Numerical Solutions of Equations (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

The diagram below shows part of the graph y = f(x) where f(x) = 2x cos (3x) - 1.

(i) Find f(1.6) and f(1.7), giving your answers to three significant figures.

(ii) Briefly explain the significance of your results from part (i).

One of the solutions to the equation f(x) = 0 is x = 2.55, correct to three significant figures.

(i) Write down the upper and lower bound of 2.55.

(ii) Hence, use the sign change rule to confirm that this is a solution (to three significant figures) to the equation f(x) = 0.

Show that the equation x³ + 3 = 5x can be rewritten as x =

Starting with x_o = 1.8, use the iterative formula

X_n+1 =

to find a root of the equation x³ + 3 = 5x, correct to two decimal places.

Part of the graph of y = tan θ is shown below, where θ is measured in radians.

Explain why the change of sign rule would fail if attempting to locate a root of the function f(θ ) = tan θ using the values of θ = 1.55 and θ = 1.65.

The diagram below shows the graphs of y = x and y = In(x - 1) + 3.

The iterative formula X_n+1 = In (x_n - 1) + 3 is to be used to find an estimate for a root, ⍺, of the function f(x).

Write down an expression for f(x).

Using an initial estimate, x_o = 2, show, by adding to the diagram above, which of the two points (S or T) the sequence of estimates x₁, x₂, x₃, ... will converge to. Hence deduce whether ⍺ is the x-coordinate of point S or point T.

Find the estimates x_1,x₂, x₃ and x₄, giving each to three decimal places.

Confirm that ⍺ = 4.146 correct to three decimal places.

The village of Greendale lies on a straight road, as modelled by the line y = x on the graph below. To ease rush hour congestion, a bypass is to be built around Greendale. The path of the bypass is modelled by the equation y = 6

The bypass runs from the origin to the point P(p, p).

On the diagram show how using the iterative formula x_n+1 = 6 with 0 < x_o < p will lead to convergence at the point P.

Use the iterative method X_n+1 = 6 with x_o = 5 to find the value x_n+1 of p, correct to two decimal places.

Use the sign change rule with the function f(x) = x - 6 to show your answer to part (b) is correct to two decimal places.

The game of Curveball is played on a flat table.
A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest in the winning zone.

A particular player decides to roll the ball at an angle of 45°.
This is illustrated by the graph below with the ball being rolled from the origin and the shaded area being the winning zone.

The boundary of the winning zone is given by part of the curve with equation

y = sin² (e^x).

Use the iterative formula Х_п+1 = sin² (e^x) with initial starting value x_o = 0.5, to show that the x-coordinate of the point where this player's ball should cross the winning zone boundary is 0.497 to three significant figures.

Use your answer to part (a) to find the minimum distance the ball should travel for this player to win Curveball.

According to legend, a unicorn can heal an injury almost instantly by touching it with its horn.

When a unicorn touches a cut in human skin of length L mm, it will heal according to the model

f(t) = Le^-t - t t ≥ 0

where f is the length of the cut in millimetres, at time t seconds after the unicorn has touched the injury with its horn.

(i) Write down the value f would be when the cut is completely healed.

(ii) Show that, for a cut in human skin of length 5 mm the equation f(t) = 0 can be rearranged into the form

t =-In ()

Use the iterative formula t_n+1 =- In (), with initial value t_o = 1, to find how nany seconds it takes a cut of size 5 mm to heal once a unicorn has touched it with ts horn.

(i) Write down the values of the estimates t₁, t₂ and t₃ to four decimal places.

(i) Give your final answer to two significant figures.

(ii) State the number of iterations required for convergence to two significant figures.

Briefly explain why the model should also restrict the range of f(t) to be greater than or equal to zero?

The diagram below shows part of the function y = f(x) where f(x) = 3x² sin²x - 2.

Correct to three significant figures, f(0.9) = -0.509 and f(3.4) = 0.265.

Explain why using the sign change rule with these values would not necessarily be helpful in finding the root close to x = 0.98.

Using suitable values of x, show that there is a root close to x = 0.98.

Show that the root close to x = 0.98 is 0.982, correct to three significant figures.

The diagram below shows a sketch of the graphs y = x, and y =

An iterative formula is used to find roots to the equation x³ - 3x² - 2x + 1 = 0.
On the diagram above show that the iterative formula

x_n+1 =

would converge to the root close to x = 3.5 when using a starting value of x_o = 0.5.

(i) Use x_o = 0.5 in the iterative formula from part (a) to find three further approximations to the root close to x = 3.5.
Give each approximation correct to three significant figures.

(ii) Comment on your approximations and what they suggest about convergence to the root close to x = 3.5.

Confirm that the root close to x = 3.5 is 3.49 correct to three significant figures.

The diagrams below show the graphs of four different functions.

Match each graph above with the correct statement below.

(1). The sign change rule with values of x = 2 and x = 4 would indicate a root but has failed due to the discontinuity (asymptote) at x = 3.

(2). The sign change rule with values of x = 1 and x = 5 would indicate no root but has failed because there are two roots in the interval (1,5).

(3). The sign change rule with values of x = 3 and x = 5 would indicate no root but fail as there are two roots in the interval (3, 5).

(4). The sign change rule with values of x = 3 and x = 5 would indicate no root but has failed to find the root ⍺ as the graph has a turning point at x = ⍺.

The diagram below shows the graphs of y = x and y = g(x).

Show on the diagram, using the value of x_o indicated, how an iterative process will lead to a sequence of estimates that converge to the x-coordinate of the point P.
Mark the estimates x₁, and x₂ on your diagram.

By finding a suitable iterative formula, use x_o = 2 to estimate a root to the equation x - sin 0.8x = 2.5 correct to two significant figures.

Confirm that your answer to part (b) is correct to two significant figures.

The village of Crinkley Bottom lies on a straight road, as modelled by the line y = x on the graph below. Rush hour traffic causes much air pollution in the village so to improve the air quality around Crinkley Bottom a bypass is to be built.
The path of the bypass is modelled by part of the equation y = x² sin x.

The bypass is to be built with a roundabout south of the village at the origin and a northern roundabout which re-joins the road through Crinkley Bottom at the point P (p,p).

On the diagram show how using the iterative formula x_n+1 = x_n²sin x with 0 < x_o < p will lead to convergence at the southern roundabout

Use the alternative iterative method

x_n+1 =

with x_o = 1, to find the position of the roundabout at P to four significant figures.

Verify that your answer to part (b) is correct to four significant figures.

The game of Logball is played on a flat table.
A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

A particular player decides to roll the ball at an angle of 45°, as illustrated in the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

The lower boundary of the winning zone has equation y = 3 - In(x + 1)² x, y ≥ 0

The upper boundary of the winning zone has equation y = 4 - In(x + 1)² x, y ≥ 0

Using an appropriate iterative formula with initial value x_o = 1.8, find the minimum listance this plaver's ball needs to travel to stop within the winning zone. Give your answer to two significant figures.

Using another iterative formula with initial value x_o = 2.5, find the maximum distance this player's ball can travel yet remain within the winning zone. Give your answer to two significant figures.

According to legend, unicorn tears have magical healing powers.
When a unicorn tear is applied to a bruise of size A mm² it will heal according to the model

f(t) = Ae^-0.15t - 0.2t t ≥ 0

where f is the area of the bruise, in square millimetres, at time t seconds after the unicorn tear is applied.

Show that, for a bruise of initial size 20 mm?, the equation f(t) = 0 can be rearranged into the form

t=

Using the equation from part (a) as an iterative formula and initial value t_o = 12, find how many seconds it takes a bruise of size 20 mm² to heal once a unicorn tear is applied. Give your answer to three significant figures.

It is rumoured that a unicorn tear can heal bruises one-hundred-thousand times faster than they would heal naturally. Approximately how many days would it take a bruise of initial size 20 mm² to heal without a unicorn tear?

The diagram below shows part of the graph with equation f(x) = x tan(- x) - 3.

A student searches for a root of the equation f(x) = 0. They find that f(1.5) = -24.2 and that f(1.6) = 51.8.
The student concludes that there is a root in the interval 1.5 < x < 1.6.
Explain why the student's conclusion is incorrect.

Verify that x = 0 is a solution to the equation f(x) + 3 = 0.

Explain why the sign change rule would fail if searching for the root x = 0 of the equation f(x) + 3 = 0.

The function, f(x) is defined by f(x) = -x+1

Show that the equation f(x) = 0 can be written in the form

x = e^-x + 1

On the same diagram sketch the graphs of y = x and y = e^-x + 1.

The equation f(x) = 0 has a root, a, close to x = 1.
The iterative formula x_n+1 = +1 with x_o = 2 is to be used to find ⍺ correct to three significant figures

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root ⍺.

(i) Find the values of x₁, x₂ and x₃, giving each correct to three significant figures.

(ii) How many iterations are required before x_n, and x_n-1, agree to two decimal places?

The root ⍺ lies in the interval p < x < q.
Write down the values of p and q such that a can be deduced accurate to two decimal places from the interval.

Sketch three separate graphs with values of x = p and x = q, to show how the sign change rule would fail to find a root ⍺ in the interval (p, q) for the following reasons:

(i) Sign change rule indicates a root but there isn't one due to a discontinuity in the graph.
(ii) Sign change rule indicates no root but there is a root at a turning point.
(iii) Sign change rule indicates no root but there are in fact two roots in the interval (p, q).

On each diagram, clearly labelled p, q and the root ⍺.

Sketch two separate diagrams to show how an iterative formula of the form

x_{n+ 1} = g(x_n)

can diverge in two different ways when being used to find an estimate for a root to the equation f(x) = 0.

The village of Camberwick Green lies on a straight road, as modelled by the line y = x on the graph below. Rush hour traffic through the village causes both congestion and air pollution. To ease congestion and improve the air quality around Camberwick Green a bypass, modelled by part of the equation y = 1 + 2 In(x²), is to be built.

The bypass is to be built with a roundabout south of the village at point S and a northern roundabout which re-joins the road through Camberwick Green at the point P(p, p).

Write down the coordinates of the southern roundabout at point S.

(i) Show on the diagram how an iterative method with a starting value (x₀) in the interval (1, p) will converge to the northern roundabout at point P.

(ii) Use a suitable iterative formula with an appropriate initial value (x₀) to find the value of p to five significant figures.

Find the length of road through Camberwick Green that will benefit from the construction of the bypass. Decide on an appropriate unit of measurement.

The game of Funcball is played on a flat table. A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

The winning zone is modelled by the function

f(x) = In (3x + 4) - 0.25x².

The lower boundary of the winning zone has equation y = f(x), x, y ≥ 0
The upper boundary of the winning zone has equation y = f(x) + 1, x, y ≥ 0

A particular player decides to roll the ball at an angle of 45°, as illustrated by the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

Using iterative formulas with initial values x_o = 1.5 and x_o = 2.1 as appropriate, find the exact distances between which the ball must stop for this player to win Funcball. Give your answers to two significant figures.

According to legend, unicorn tears have magical healing powers. Without unicorn tears, a bruise of initial size A mm? will heal according to the model

f(t) = Ae^-0.25t - 0.1t t ≥ 0

where f is the area of the bruise, in square millimetres, at time t days since the bruise first appeared.

Use the iterative formula

t_n+1 = 4In

with to = 10 to find how many days it takes a bruise to heal without unicorn tears.
Give your answer to three significant figures.

With unicorn tears, a bruise of the same initial size will heal according to the model

u(T) = ATe^-T - 0.1T T ≥ 0

where time T is measured in seconds.

(i) Find the initial size of the bruise considered in part (a).

(ii) Find how many seconds it takes the same size bruise to heal using unicorn tears.

Using your answers from parts (a) and (b), work out approximately how many times quicker a bruise heals when unicorn tears are used.