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DifficultyAll Level 2 Level 3 Level 4 Level 5

Locating roots by change of sign

39 questions

LessonNot started

In this question you must show detailed reasoning.

The diagram shows the curve with equation

y = \dfrac{2x - 5}{(x + 1)^2}

for

x > -1

. The tangent to the curve at the point

P

has gradient

1

.(a) Show that the

x

-coordinate of

P

satisfies the equation

x^3 + 3x^2 + 5x - 11 = 0

.

(5 marks)
(b) Show by calculation that the

x

-coordinate of

P

lies between

1

and

1.2

.

(2 marks)
(c) Show that the iteration

x_{n+1} = \frac{11 - x_n^3 - 3x_n^2}{5}

cannot converge to the

x

-coordinate of

P

whatever starting value is used.

(2 marks)
(d) Use the Newton-Raphson method, with initial value

1

, to determine the coordinates of

P

correct to 5 decimal places.

(5 marks)

●●●●●Level 514 marksStart

A curve has equation

y = k\cos x + x - x^2

, where

x

is in radians and

k

is a positive constant.
(a) Given that the curve has a point of inflection where

x = \tfrac23\pi

, show that

k = 4

.

(5 marks)

It is also given that the curve crosses the positive

x

-axis at exactly one point.
(b) Show by calculation that the

x

-coordinate of this point lies between

1.2

and

1.6

.

(2 marks)
(c) Starting with

x_0 = 1.4

, use the Newton-Raphson method to find this

x

-coordinate correct to 5 decimal places. Show the result of each iteration to 6 decimal places.

(3 marks)
(d) By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.

(1 mark)

●●●●●Level 511 marksStart

The diagram shows the cross-section of a horizontal pipe, which is a circle with centre

O

and radius

50

cm. The pipe contains some water. In the cross-section, the surface of the water is the chord

AB

and the angle

AOB

\theta

radians. The water occupies the shaded segment bounded by the chord

AB

and the arc

AB

, which has area

900

^2

.(a) Show that

\theta = \sin\theta + 0.72

.

(3 marks)
(b) Let

\text{F}(\theta) = \sin\theta + 0.72

.

By considering the value of

\text{F}'(\theta)

where

\theta = 1.7

, explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that

1.7

is sufficiently close to the root.

(2 marks)
(c) Use the iterative formula

\theta_{n+1} = \sin\theta_n + 0.72

with a starting value of

1.7

to find the value of

\theta

correct to 4 significant figures.

You should show the result of each iteration.

(3 marks)
(d) Use a change of sign method to show that the value of

\theta

found in part (c) is correct to 4 significant figures.

(3 marks)

●●●●●Level 511 marksStart

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

Figure 4 shows part of the curve with equation

y = \mathrm{f}(x)

.Figure 4
(a) Show that the equation

\mathrm{f}(x) = 0

has a root

\alpha

in the interval

[1.8, 1.9]

. (2 marks)
(b) Taking

x_0 = 1.9

, apply the Newton–Raphson procedure twice to

\mathrm{f}(x)

to obtain an approximation for

\alpha

, giving your answer to

5

decimal places. (4 marks)
(c) Explain, with reference to the gradient of the curve, why

x_0 = 1.15

would be a poor choice of first approximation for the Newton–Raphson procedure. (2 marks)
(d) An alternative approach uses the iterative formula

x_{n+1} = \sqrt[3]{4x_n - 1}

Using this formula with

x_0 = 1.9

, find the value of

x_2

, giving your answer to

4

decimal places. (2 marks)

●●●●●Level 510 marksStart

(a) The equation

x^4 = \mathrm{e}^{9 - 2x}

has a single solution,

x = \alpha

By considering a suitable change of sign, show that

\alpha

lies between

2

and

3

(2 marks)
(b) Show that the equation

x^4 = \mathrm{e}^{9 - 2x}

can be rearranged to give

x = 4.5 - 2\ln x

(3 marks)
(c) (i) Use the iterative formula

x_{n+1} = 4.5 - 2\ln x_n

with

x_1 = 3

, to find

x_2

x_3

and

x_4

Give your answers to three decimal places.

(ii) Figure 1 below shows a sketch of parts of the graphs of

y = 4.5 - 2\ln x

and

y = x

On Figure 1, draw a staircase or cobweb diagram to show how convergence takes place.

Label, on the

x

-axis, the positions of

x_2

x_3

and

x_4

(iii) Explain why the iterative formula

x_{n+1} = 4.5 - 2\ln x_n

fails to converge to

\alpha

when the starting value is

x_1 = \mathrm{e}^{2.25}

(5 marks)

●●●●●Level 510 marksStart

The diagram shows a sector of a circle

OAB

.

The point

C

lies on

OB

such that

AC

is perpendicular to

OB

.

Angle

AOB

\theta

radians.(a) Given the area of the sector

OAB

\sqrt{2}

times the area of the triangle

OAC

, show that

\sin 2\theta = \sqrt{2}\,\theta

(4 marks)
(b) Use a suitable change of sign to show that a solution to the equation

\sin 2\theta = \sqrt{2}\,\theta

lies in the interval given by

\theta \in \left[\dfrac{\pi}{5}, \dfrac{\pi}{4}\right]

(2 marks)
(c) The Newton-Raphson method is used to find an approximate solution to the equation

\sin 2\theta = \sqrt{2}\,\theta

(c) (i) Using

\theta_1 = \dfrac{\pi}{4}

as a first approximation for

\theta

, apply the Newton-Raphson method twice to find the value of

\theta_3

Give your answer to three decimal places.

(ii) Explain how a more accurate approximation for

\theta

can be found using the Newton-Raphson method.

(iii) Explain why using

\theta_1 = \dfrac{\pi}{8}

as a first approximation in the Newton-Raphson method does not lead to a solution for

\theta

.

(6 marks)

●●●●●Level 512 marksStart

Iterative methods (x = g(x))

33 questions

LessonNot started

In this question you must show detailed reasoning.

The diagram shows the curve with equation

y = \dfrac{2x - 5}{(x + 1)^2}

for

x > -1

. The tangent to the curve at the point

P

has gradient

1

.(a) Show that the

x

-coordinate of

P

satisfies the equation

x^3 + 3x^2 + 5x - 11 = 0

.

(5 marks)
(b) Show by calculation that the

x

-coordinate of

P

lies between

1

and

1.2

.

(2 marks)
(c) Show that the iteration

x_{n+1} = \frac{11 - x_n^3 - 3x_n^2}{5}

cannot converge to the

x

-coordinate of

P

whatever starting value is used.

(2 marks)
(d) Use the Newton-Raphson method, with initial value

1

, to determine the coordinates of

P

correct to 5 decimal places.

(5 marks)

●●●●●Level 514 marksStart

The diagram shows the cross-section of a horizontal pipe, which is a circle with centre

O

and radius

50

cm. The pipe contains some water. In the cross-section, the surface of the water is the chord

AB

and the angle

AOB

\theta

radians. The water occupies the shaded segment bounded by the chord

AB

and the arc

AB

, which has area

900

^2

.(a) Show that

\theta = \sin\theta + 0.72

.

(3 marks)
(b) Let

\text{F}(\theta) = \sin\theta + 0.72

.

By considering the value of

\text{F}'(\theta)

where

\theta = 1.7

, explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that

1.7

is sufficiently close to the root.

(2 marks)
(c) Use the iterative formula

\theta_{n+1} = \sin\theta_n + 0.72

with a starting value of

1.7

to find the value of

\theta

correct to 4 significant figures.

You should show the result of each iteration.

(3 marks)
(d) Use a change of sign method to show that the value of

\theta

found in part (c) is correct to 4 significant figures.

(3 marks)

●●●●●Level 511 marksStart

The diagram shows a sector

OAB

of a circle with centre

O

and radius

r

. The angle

AOB

\theta

radians, where

0 < \theta < \tfrac12\pi

. The tangent to the circle at

A

meets the line

OB

extended at

T

. The region

R

is bounded by the arc

AB

and the lines

BT

and

AT

. The areas of region

R

and sector

OAB

are in the ratio

6 : 5

.(a) Show that

\theta = \tan^{-1}(2.2\theta)

.

(4 marks)

The equation

\theta = \tan^{-1}(2.2\theta)

has only one root for

0 < \theta < \tfrac12\pi

.
(b) This root can be found by using the iterative formula

\theta_{n+1} = \tan^{-1}(2.2\theta_n)

with a starting value of

\theta_1 = 1

Write down the values of $\theta_2$ , $\theta_3$ and $\theta_4$ .

Hence find the value of this root correct to 3 significant figures.

(3 marks)

The diagram shows the graph of

y = \tan^{-1}(2.2\theta)

and the line

y = \theta

, for

0 \le \theta \le \tfrac12\pi

.(c) • Use this diagram to show how the iterative process used in (b) converges to this root.

State the type of convergence.

(3 marks)
(d) Draw a suitable diagram to show why using an iterative process with the formula

\theta_{n+1} = \tfrac{5}{11}\tan\theta_n

does not converge to the root found in (b).

(2 marks)

●●●●●Level 512 marksStart

Two new podcasts,

A

and

B

, are launched on the same day. Each podcast is downloaded

500

times in Week 1. The number of downloads of podcast

A

will increase by

60

for every subsequent week. The number of downloads of podcast

B

will increase by

7\%

for every subsequent week.
(a) Verify that in Week 10 podcast

A

is downloaded

121

more times than podcast

B

, correct to the nearest whole number.

(3 marks)

Let Week

X

be the first week in which podcast

B

is downloaded more times than podcast

A

.
(b) Show that

X

satisfies the inequality

X > 1 + \log_{1.07}(0.12X + 0.88)

.

(3 marks)
(c) Use the iterative formula

x_{n+1} = 1 + \log_{1.07}(0.12x_n + 0.88)

with

x_1 = 20

to find the value of

X

.

You should show the result of each iteration.

(3 marks)
(d) (i) Give a reason why the model for podcast

A

may not be realistic.

(ii) Give a different reason why the model for podcast

B

may not be realistic.

(2 marks)

●●●●●Level 511 marksStart

Figure 1

A group of red squirrels is released into a forest and the number of red squirrels in the forest is then monitored.

The number of red squirrels,

N

, in the forest is modelled by the equation

N = 180 + 300\mathrm{e}^{-0.1t} - 420\mathrm{e}^{-0.3t}

where

t

months is the time after the squirrels are released.

Figure 1 is a sketch of

N

against

t

.

Use the equation of the model to answer parts (a) to (e).
(a) State the number of red squirrels released into the forest.

(1 mark)

In the long term, the number of red squirrels in the forest approaches

L

.
(b) State the value of

L

.

(1 mark)

The number of red squirrels in the forest reaches its maximum value after

T

months.
(c) Find the value of

T

, giving your answer to 3 decimal places.

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

(5 marks)

The number of red squirrels in the forest is

250

for the second time after

M

months.
(d) Show that

M

is a solution of the equation

t = 10\ln\left(\frac{30}{7 + 42\mathrm{e}^{-0.3t}}\right)

(2 marks)

Using the iteration formula

t_{n+1} = 10\ln\left(\frac{30}{7 + 42\mathrm{e}^{-0.3t_n}}\right) \qquad \text{with } t_1 = 13

(e) (i) find, to 4 decimal places, the value of

t_2

(ii) find, to 4 decimal places, the value of

M

(3 marks)

●●●●●Level 512 marksStart

Figure 1A frog jumps from a point

O

on horizontal ground. Figure 1 shows the path of the frog from

O

until it lands, a horizontal distance of

d

cm from

O

.

When the frog has travelled a horizontal distance of

x

cm, its height above the ground is

h

cm.

The frog is modelled as a particle travelling in a vertical plane above horizontal ground.

The path of the frog is modelled by the equation

h = x - 0.25x\text{e}^{0.04x} \qquad 0 \leqslant x \leqslant d

Use the model to answer parts (a),
(b) and (c).
(a) Find the value of

d

, giving your answer to 2 decimal places.

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

(3 marks)
(b) Show that the maximum value of

h

occurs when

x = 25\ln\left(\dfrac{100}{25 + x}\right)

(4 marks)

Using the iteration formula

x_{n+1} = 25\ln\left(\dfrac{100}{25 + x_n}\right) \qquad x_1 = 15

(c) (i) find the value of

x_2

to 2 decimal places,

(ii) find, by repeated iteration, the horizontal distance travelled by the frog before it reaches its maximum height. Give your answer to 2 decimal places.

(3 marks)

●●●●●Level 510 marksStart

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

Figure 4 shows part of the curve with equation

y = \mathrm{f}(x)

.Figure 4
(a) Show that the equation

\mathrm{f}(x) = 0

has a root

\alpha

in the interval

[1.8, 1.9]

. (2 marks)
(b) Taking

x_0 = 1.9

, apply the Newton–Raphson procedure twice to

\mathrm{f}(x)

to obtain an approximation for

\alpha

, giving your answer to

5

decimal places. (4 marks)
(c) Explain, with reference to the gradient of the curve, why

x_0 = 1.15

would be a poor choice of first approximation for the Newton–Raphson procedure. (2 marks)
(d) An alternative approach uses the iterative formula

x_{n+1} = \sqrt[3]{4x_n - 1}

Using this formula with

x_0 = 1.9

, find the value of

x_2

, giving your answer to

4

decimal places. (2 marks)

●●●●●Level 510 marksStart

(a) The equation

x^4 = \mathrm{e}^{9 - 2x}

has a single solution,

x = \alpha

By considering a suitable change of sign, show that

\alpha

lies between

2

and

3

(2 marks)
(b) Show that the equation

x^4 = \mathrm{e}^{9 - 2x}

can be rearranged to give

x = 4.5 - 2\ln x

(3 marks)
(c) (i) Use the iterative formula

x_{n+1} = 4.5 - 2\ln x_n

with

x_1 = 3

, to find

x_2

x_3

and

x_4

Give your answers to three decimal places.

(ii) Figure 1 below shows a sketch of parts of the graphs of

y = 4.5 - 2\ln x

and

y = x

On Figure 1, draw a staircase or cobweb diagram to show how convergence takes place.

Label, on the

x

-axis, the positions of

x_2

x_3

and

x_4

(iii) Explain why the iterative formula

x_{n+1} = 4.5 - 2\ln x_n

fails to converge to

\alpha

when the starting value is

x_1 = \mathrm{e}^{2.25}

(5 marks)

●●●●●Level 510 marksStart

The Newton–Raphson method

29 questions

LessonNot started

In this question you must show detailed reasoning.

The diagram shows the curve with equation

y = \dfrac{2x - 5}{(x + 1)^2}

for

x > -1

. The tangent to the curve at the point

P

has gradient

1

.(a) Show that the

x

-coordinate of

P

satisfies the equation

x^3 + 3x^2 + 5x - 11 = 0

.

(5 marks)
(b) Show by calculation that the

x

-coordinate of

P

lies between

1

and

1.2

.

(2 marks)
(c) Show that the iteration

x_{n+1} = \frac{11 - x_n^3 - 3x_n^2}{5}

cannot converge to the

x

-coordinate of

P

whatever starting value is used.

(2 marks)
(d) Use the Newton-Raphson method, with initial value

1

, to determine the coordinates of

P

correct to 5 decimal places.

(5 marks)

●●●●●Level 514 marksStart

A curve has equation

y = k\cos x + x - x^2

, where

x

is in radians and

k

is a positive constant.
(a) Given that the curve has a point of inflection where

x = \tfrac23\pi

, show that

k = 4

.

(5 marks)

It is also given that the curve crosses the positive

x

-axis at exactly one point.
(b) Show by calculation that the

x

-coordinate of this point lies between

1.2

and

1.6

.

(2 marks)
(c) Starting with

x_0 = 1.4

, use the Newton-Raphson method to find this

x

●●●●●Level 511 marksStart

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

Figure 4 shows part of the curve with equation

y = \mathrm{f}(x)

.Figure 4
(a) Show that the equation

\mathrm{f}(x) = 0

has a root

\alpha

in the interval

[1.8, 1.9]

. (2 marks)
(b) Taking

x_0 = 1.9

, apply the Newton–Raphson procedure twice to

\mathrm{f}(x)

to obtain an approximation for

\alpha

, giving your answer to

5

decimal places. (4 marks)
(c) Explain, with reference to the gradient of the curve, why

x_0 = 1.15

would be a poor choice of first approximation for the Newton–Raphson procedure. (2 marks)
(d) An alternative approach uses the iterative formula

x_{n+1} = \sqrt[3]{4x_n - 1}

Using this formula with

x_0 = 1.9

, find the value of

x_2

, giving your answer to

4

decimal places. (2 marks)

●●●●●Level 510 marksStart

The diagram shows a sector of a circle

OAB

.

The point

C

lies on

OB

such that

AC

is perpendicular to

OB

.

Angle

AOB

\theta

radians.(a) Given the area of the sector

OAB

\sqrt{2}

times the area of the triangle

OAC

, show that

\sin 2\theta = \sqrt{2}\,\theta

(4 marks)
(b) Use a suitable change of sign to show that a solution to the equation

\sin 2\theta = \sqrt{2}\,\theta

lies in the interval given by

\theta \in \left[\dfrac{\pi}{5}, \dfrac{\pi}{4}\right]

(2 marks)
(c) The Newton-Raphson method is used to find an approximate solution to the equation

\sin 2\theta = \sqrt{2}\,\theta

(c) (i) Using

\theta_1 = \dfrac{\pi}{4}

as a first approximation for

\theta

, apply the Newton-Raphson method twice to find the value of

\theta_3

Give your answer to three decimal places.

(ii) Explain how a more accurate approximation for

\theta

can be found using the Newton-Raphson method.

(iii) Explain why using

\theta_1 = \dfrac{\pi}{8}

as a first approximation in the Newton-Raphson method does not lead to a solution for

\theta

.

(6 marks)

●●●●●Level 512 marksStart

The trapezium rule

29 questions

LessonNot started

(a) Use the substitution

u = 2x - 3

to find

\displaystyle\int x\sqrt{2x - 3}\; dx

,

giving your answer in terms of

x

. (4 marks)

The finite region

R

is bounded by the curve

y = x\sqrt{2x - 3}

, the

x

-axis and the lines

x = 2

and

x = 6

.(b) Hence find the exact area of

R

. (3 marks)
(c) Use the trapezium rule, with four strips of equal width, to estimate the area of

R

. Explain, with reference to the shape of the curve, whether your estimate is an over-estimate or an under-estimate, and confirm that this is consistent with your answer to (b). (3 marks)

●●●●●Level 510 marksStart

The region bounded by the curve

y = (x - 6)\ln x

and the

x

-axis is shaded in the diagram below.(a) Use the trapezium rule with

5

ordinates to find an estimate for the area of the shaded region.

Give your answer correct to three significant figures.

(3 marks)
(b) Show that the exact area is given by

18\ln 6 - \frac{85}{4}

Fully justify your answer.

(6 marks)

●●●●●Level 59 marksStart