[Maximum number: 5]

The sequence of values given by the iterative formula

x_{n+1}=\frac{4}{x_{n}^{2}}+\frac{2 x_{n}}{3},

with initial value $x_{1}=2$ , converges to $\alpha$ .

(a)

Use this iterative formula to find $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

[ 3 ]

(b)

State an equation that is satisfied by $\alpha$ , and hence find the exact value of $\alpha$ .

[ 2 ]

[Maximum number: 5]

The sequence of values given by the iterative formula

x_{n+1}=\frac{7 x_{n}}{8}+\frac{5}{2 x_{n}^{4}}

with initial value $x_{1}=1.7$ , converges to $\alpha$ .

(a)

Use this iterative formula to determine $\alpha$ correct to 2 decimal places, giving the result of each iteration to 4 decimal places.

[ 3 ]

(b)

State an equation that is satisfied by $\alpha$ and hence show that $\alpha=\sqrt[5]{20}$ .

[ 2 ]

[Maximum number: 5]

The diagram shows the curve $y=x^{4}+2 x-9$ . The curve cuts the positive x-axis at the point P.

(a)

Verify by calculation that the x-coordinate of P lies between 1.5 and 1.6.

[ 2 ]

(b)

Show that the x-coordinate of P satisfies the equation

x=\sqrt[3]{\left(\frac{9}{x}-2\right) .}

(c)

Use the iterative formula

x_{n+1}=\sqrt[3]{\left(\frac{9}{x_{n}}-2\right)}

to determine the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

[Maximum number: 5]

The sequence of values given by the iterative formula

x_{n+1}=\frac{x_{n}\left(x_{n}^{3}+100\right)}{2\left(x_{n}^{3}+25\right)},

with initial value $x_{1}=3.5$ , converges to $\alpha$ .

(a)

Use this formula to calculate $\alpha$ correct to 4 decimal places, showing the result of each iteration to 6 decimal places.

[ 3 ]

(b)

State an equation satisfied by $\alpha$ and hence find the exact value of $\alpha$ .

[ 2 ]

[Maximum number: 3]

In the diagram, A B C is a triangle in which angle A B C is a right angle and B C=a. A circular arc, with centre C and radius a, joins B and the point M on A C. The angle A C B is $\theta$ radians. The area of the sector C M B is equal to one third of the area of the triangle A B C.

(a)

This equation has one root in the interval $0<\theta<\frac{1}{2} \pi$ . Use the iterative formula

\theta_{n+1}=\tan ^{-1}\left(3 \theta_{n}\right)

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

(a)

By sketching a suitable pair of graphs, show that the equation $x^{3}=3-x$ has exactly one real root.

[ 2 ]

(b)

Show that if a sequence of real values given by the iterative formula

x_{n+1}=\frac{2 x_{n}^{3}+3}{3 x_{n}^{2}+1}

converges, then it converges to the root of the equation in part (i).

[ 2 ]

(c)

Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

[ 3 ]

(a)

By sketching a suitable pair of graphs, show that the equation

x^{3}=11-2 x

has exactly one real root.

[ 2 ]

(b)

Use the iterative formula

x_{n+1}=\sqrt[3]{ }\left(11-2 x_{n}\right)

to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

[ 3 ]

[Maximum number: 5]

The curve with equation $y=\frac{\ln x}{3+x}$ has a stationary point at x=p.

(a)

By sketching suitable graphs, show that the equation in part (i) has only one root.

[ 2 ]

(b)

It is given that the equation in part (i) can be written in the form $x=\frac{3+x}{\ln x}$ . Use an iterative formula based on this rearrangement to determine the value of p correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

[Maximum number: 7]

The equation $x^{3}-x^{2}-6=0$ has one real root, denoted by $\alpha$ .

(a)

Find by calculation the pair of consecutive integers between which $\alpha$ lies.

[ 2 ]

(b)

Show that, if a sequence of values given by the iterative formula

x_{n+1}=\sqrt{ }\left(x_{n}+\frac{6}{x_{n}}\right)

converges, then it converges to $\alpha$ .

[ 2 ]

(c)

Use this iterative formula to determine $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

[ 3 ]

[Maximum number: 7]

The equation $x^{3}-x^{2}-6=0$ has one real root, denoted by $\alpha$ .

(a)

Find by calculation the pair of consecutive integers between which $\alpha$ lies.

[ 2 ]

(b)

Show that, if a sequence of values given by the iterative formula

x_{n+1}=\sqrt{ }\left(x_{n}+\frac{6}{x_{n}}\right)

converges, then it converges to $\alpha$ .

[ 2 ]

(c)

Use this iterative formula to determine $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

[ 3 ]