Solving Equations (AQA A Level Maths: Pure): Exam Questions

The table below shows corresponding values of $x$ and $y$ for $y=\sqrt{\ln x}$.

Use the trapezium rule with all the values in the table to find an estimate for

${\int }_2^4\sqrt{\ln x}dx$

State, with a reason, whether your answer to part (a) is an overestimate or an underestimate of the value of the integral.

The graph of $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=2x-{\left(\ln x\right)}^3-3x>0$

is shown below, where $\alpha$ and $\beta$ are solutions to the equation $\mathrm{f}\left(x\right)=0$

The Newton-Raphson method is to be used to estimate the values of $\alpha$ and $\beta$.

Indicate on the diagram the starting value, $x_0$, that would lead to the Newton-Raphson method failing to find either solution, $\alpha$ or $\beta$.

[You do not need to calculate the value of $x_0$.]

Find an expression for $\mathrm{f}\text{'}\left(x\right)$.

Apply the Newton-Raphson method with $x_0=1$ to find $\beta$.

Give your answer to 5 significant figures.

The curve with equation $y=\sqrt{{\mathrm{e}}^{-x}+2x}$ is shown below.

The shaded area is represented by

${\int }_2^8\sqrt{{\mathrm{e}}^{-x}+2x}dx$

Use the trapezium rule with 6 equally spaced trapezia to find an estimate for the area of the shaded region.

Give your answer to 3 significant figures.

Explain how the accuracy of the estimate in part (a) can be improved.

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

Show that the equation $\mathrm{f}\left(x\right)=0$ has a root $\alpha$ in the interval $1<x<2$.

The iterative formula

$x_{n+1}=\sqrt{4-\frac{1}{x_n}}$

is used with $x_1=1.5$ to find an approximate value for $\alpha$.

Calculate the values of $x_2$ and $x_3$, giving your answers to 4 decimal places.

Show that if the iteration converges to a limit $\alpha$, then $\alpha$ must be a root of the equation $\mathrm{f}\left(x\right)=0$.

The diagram below shows part of the graph $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=2x\cos \left(3x\right)-1$

Show that a solution to the equation $\mathrm{f}\left(x\right)=0$ exists in the interval $1.6<x<1.7$

Using a suitable interval. show that $x=2.55$ is a solution to the equation $\mathrm{f}\left(x\right)=0$, correct to 3 significant figures.

Sketch two separate diagrams to show how the trapezium rule can lead to either an underestimate or an overestimate.

Use the trapezium rule with step size $h=0.25$ to find an estimate for the area bounded by the curve with equation $y=1+0.3x^2\sin x$ and the lines $x=1$, $x=2$ and the $x$-axis.

Give your answer to 3 significant figures.

The function $\mathrm{f}\left(x\right)$ is defined as

 $\mathrm{f}\left(x\right)=x^2-\ln (x+2)x>0$

Show that there is a solution to the equation $\mathrm{f}\left(x\right)=0$ in the interval $1<x<1.2$

Find an expression for $\mathrm{f}\text{'}\left(x\right)$

Taking $x_0=1$ as the first approximation, apply the Newton-Raphson method repeatedly to find a solution to the equation $\mathrm{f}\left(x\right)=0$ in the interval $\left[1,1.2\right]$.

Give your answer correct to 3 decimal places.

The trapezium rule is to be used to approximate

${\int }_4^8\mathrm{f}\left(x\right)\mathrm{d}x$

The table below shows values of $x$ and $\mathrm{f}\left(x\right)$.

Using the values in the table, find an estimate for the integral using

(i) 2 trapezia,
(ii) 4 trapezia,
(iii) 8 trapezia.

Explain which estimate from part (a) is likely to be the most accurate approximation of

${\int }_4^8\mathrm{f}\left(x\right)\mathrm{d}x$

The diagram below shows part of the curve with the equation $y=2^{\ln x}$

The trapezium rule is used to estimate the area of the shaded region, given by

${\int }_5^{10}{2}^{\ln x}\mathrm{d}x$

Four equally spaced trapezia are used, each of width $h$.

Find $h$.

Complete the table of values below, correct to 3 significant figures.

Using the trapezium rule with all the values of $y$ in the table, find an estimate for

${\int }_5^{10}{2}^{\ln x}\mathrm{d}x$

State whether your answer to part (c) is an overestimate or an underestimate.

The diagram below shows part of the graph with equation $y=3x-{\mathrm{e}}^{x^2}.$

The area between the curve and the $x$-axis from $x=0.5$ to $x=1$ is shaded.

Use the trapezium rule with 5 equally spaced trapezia to find an estimate for the area of the shaded region.

Give your answer to 3 significant figures.

The figure below shows the line $y=x$ and the curve $y=\ln \left(x-1\right)+3$

The iteration formula

$x_{n+1}=\ln \left(x_n-1\right)+3$

with $x_0=2$ is used to find an estimate for a root, $\alpha$, of the equation$\mathrm{f}\left(x\right)=0$.

Write down an expression for $\mathrm{f}\left(x\right)$.

Draw a staircase diagram on the graph in part (a) to determine whether the iteration formula starting with $x_0=2$ finds an approximation for the $x$-coordinate of point $S$ or the $x$-coordinate of point $T$.

Find the values of the estimates $x_1,x_2,x_3$ and $x_4$, giving each answer to 3 decimal places.

Using a suitable interval, show that $\alpha =4.146$ to 3 decimal places.

The diagram below shows part of the curve $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=3x^2{\sin }^{2}x-2-\frac{3\pi }{2}<x<\frac{3\pi }{2}$

You are given that $\mathrm{f}\left(0.9\right)=-0.509$ and$\mathrm{f}(3.4)=0.265$, to 3 significant figures,

A student wishes to find an estimate of the root of $\mathrm{f}\left(x\right)=0$ that is close to $x=0.98$

Explain why a change of sign in the interval $\left[0.9,3.4\right]$ is not necessarily helpful to the student.

Using a suitable interval, 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 3 significant figures.

The graph below shows a sketch of the line $y=x$ and the curve  $y=\sqrt[3]{3x^2+2x-1}$

An iteration formula is used to find the three roots of the equation

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

Draw a staircase diagram on the graph above to show that the iteration formula

$x_{n+1}=\sqrt[3]{3{x_n}^2+2x_n-1}$

with a starting value of $x_0=0.5$ converges to the largest positive root of the equation.

(i) Use the iteration formula from part (a) with $x_0=0.5$ to find $x_1,x_2$ and $x_3$, to 3 significant figures.

(ii) If the root is close to $x=3.5$, describe the speed of convergence to the root of the values $x_1,x_2$ and $x_3$.

Show that the root close to $x=3.5$ is $3.49$, correct to 3 significant figures.

The function $\mathrm{f}\left(x\right)$ is defined as

 $\mathrm{f}\left(x\right)=\sin 3x-\ln 2xx>0$

where $x$ is in radians.

Apply the Newton-Raphson method with a starting approximation of $x_0=0.8$ to find a solution to the equation

$\sin 3x=\ln 2x$

giving your answer correct to 4 decimal places.

The graph of $y=\mathrm{f}\left(x\right)$ has a local maximum point with coordinates $\left(\beta ,\mathrm{f}\left(\beta \right)\right)$.

Describe what happens when applying the Newton-Raphson method with a starting approximation of $x_0=\beta$.

Graphs of $y=\mathrm{f}\left(x\right)$ for four different functions are shown below.

Match each graph above with the correct statement below:

1. $\mathrm{f}\left(x\right)$ is not continuous and it is possible to have an interval showing no sign change that contains exactly one root

2. $\mathrm{f}\left(x\right)$ is not continuous and it is possible to have an interval showing no sign change that contains more than one root

3. $\mathrm{f}\left(x\right)$ is continuous and it is possible to have an interval showing no sign change that contains exact one root

4. $\mathrm{f}\left(x\right)$ is continuous and it is possible to have an interval showing no sign change that contains more than one root

The diagram below shows the line $y=x$ and the curve $y=\mathrm{g}\left(x\right)$.

Draw a cobweb diagram on the graph above, using the starting approximation $x_0$ indicated.

You must show

• the first two estimates, $x_1$ and $x_2$

• convergence to a root of the equation $x-\mathrm{g}\left(x\right)=0$

By forming a suitable iteration formula with $x_0=2$ and using repeated iteration, find a root of the equation

$x-\sin 0.8x=2.5$

correct to 2 significant figures.

Using a suitable interval and a suitable function that should be stated, show that your answer to part (b) is correct to 2 significant figures.

The diagram below shows part of the graph of $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=0.3{\mathrm{e}}^{\sin x}-0.5$

Two roots of the equation $\mathrm{f}\left(x\right)=0$ are shown, $\alpha$ and $\beta$.

Write down the $x$-coordinate of the maximum point, $M$, on the graph.

The Newton-Raphson method is applied to find an estimate for the root $\beta$.

The starting approximation, $x_0$, is the smallest positive integer value greater than the $x$-coordinate of the maximum point $M$.

Find the first four estimates and use a suitable interval to find $\beta$ correct to 5 significant figures.

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$ where the function $\mathrm{f}\left(x\right)$ is defined by

$\mathrm{f}\left(x\right)=10-5x^2-\frac{1}{2x+4}x>-2$

The function $\mathrm{f}\left(x\right)$ has a positive root close to $x=1.4$

An iteration formula is given by

where $k$ is an integer to be found.

Use repeated iteration to find an estimate of the positive root, to 6 decimal places, using a starting value of $x_0=1.4$

Apply the Newton-Raphson method with $x_0=1.4$ to find an estimate of the positive root, to 6 decimal places.

Compare the rates at which the estimates converge between the different methods in part (a) and part (b).

The function, $\mathrm{f}\left(x\right)$, is defined by

$\mathrm{f}\left(x\right)=\frac{1}{{\mathrm{e}}^x}-x+1x\in \mathrm{ℝ}$

Show that the equation $\mathrm{f}\left(x\right)=0$ can be written in the form

$x={\mathrm{e}}^{ax}+b$

where $a$ and $b$ are integers to be found.

On the same diagram, sketch the graphs of $y=x$ and $y={\mathrm{e}}^{ax}+b$, using your values of $a$ and $b$ from part (a).

The equation $\mathrm{f}\left(x\right)=0$ has a root, $\alpha$, close to $x=1$.

Draw a cobweb diagram on the graph in part (b) to show how the iteration formula

$x_{n+1}=e^{a{x}_n}+b$

with $x_0=2$ converges to $\alpha$.

(i) Find the values of$x_1,x_2$and $x_3$, giving each answer correct to 3 significant figures.

(ii) How many iterations, $n$, are required before $x_n$ and $x_{n-1}$ agree with each other to 2 decimal places?

The root $\alpha$ lies in the interval $p<x<q$.

Find the values of $p$ and $q$ that give the largest interval such that $\alpha$ can be found to 2 decimal places.

Apply the Newton-Raphson method with $x_0=1.5$ to find a solution to equation

$x^5-2x^4+3x^3-4x^2+1=0$

correct to 4 significant figures.

Show that there is a solution to the equation in the interval [0.605 , 0.615].

Without further calculation, state the value of this solution to the highest degree of accuracy possible.

The diagram below shows the graph of $y=4-2x^{\ln x}$ where $x>0$.

Use the trapezium rule in step sizes of 0.2 to find an estimate of the integral

${\int }_1^2\left(4-2x^{\ln x}\right)dx$

to 3 significant figures.

The diagram below shows part of the graph with equation $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=x\tan \left(\pi -x\right)-3$

A student searches for a root of the equation $\mathrm{f}\left(x\right)=0$.

They find that $\mathrm{f}\left(1.5\right)<0$ and $\mathrm{f}\left(1.6\right)>0$. They then conclude that there is a root in the interval $1.5<x<1.6$.

Explain why the student’s conclusion is not correct.

Explain why a change of sign method would fail when searching for the root$x=0$ of the equation

$\mathrm{f}\left(x\right)+3=0$

The function $\mathrm{f}\left(x\right)$ is defined as

  $\mathrm{f}\left(x\right)=-3+5\cos x\sin 2xx\in \mathrm{ℝ}$

Show that the Newton-Raphson formula can be written as

$x_{n+1}=x_n-\frac{-3+5\cos x_n\sin 2x_n}{a\cos x_n\left(1-b{\sin }^2{x}_n\right)}$

where $a$ and $b$ are integers to be found.

Apply the Newton-Raphson method with $x_0=0.3$ to find, to 5 significant figures, a solution of the equation

$10{\cos }^{2}x\sin x=3$

A student wants to use the Newton-Raphson method to find a solution to the equation

$\mathrm{f}\left(x\right)=-3$

in the range $x>0$.

A teacher tells the student that the Newton-Raphson method is not necessary.

Explain why and find a solution to the equation, giving the highest accuracy possible.

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$ where $\mathrm{f}\left(x\right)$ is defined by

  $\mathrm{f}\left(x\right)=5x+\frac{2}{x^2}-12x>0$

The equation $\mathrm{f}\left(x\right)=0$ has a solution close to $x=0.4$

Use repeated iteration with $x_0=0.4$ to find this solution to 4 decimal places.

You must state your iteration formula clearly.

Apply the Newton-Raphson method to find the same root, using a different starting value of $x_0=0.5$

Give your answer to 4 decimal places.

Compare the rates at which the estimates converge between the different methods in part (a) and part (b).

The figure below shows a sketch of the curve $y=x{\left(x-6\right)}^2$.

The coordinates of the local maximum point are $\left(2,32\right)$.

Use the trapezium rule with 4 equally-spaced trapezia to find an approximation to

${\int }_1^{5}x{\left(x-6\right)}^{2}dx$

By replacing the trapezia with rectangles that fit above the curve, find an upper bound for the area shaded.

Use rectangles that fit below the curve to find a lower bound.