.

(ii) Hence state the least number of mice on the farm during the year, according to the model.

(2 marks)

The ecologist finds that the number of mice on the farm is actually lowest in October.
(c) Use this information to comment on the model for the number of mice.

(2 marks)

The number of owls,

N

, on the farm during the same year is modelled by

N = 40 + 15\sin(30t + 60)^\circ

The number of owls is at its lowest when

t = T

(d) Use the two models to find the number of mice on the farm when

t = T

(4 marks)

●●●●●Level 511 marksStart

(a) Sketch the graph of

y = x^2(3x + a)

where

a

is a positive constant.

(3 marks)
(b) The polynomial

\text{p}(x)

is given by

\text{p}(x) = x^2(3x + a) + 8

(b) (i) It is given that

(x + 2)

is a factor of

\text{p}(x)

Use the factor theorem to show that

a = 4

(ii) State the transformation which maps the curve with equation

y = x^2(3x + 4)

onto the curve with equation

y = x^2(3x + 4) + 8

(iii) The polynomial

\text{p}(x)

can be written as

(x + 2)(3x^2 + bx + c)

Without finding the values of

b

and

c

, use your answers to parts (a) and (b)(ii) to explain why

b^2 < 12c

(6 marks)

●●●●●Level 59 marksStart

(a) Given that

8\sin\theta + 6\cos\theta \equiv R\sin(\theta + \alpha)

where

R > 0

and

0 < \alpha < \dfrac{\pi}{2}

, find the exact value of

R

and find

\alpha

, in radians, correct to 3 decimal places.

(3 marks)

A scientist measures the temperature of the water in two garden ponds throughout one year.

For the first pond, the water temperature,

S\ ^\circ

C, is modelled by

S = A - 8\sin\left(\frac{\pi t}{6}\right) - 6\cos\left(\frac{\pi t}{6}\right)

where

t

is the number of months after the start of the year and

A

is a constant.

According to the model, the highest water temperature in the first pond during the year is

23\ ^\circ

C
(b) (i) Find a complete equation for the model.

(ii) Hence state the lowest water temperature in the first pond during the year, according to the model.

(2 marks)

The scientist observes that the water temperature in the first pond is actually at its lowest towards the end of February.
(c) Use this information to comment on the model for the first pond.

(2 marks)

For the second pond, the water temperature,

P\ ^\circ

C, during the same year is modelled by

P = 12 + 9\cos\left(\frac{\pi t}{6} - 2.3\right)

The water temperature in the second pond is at its highest when

t = T

(d) Use the two models to find the water temperature in the first pond when

t = T

, giving your answer to 3 significant figures.

(4 marks)

●●●●●Level 511 marksStart

The diagram shows the curve

y = f(x)

, which has a maximum point at

(-2, 5)

and crosses the

x

-axis at

(-5, 0)

and

(1, 0)

.On separate diagrams, sketch
(a)

y = f(-x)

and (b)

y = 2f(x)

. In each case give the coordinates of the new maximum point and of the points where the curve crosses the

x

-axis.

●●●●●Level 54 marksStart

The curve

y = f(x)

has a minimum point at

(1, -2)

. The curve is transformed to

y = -f(x - 3)

. Describe the two transformations involved, and state the coordinates of the turning point of the new curve, stating whether it is a maximum or a minimum.

●●●●●Level 53 marksStart

(a) Express

6\cos 2x + 5\sin 2x

in the form

R\cos(2x - \alpha)

, where

R > 0

and

0 < \alpha < \tfrac12\pi

. Give the exact value of

R

and give the value of

\alpha

correct to 4 significant figures.

(3 marks)
(b) Give full details of a sequence of three transformations needed to transform the curve

y = \cos x

to the curve

y = 6\cos 2x + 5\sin 2x

.

(4 marks)
(c) Determine the greatest value of

6\cos 2x + 5\sin 2x

x

varies and give the smallest positive value of

x

for which it occurs.

(2 marks)
(d) Determine the least value of

6\cos 2x + 5\sin 2x

x

varies and give the smallest positive value of

x

for which it occurs.

(2 marks)

●●●●●Level 511 marksStart

(a) Sketch the curve with equation

y = -\dfrac{12}{x} \qquad x \neq 0

(2 marks)

The curve with equation

y = -\dfrac{12}{x}

x \neq 0

, is transformed onto the curve

C

with equation

y = 3 - \dfrac{12}{x} \qquad x \neq 0

(b) Fully describe this transformation.

(2 marks)

The curve

C

and the line with equation

y = kx + 9

where

k

is a constant, intersect at exactly two points,

P

and

Q

.

Given that the

x

coordinate of

P

-3

(c) show that

k = \dfrac{2}{3}

(2 marks)
(d) Find, using algebra, the coordinates of

Q

.

(Solutions relying entirely on calculator technology are not acceptable.)

(4 marks)

●●●●●Level 510 marksStart

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

Figure 1 shows a sketch of the curve

C

with equation

y = \mathrm{f}(x)

, where

\mathrm{f}(x) = (2x + 3)(x - 4)^2

(a) Deduce the range of values of

x

for which

\mathrm{f}(x) \geqslant 0

(1 mark)
(b) Find

\mathrm{f}'(x)

, giving your answer in simplest form.

(3 marks)

The point

R(-1,\ 25)

lies on

C

.

Given that the tangent to

C

at the point

P

is parallel to the tangent to

C

at the point

R

(c) (i) show that the

x

coordinate of

P

satisfies the equation

3x^2 - 13x - 16 = 0

(ii) Hence find the

x

coordinate of

P

.

(4 marks)
(d) Find the point to which

R

is transformed when the curve with equation

y = \mathrm{f}(x)

is transformed to the curve with equation

(i)

y = \mathrm{f}(x + 2)

(ii)

y = \mathrm{f}\left(\tfrac12 x\right)

(2 marks)

●●●●●Level 510 marksStart

Polynomials and the factor theorem

50 questions

LessonNot started

Figure 1

Figure 1 shows a sketch of part of the curve

C_1

Given that

C_1

has equation $y = \mathrm{f}(x)$ where $\mathrm{f}(x)$ is a cubic function
cuts the $x$ -axis at $x = -2$ and touches the $x$ -axis at $x = 4$
passes through the point $(2, 24)$

(a) find

\mathrm{f}(x)

(3 marks)

The curve

C_2

has equation

y = \tfrac32(x + 2)(10 - x)

A copy of Figure 1, labelled Diagram 1, is shown below.Diagram 1
(b) On Diagram 1 sketch a graph of the curve

C_2

(2 marks)
(c) Use algebra to find the coordinates of the points where

C_1

and

C_2

intersect.

Show each stage of your working.

(5 marks)

●●●●●Level 510 marksStart

The polynomial

\text{p}(x)

is given by

\text{p}(x) = 27x^3 + 3(b - 6)x^2 - 2(b - 6)x - 8

where

b

is a constant.
(a) Use the factor theorem to prove that

(3x - 2)

is a factor of

\text{p}(x)

for all values of

b

(3 marks)
(b) The graph of

y = \text{p}(x)

meets the

x

-axis at exactly two points.
(b) (i) Sketch a possible graph of

y = \text{p}(x)

(ii) Given

\text{p}(x)

can be written as

\text{p}(x) = (3x - 2)(9x^2 + bx + 4)

find the value of

b

Fully justify your answer.

(7 marks)

●●●●●Level 510 marksStart

f(x) = 2x^3 + ax^2 + bx - 6

, where

a

and

b

are constants. Given that

(x - 2)

is a factor of

f(x)

and that

f(-1) = -12

:
(a) find the values of

a

and

b

;
(b) factorise

f(x)

completely.

●●●●●Level 56 marksStart

The polynomial

f(x) = x^3 + ax^2 + bx + 30

has

(x - 2)

and

(x + 3)

as factors.
(a) Find the values of

a

and

b

.
(b) Find the remaining linear factor.

●●●●●Level 54 marksStart

f(x) = 3x^3 - 5x^2 - 26x - 8

.
(a) Show that

(x + 2)

is a factor of

f(x)

. (2 marks)
(b) Factorise

f(x)

completely. (3 marks)
(c) Hence solve

f(x) = 0

. (2 marks)

●●●●●Level 57 marksStart

f(x) = x^3 + ax^2 - 5x + b

. Given that

(x - 1)

is a factor of

f(x)

, and that the remainder when

f(x)

is divided by

(x + 2)

-15

, find the values of

a

and

b

. (6 marks)

●●●●●Level 56 marksStart

(a) Sketch the graph of

y = x^2(3x + a)

where

a

is a positive constant.

(3 marks)
(b) The polynomial

\text{p}(x)

is given by

\text{p}(x) = x^2(3x + a) + 8

(b) (i) It is given that

(x + 2)

is a factor of

\text{p}(x)

Use the factor theorem to show that

a = 4

(ii) State the transformation which maps the curve with equation

y = x^2(3x + 4)

onto the curve with equation

y = x^2(3x + 4) + 8

(iii) The polynomial

\text{p}(x)

can be written as

(x + 2)(3x^2 + bx + c)

Without finding the values of

b

and

c

, use your answers to parts (a) and (b)(ii) to explain why

b^2 < 12c

(6 marks)

●●●●●Level 59 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve

C

has equation

y^3 + x^2y = 10

(a) Find

\dfrac{dy}{dx}

in terms of

x

and

y

.

(5 marks)

The point

P(3, 1)

lies on

C

.
(b) Show that the normal to

C

P

has equation

y = 2x - 5

(2 marks)
(c) Prove that the normal to

C

P

does not meet

C

again.

You should use algebra for your proof and make your reasoning clear.

(5 marks)

●●●●●Level 512 marksStart

f(x) = x^{3} - 3x^{2} - 10x + 24

.
(a) Given that

f(2) = 0

, factorise

f(x)

completely. (4 marks)
(b) Hence solve the inequality

f(x) > 0

. (2 marks)

●●●●●Level 56 marksStart

In this question you must show detailed reasoning.
(a) (i) Use the formula for

\cos(A + B)

, and the double angle formulae, to show that

\cos 3\theta \equiv 4\cos^3\theta - 3\cos\theta

.

(ii) By writing

-\cos 2\theta

\cos(180^\circ - 2\theta)

, solve the equation

\cos 3\theta + \cos 2\theta = 0

for

0^\circ \le \theta \le 180^\circ

.

(5 marks)
(b) (i) Show that

4x^3 + 2x^2 - 3x - 1 = (x + 1)(4x^2 - 2x - 1)

.

(ii) Hence find the exact roots of the equation

4x^3 + 2x^2 - 3x - 1 = 0

.

(3 marks)
(c) Use the results from parts (a)(ii) and (b)(ii) to show that

\cos 36^\circ = \dfrac{1 + \sqrt5}{4}

.

(2 marks)

●●●●●Level 510 marksStart

Figure 1

Figure 1 shows a sketch of part of the curve

C

with equation

y = \mathrm{f}(x)

, where

\mathrm{f}(x) = (x + 3)(2x - 1)(x - 4)

(a) Use the given information to state the values of

x

for which

\mathrm{f}(x) > 0

(2 marks)
(b) Expand

(x + 3)(2x - 1)(x - 4)

, writing your answer as a polynomial in simplest form.

(3 marks)

The straight line

l

is the tangent to

C

at the point where

C

cuts the

y

-axis.

Given that

l

cuts

C

at the point

P

, as shown in Figure 1,
(c) find, using algebra, the

x

coordinate of

P

(Solutions relying on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 510 marksStart

f(x) = x^3 - 2x^2 - 5x + 6

.
(a) Factorise

f(x)

completely.
(b) Hence solve the inequality

f(x) \ge 0

●●●●●Level 55 marksStart

Algebraic fractions

30 questions

LessonNot started

(a) Solve the inequality

\dfrac{x + 4}{x - 2} \leqslant 3

(5 marks)
(b) Hence, or otherwise, solve the inequality

\dfrac{|x| + 4}{|x| - 2} \leqslant 3

(3 marks)

●●●●●Level 58 marksStart

In this question you must show detailed reasoning.

Figure 1 shows a closed box in the shape of a cuboid with a square base of side

x

cm and height

h

cm. The total surface area of the box is

600

cm².Figure 1 (not accurately drawn)
(a) Show that the volume

V

cm³ of the box is given by

V = 150x - \dfrac{x^3}{2}

. (3 marks)
(b) Find the value of

x

for which

V

is a maximum, and find the maximum volume. You must show that your value gives a maximum. (5 marks)
(c) Find the corresponding value of

h

, and describe the shape of the box in this case. (2 marks)

●●●●●Level 510 marksStart

The functions

f

and

g

are defined by

f(x) = \frac{5x + 3}{x + 1} \qquad x \in \mathbb{R},\ x > -1

g(x) = 2x - 9 \qquad x \in \mathbb{R}

(a) Find

fg(6)

.

(2 marks)
(b) Show that

f(x) = A + \dfrac{B}{x + 1}

, where

A

and

B

are constants to be found.

(2 marks)
(c) Hence state the range of

f

.

(1 mark)
(d) Find

f^{-1}(x)

, stating the domain of

f^{-1}

.

(3 marks)

The function

h

is defined by

h(x) = x^2 - 8x + k \qquad x \in \mathbb{R}

where

k

is a constant.
(e) Find the range of values of

k

for which the equation

g(x) = h(x)

has no real solutions.

(3 marks)

●●●●●Level 511 marksStart

The function f is defined by

\text{f}(x) = \dfrac{3x^2 + 13x + 4}{x^2 + 3x - 4} - \dfrac{2}{x - 1} \qquad x \in \mathbb{R},\ x > 1

(a) Show that

\text{f}(x) = \dfrac{3x - 1}{x - 1}

(3 marks)
(b) Show, using calculus, that f is a decreasing function. You must make your reasoning clear.

(3 marks)

The function g is defined by

\text{g}(x) = 4\ln(x + 2) - 1 \qquad x \geqslant -1

(c) Find

\text{g}^{-1}

(3 marks)
(d) Find the exact value of

a

for which

\text{gf}(a) = 7

(Solutions relying on calculator technology are not acceptable.)

(4 marks)

●●●●●Level 513 marksStart

Figure 1Figure 1 shows a sketch of part of the curve

C

with equation

y = \text{f}(x)

where

\text{f}(x) = \dfrac{x^2 + 4x - 5}{x + 1} \qquad x > -1

(a) Find

\text{f}'(x)

, giving the answer in simplest form.

(3 marks)

The line

l

is the normal to

C

at the point

P(3, 4)

(b) Show that an equation for

l

2x + 3y = 18

(3 marks)
(c) Write

\text{f}(x)

in the form

Ax + B + \dfrac{D}{x + 1}

where

A

B

and

D

are constants.

(3 marks)

The region

R

, shown shaded in Figure 1, is bounded by

C

l

and the

x

-axis.
(d) Use algebraic integration to find the exact area of

R

, giving your answer in the form

p + q\ln 2

, where

p

and

q

are rational constants.

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 514 marksStart

Partial fractions

46 questions

LessonNot started

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) Express

\dfrac{10}{(M + 2)(8 - M)}

in partial fractions.

(2 marks)

A crystal is grown by placing a small seed crystal in a solution.

The mass,

M

grams, of the crystal,

t

hours after the seed crystal is placed in the solution, is modelled by the differential equation

\frac{dM}{dt} = \frac{1}{50}(M + 2)(8 - M) \qquad 0 \le M < 8,\ t \ge 0

Given that the seed crystal has mass

0.5

grams,
(b) solve the differential equation to show that

M = \frac{8e^{0.2t} - 6}{e^{0.2t} + 3}

(5 marks)
(c) Find, according to the model, the time taken for the mass of the crystal to reach

6

grams. Give your answer in hours and minutes, to the nearest minute.

(2 marks)
(d) Deduce the maximum mass of the crystal, according to the model.

(1 mark)

●●●●●Level 510 marksStart

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

.
(a) Express

f(x)

in partial fractions. (3 marks)
(b) Hence show that, for small

x

f(x) \approx 2 - 2x + 6x^2

. (6 marks)
(c) State the range of values of

x

for which the full expansion is valid, justifying your answer. (3 marks)

●●●●●Level 512 marksStart

A biologist is studying the population of fish in a newly stocked lake. The number of fish,

N

thousand, after

t

months is modelled by the differential equation

\dfrac{\mathrm{d}N}{\mathrm{d}t} = kN(8 - N)

where

k

is a positive constant. When the lake was stocked,

t = 0

and

N = 2

.
(a) Show that

N = \dfrac{8}{1 + 3e^{-8kt}}

. (6 marks)
(b) After

10

months there are

4000

fish in the lake. Find the exact value of

k

. (2 marks)
(c) Find the time taken, according to the model, for the population to reach

6000

fish. (2 marks)
(d) The biologist claims that, according to the model, the population will eventually reach

8000

fish.

Use the model to explain whether this claim is correct. (2 marks)

●●●●●Level 512 marksStart

\mathrm{f}(x) = \dfrac{9}{(x - 1)(x + 2)^2}

(a) Express

\mathrm{f}(x)

in partial fractions. (5 marks)
(b) Hence find

\displaystyle\int \mathrm{f}(x)\;\mathrm{d}x

. (4 marks)
(c) Hence show that

\int_2^5 \mathrm{f}(x)\;\mathrm{d}x = \ln\dfrac{16}{7} - \dfrac{9}{28}

(3 marks)

●●●●●Level 512 marksStart

(a) Express

\dfrac{3x^{2}+6}{(x+2)(x-1)^{2}}

in partial fractions. (5 marks)
(b) Hence show that

\displaystyle\int_{2}^{4}\dfrac{3x^{2}+6}{(x+2)(x-1)^{2}}\,dx = \ln\dfrac{27}{4} + 2

. (3 marks)

●●●●●Level 58 marksStart

In a chemical reaction two substances combine to form a product. At time

t

minutes after the reaction starts, the mass of product formed is

x

grams. The reaction is modelled by the differential equation

\dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{1}{4}(2 - x)(4 - x), \qquad 0 \le x < 2

and

x = 0

when

t = 0

.
(a) Express

\dfrac{1}{(2 - x)(4 - x)}

in partial fractions. (2 marks)
(b) Hence show that

x = \dfrac{4\left(\mathrm{e}^{t/2} - 1\right)}{2\mathrm{e}^{t/2} - 1}

. (5 marks)
(c) Find the value that

x

approaches as

t

becomes large. Explain, with reference to the differential equation, why the mass of product never reaches this value. (2 marks)

●●●●●Level 59 marksStart

(a) Express

\dfrac{3x + 4}{(x + 1)(x + 2)}

in partial fractions. (4 marks)
(b) Hence show that

\displaystyle\int_{0}^{1} \dfrac{3x + 4}{(x + 1)(x + 2)} \, dx = \ln \dfrac{9}{2}

. (3 marks)

●●●●●Level 57 marksStart

A lake is stocked with fish. The number of fish in the lake, in thousands,

t

years after stocking is

P

. A model for the population is

\dfrac{\mathrm{d}P}{\mathrm{d}t} = \dfrac{1}{20}P(4 - P)

and

P = 1

when

t = 0

.
(a) Show that, according to the model,

P = \dfrac{4}{1 + 3e^{-0.2t}}

. (8 marks)
(b) Find the exact value of

t

when the model predicts that the population reaches

3000

fish, giving your answer in the form

a\ln 3

where

a

is an integer. (2 marks)
(c) State what the model predicts about the fish population in the long term, and give one criticism of the model. (2 marks)

●●●●●Level 512 marksStart

In a chemical reaction, two substances combine to form a product. The mass,

m

grams, of product that has formed

t

minutes after the reaction starts is modelled by the differential equation

\dfrac{\mathrm{d}m}{\mathrm{d}t} = k(3 - m)(6 - m)

\qquad 0 \le m < 3

,

where

k

is a positive constant. When

t = 0

m = 0

.
(a) Solve the differential equation to show that

m = \dfrac{6\left(e^{3kt} - 1\right)}{2e^{3kt} - 1}

. (7 marks)
(b) Given that

1

gram of product has formed after

5

minutes, find the time taken for

2

grams of product to form. Give your answer to the nearest tenth of a minute. (3 marks)
(c) Explain why, according to this model, the mass of product formed can never reach

3

grams. (1 mark)

●●●●●Level 511 marksStart

A lake is stocked with fish. The population of fish is modelled by the differential equation

\dfrac{\mathrm{d}P}{\mathrm{d}t} = \dfrac{1}{20}P(10 - P), \qquad 0 < P < 10

where

P

is the number of fish, in thousands, and

t

is the time in years after the lake was stocked. When

t = 0

P = 2

.
(a) Solve the differential equation to show that

P = \dfrac{10}{1 + 4e^{-t/2}}

(8 marks)
(b) Find, in exact form, the value of

t

at which the population first reaches

8000

fish. (3 marks)
(c) State, with a reason, the long-term population of fish predicted by this model. (1 mark)

●●●●●Level 512 marksStart

A lake is stocked with fish. The number of fish in the lake,

P

thousand,

t

years after stocking is modelled by the differential equation

\dfrac{\mathrm{d}P}{\mathrm{d}t} = \dfrac{P(6 - P)}{12}

\qquad 0 < P < 6

.

Initially there are

1000

fish in the lake.
(a) Express

\dfrac{1}{P(6 - P)}

in partial fractions. (2 marks)
(b) Hence solve the differential equation, showing that

P = \dfrac{6}{1 + 5e^{-\frac{t}{2}}}

. (5 marks)
(c) Find, to 3 significant figures, the time taken for the number of fish to reach

3000

. (2 marks)
(d) State, with a reason, what the model predicts about the number of fish in the lake in the long term. (1 mark)

●●●●●Level 510 marksStart

(a) Express

\dfrac{24\,000}{V(600 - V)}

in partial fractions.

(2 marks)

A food scientist is testing a new bread dough.

The volume of a piece of the dough,

V\ \text{cm}^3

t

minutes after the piece is left to rise, is modelled by the differential equation

\frac{dV}{dt} = \frac{V(600 - V)}{24\,000}\qquad 0 < V < 600,\ t \geqslant 0

Given that the volume of the piece of dough was

400\ \text{cm}^3

when it was left to rise,
(b) find, according to the model, the time taken, in minutes, for the volume of the piece of dough to reach

550\ \text{cm}^3

. Give your answer to the nearest minute.

(5 marks)
(c) Show that

V = \frac{A}{B + e^{-kt}}

where

A

B

and

k

are constants to be found.

(3 marks)

The food scientist states that, according to the model, the volume of the piece of dough approaches a limiting value

L\ \text{cm}^3

.
(d) Find the value of

L

, giving a reason for your answer.

(2 marks)

●●●●●Level 512 marksStart

(a) Express

\dfrac{10}{(t + 1)(t + 6)}

in partial fractions.

(3 marks)

A marine biologist is monitoring a sponge growing on a rock ledge.

The volume of the sponge,

V\ \text{cm}^3

t

weeks after monitoring began, is modelled by the differential equation

\frac{dV}{dt} = \frac{10V}{(t + 1)(t + 6)}\qquad V > 0,\ t \geqslant 0

Given that the volume of the sponge was

25\ \text{cm}^3

when monitoring began,
(b) solve the differential equation to show that

V = \frac{900(t + 1)^2}{(t + 6)^2}

(5 marks)
(c) Find, according to the model, the time taken for the volume of the sponge to reach

400\ \text{cm}^3

.

(2 marks)
(d) State a limitation of the model.

(1 mark)

●●●●●Level 511 marksStart

Find the general solution of the differential equation

(x^3 + x^2 - 6x)\frac{dy}{dx} = (18x - 6)y.

Give your answer in the form

y = \text{f}(x)

.

(9 marks)

●●●●●Level 59 marksStart

(a) Given that

\frac{10}{25 - 4x^2} \equiv \frac{A}{5 - 2x} + \frac{B}{5 + 2x}

find the values of

A

and

B

(3 marks)
(b) A farmer has an empty water trough in the shape of a cuboid. The trough has length

2.5

metres, width

1

metre and depth

0.6

metres.

The trough has a small crack in its base. Water is poured into the trough at a rate of

0.05

cubic metres per minute.

At time

t

minutes after the trough starts to be filled, the depth of water is

d

metres and water leaks out at a rate of

0.05d^2

cubic metres per minute.

At time

t

minutes after the trough starts to be filled, the volume of water in the trough is

V

cubic metres.
(b) (i) Show that

\frac{dV}{dt} = \frac{25 - 4V^2}{500}

(ii) Hence, find

t

in terms of

V

(iii) Determine how long it takes to fill the trough with water.

Give your answer to the nearest minute.

(11 marks)

●●●●●Level 514 marksStart

A biologist is monitoring a colony of seals. At time

t

years after monitoring began, the number of seals in the colony is

N

hundred.

The biologist models the growth of the colony by the differential equation

\dfrac{\mathrm{d}N}{\mathrm{d}t} = \dfrac{N(10 - N)}{50}

When monitoring began there were

200

seals in the colony.
(a) Solve the differential equation to show that, according to the model,

N = \dfrac{10}{1 + 4\mathrm{e}^{-0.2t}}

(7 marks)
(b) Find, to 1 decimal place, the number of years after monitoring began at which the model predicts the colony will reach

800

seals. (2 marks)
(c) Explain why, according to this model, the colony can never reach

1000

seals. (1 mark)

●●●●●Level 510 marksStart

(a) Express

\frac{200}{S(40 - S)}

in partial fractions.

(2 marks)

A gardener is studying the growth of a weed on the surface of a pond.

The area of the surface of the pond covered by the weed,

S

^2

t

days after the weed was first seen, is modelled by the differential equation

\frac{dS}{dt} = \frac{1}{200}S(40 - S) \qquad 0 < S < 40,\ t \geqslant 0

Given that the weed covered

8

^2

of the surface of the pond when it was first seen,
(b) find, according to the model, the time taken for the weed to cover

20

^2

of the surface of the pond. Give your answer in days, to one decimal place.

(5 marks)
(c) Show that

S = \frac{A}{B + e^{-kt}}

where

A

B

and

k

are constants to be found.

(3 marks)

As

t

increases, the area of the surface of the pond covered by the weed approaches a limiting value

L

^2

.
(d) Find the value of

L

, giving a reason for your answer.

(2 marks)

●●●●●Level 512 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Given that

\dfrac{3x^2 - 6x + 9}{(x + 1)(x - 2)^2} \equiv \dfrac{A}{x + 1} + \dfrac{B}{x - 2} + \dfrac{C}{(x - 2)^2}

(a) find the values of the constants

A

B

and

C

.

(4 marks)
(b) Hence find the exact value of

\int_3^5 \dfrac{3x^2 - 6x + 9}{(x + 1)(x - 2)^2}\,dx

giving your answer in the form

p\ln q + r

where

p

q

and

r

are rational numbers.

(6 marks)

●●●●●Level 510 marksStart

(a) Express

\dfrac{1}{N(6 - N)}

in partial fractions.

(2 marks)

The number of midges near a lake,

N

thousand, is modelled by the differential equation

\dfrac{dN}{dt} = \dfrac{N(6 - N)\sin\left(\dfrac{t}{8}\right)}{12}

where

t

days is the time after the start of the observations.

Given that, at the start of the observations, there are

2000

midges near the lake,
(b) solve the differential equation to show that, according to the model,

N = \dfrac{6}{1 + 2\text{e}^{k\left(\cos\left(\frac{t}{8}\right) - 1\right)}}

where

k

is a constant to be found.

(6 marks)
(c) Hence find, according to the model, the time taken for the number of midges to reach its maximum for the 3rd time.

Give your answer to the nearest day.

(Solutions relying entirely on calculator technology are not acceptable.)

(2 marks)

●●●●●Level 510 marksStart

The gradient of the curve

y = \text{f}(x)

, for

x > -2

, is given by the differential equation

2(x + 2)\frac{dy}{dx} = (y - 1)(y + 3)

and the curve passes through the point

(0, -1)

. By solving this differential equation show that

\text{f}(x) = \frac{ax^2 + bx - 8}{x^2 + 4x + 8},

where

a

and

b

are integers to be determined.

(9 marks)

●●●●●Level 59 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1Figure 1 shows a sketch of part of the curve defined by the parametric equations

x = t^2 - 1 \qquad y = \dfrac{5}{(t + 1)(t + 4)} \qquad t \geqslant 0

The region

R

, shown shaded in Figure 1, is bounded by the curve, the

y

-axis, the

x

-axis and the line with equation

x = 8

The curve meets the

y

-axis where

t = a

and meets the line with equation

x = 8

where

t = b

.
(a) Find the value of

a

and the value of

b

.

(2 marks)
(b) Show that the area of region

R

is given by

M\int_a^b \dfrac{t}{(t + 1)(t + 4)}\,dt

where

M

is a constant to be found.

(5 marks)
(c) (i) Write

\dfrac{t}{(t + 1)(t + 4)}

in partial fractions.

(ii) Use algebraic integration to find the exact area of

R

, giving your answer in the form

p\ln\dfrac{7}{5} + q\ln 2

where

p

and

q

are rational constants.

(6 marks)

●●●●●Level 513 marksStart

The modulus function

40 questions

LessonNot started

Solve

|3x - 2| = |x + 4|

●●●●●Level 53 marksStart

(a) Solve the inequality

\dfrac{x + 4}{x - 2} \leqslant 3

(5 marks)
(b) Hence, or otherwise, solve the inequality

\dfrac{|x| + 4}{|x| - 2} \leqslant 3

(3 marks)

●●●●●Level 58 marksStart

Solve

|x - 1| = |2x + 3|

. (4 marks)

●●●●●Level 54 marksStart

Solve the inequality

|3x - 1| < x + 3

. (5 marks)

●●●●●Level 55 marksStart

Solve the equation

\left|x^2 - 4\right| = 3x

showing all stages of your working. (8 marks)

●●●●●Level 58 marksStart

Solve

|x + 1| = 2x - 1

●●●●●Level 53 marksStart

The functions

\mathrm{f}

and

\mathrm{g}

are defined by

\mathrm{f}(x) = 2\left|x - 3\right| - 4, \qquad x \in \mathbb{R}

\mathrm{g}(x) = \ln(x + 2), \qquad x \in \mathbb{R}, \; x > -2

(a) Sketch the graph of

y = \mathrm{f}(x)

, showing the coordinates of the vertex and of any points where the graph meets the coordinate axes, and state the range of

\mathrm{f}

. (3 marks)
(b) Solve the equation

\mathrm{fg}(x) = 2

, giving your answers in exact form. (4 marks)

The function

\mathrm{h}

is defined by

\mathrm{h}(x) = 2\left|x - 3\right| - 4, \qquad x \geqslant k

where

k

is a constant, and

\mathrm{h}^{-1}

exists.
(c) State the least possible value of

k

, and find

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

, stating its domain. (3 marks)

●●●●●Level 510 marksStart

The number of passengers carried each day by two bus routes is being monitored.

The number of passengers,

N_R

, in hundreds, carried each day by route R is modelled by the equation

N_R = |t - 7| + 6 \qquad t \geqslant 0

where

t

is the time in years since the two routes opened.

The number of passengers,

N_S

, in hundreds, carried each day by route S is modelled by the equation

N_S = 18 - |2t - 17| \qquad t \geqslant 0

where

t

is the time in years since the two routes opened.

Over a 16-year period the graph of

N_R

is a V-shape with a single minimum point, and the graph of

N_S

t = T

a retail park opened close to route R and the number of passengers carried each day by route R began to increase.
(b) Suggest a value for

T

, giving a reason for your answer.

(2 marks)
(c) Find the range of values of

t

for which

N_R > N_S

giving your answer in set notation.

(5 marks)
(d) State a limitation of the model used for route S.

(1 mark)

●●●●●Level 510 marksStart

Figure 1Figure 1 shows a sketch of the graph of

C_1

with equation

y = 8 - |2x - 6|

(a) Write down the coordinates of

(i) the vertex of

C_1

(ii) the intersection of

C_1

with the

y

-axis.

(2 marks)
(b) Find the

x

coordinates of the intersections of

C_1

with the

x

-axis.

(2 marks)

Diagram 1 is a copy of Figure 1.
(c) On Diagram 1, sketch the curve

C_2

with equation

y = \dfrac{1}{2}x^2 - 2

Identify clearly the coordinates of any points of intersection of

C_2

with the coordinate axes.

(3 marks)
(d) Using algebra, find the coordinates of the points of intersection of

C_1

and

C_2

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)

●●●●●Level 512 marksStart

Solve

|2x - 3| = x + 1

●●●●●Level 53 marksStart

The diagram shows part of the graph of

y = f(x)

, where

f(x) = 2|x - 1| + 3

x \ge 0

.(a) State the range of

f(x)

.
(b) Solve the equation

f(x) = x + 4

.
(c) Given that the equation

f(x) = k

, where

k

is a constant, has two distinct roots, state the set of possible values of

k

●●●●●Level 56 marksStart

The diagram shows the graph of

y = |3x - 6|

.(a) State the coordinates of the points where the graph meets the axes.

(2 marks)
(b) Given that the graphs of

y = |3x - 6|

and

y = ax + 3

have two distinct points of intersection, determine

(i) the set of possible values of

a

,

(ii) the

x

-coordinates of the points of intersection of these graphs, giving your answers in terms of

a

.

(7 marks)

●●●●●Level 59 marksStart

Figure 1Figure 1 shows a sketch of part of the graph with equation

y = \text{f}(x)

, where

\text{f}(x) = 3k - |kx - 6| \qquad x \in \mathbb{R}

and

k

is a constant with

k > 2

The graph

cuts the $y$ -axis at the point $A$
has a vertex at the point $P$

(a) Find, in simplest form in terms of

k

,

(i) the

y

coordinate of

A

(ii) the coordinates of

P

(3 marks)
(b) Find, in terms of

k

, the range of values of

x

which satisfy

3k - |kx - 6| \geqslant 2k

(3 marks)

Given that the line with equation

y = 2x + 5

intersects the graph of

y = \text{f}(x)

at 2 distinct points,
(c) find the range of values of

k

.

(4 marks)

●●●●●Level 510 marksStart

Functions, domain and range

15 questions

LessonNot started

The functions

f

and

g

are defined by

f(x) = \frac{5x + 3}{x + 1} \qquad x \in \mathbb{R},\ x > -1

g(x) = 2x - 9 \qquad x \in \mathbb{R}

(a) Find

fg(6)

.

(2 marks)
(b) Show that

f(x) = A + \dfrac{B}{x + 1}

, where

A

and

B

are constants to be found.

(2 marks)
(c) Hence state the range of

f

.

(1 mark)
(d) Find

f^{-1}(x)

, stating the domain of

f^{-1}

.

(3 marks)

The function

h

is defined by

h(x) = x^2 - 8x + k \qquad x \in \mathbb{R}

where

k

is a constant.
(e) Find the range of values of

k

for which the equation

g(x) = h(x)

has no real solutions.

(3 marks)

●●●●●Level 511 marksStart

Composite and inverse functions

45 questions

LessonNot started

The function

f

is defined by

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

\; x \neq 3

.

Show that

ff(x) = x

, and state what this tells you about

f^{-1}

. (4 marks)

●●●●●Level 54 marksStart

The function

f

is defined by

f(x) = (x - 1)^{2} + 4

for

x \geqslant 1

.
(a) Explain why

f^{-1}

exists for this domain. (1 mark)
(b) Find

f^{-1}(x)

, stating its domain. (4 marks)

●●●●●Level 55 marksStart

The functions f and g are defined for all real values of

x

f(x) = 3x^2 - 6x \quad

and

\quad g(x) = 6x - 9

.
(a) Find the range of f.

(3 marks)
(b) Give a reason why f has no inverse.

(1 mark)
(c) Given that

fg(1) = g^{-1}(a)

, where

a

is a constant, determine the value of

a

.

(4 marks)
(d) Determine the set of values of

x

for which

f(x) > g(x)

. Give your answer in set notation.

(3 marks)

●●●●●Level 511 marksStart

The functions

\mathrm{f}

and

\mathrm{g}

are defined by

\mathrm{f}(x) = 2\left|x - 3\right| - 4, \qquad x \in \mathbb{R}

\mathrm{g}(x) = \ln(x + 2), \qquad x \in \mathbb{R}, \; x > -2

(a) Sketch the graph of

y = \mathrm{f}(x)

, showing the coordinates of the vertex and of any points where the graph meets the coordinate axes, and state the range of

\mathrm{f}

. (3 marks)
(b) Solve the equation

\mathrm{fg}(x) = 2

, giving your answers in exact form. (4 marks)

The function

\mathrm{h}

is defined by

\mathrm{h}(x) = 2\left|x - 3\right| - 4, \qquad x \geqslant k

where

k

is a constant, and

\mathrm{h}^{-1}

exists.
(c) State the least possible value of

k

, and find

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

, stating its domain. (3 marks)

●●●●●Level 510 marksStart

The function

\text{f}

is defined by

\text{f}(x) = \frac{x^2 + 7}{2x + 2}

where

\text{f}

has its maximum possible domain.

The curve

y = \text{f}(x)

intersects the line

y = x

at the points

P

and

Q

as shown below.(a) State the value of

x

which is not in the domain of

\text{f}

(1 mark)
(b) Explain how you know that the function

\text{f}

is many-to-one.

(2 marks)
(c) (i) Show that the

x

-coordinates of

P

and

Q

satisfy the equation

x^2 + 2x - 7 = 0

(ii) Hence, find the exact

x

-coordinate of

P

and the exact

x

-coordinate of

Q

(3 marks)
(d) Show that

P

and

Q

are stationary points of the curve.

Fully justify your answer.

(5 marks)
(e) Using set notation, state the range of

\text{f}

(2 marks)

●●●●●Level 513 marksStart

The functions

f

and

g

are defined by

f(x) = \frac{5x + 3}{x + 1} \qquad x \in \mathbb{R},\ x > -1

g(x) = 2x - 9 \qquad x \in \mathbb{R}

(a) Find

fg(6)

.

(2 marks)
(b) Show that

f(x) = A + \dfrac{B}{x + 1}

, where

A

and

B

are constants to be found.

(2 marks)
(c) Hence state the range of

f

.

(1 mark)
(d) Find

f^{-1}(x)

, stating the domain of

f^{-1}

.

(3 marks)

The function

h

is defined by

h(x) = x^2 - 8x + k \qquad x \in \mathbb{R}

where

k

is a constant.
(e) Find the range of values of

k

for which the equation

g(x) = h(x)

has no real solutions.

(3 marks)

●●●●●Level 511 marksStart

The function

f

is defined by

f(x) = x^2 + 2

for

x \ge 0

, and

g(x) = 3x - 1

.
(a) Find

fg(x)

.
(b) Hence solve

fg(x) = 11

●●●●●Level 55 marksStart

The function f is defined by

\text{f}(x) = \dfrac{3x^2 + 13x + 4}{x^2 + 3x - 4} - \dfrac{2}{x - 1} \qquad x \in \mathbb{R},\ x > 1

(a) Show that

\text{f}(x) = \dfrac{3x - 1}{x - 1}

(3 marks)
(b) Show, using calculus, that f is a decreasing function. You must make your reasoning clear.

(3 marks)

The function g is defined by

\text{g}(x) = 4\ln(x + 2) - 1 \qquad x \geqslant -1

(c) Find

\text{g}^{-1}

(3 marks)
(d) Find the exact value of

a

for which

\text{gf}(a) = 7

(Solutions relying on calculator technology are not acceptable.)

(4 marks)

●●●●●Level 513 marksStart

The functions f and g are defined for

0 < x < \tfrac12\pi

f(x) = 2\cot^2 x \quad

and

\quad g(x) = \operatorname{cosec} x

.
(a) State the range of

(i) f,

(ii) g.

(2 marks)
(b) (i) Show that

gf(1.1) = 2.02

, correct to 3 significant figures.

(ii) Explain why

gf^{-1}(-3)

is not defined.

(3 marks)
(c) In this question you must show detailed reasoning.

Solve the equation

f(x) = 3g(x) + 3

.

(5 marks)

●●●●●Level 510 marksStart