(a)

By sketching suitable graphs, show that the equation $\mathrm{e}^{-\frac{1}{2} x}=4-x^{2}$ has one positive root and one negative root.

[ 2 ]

(b)

Verify by calculation that the negative root lies between -1 and -1.5 .

[ 2 ]

(c)

Use the iterative formula $x_{n+1}=-\sqrt{ }\left(4-e^{-\frac{1}{2} x_{n}}\right)$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

[Maximum number: 6]

The equation $x^{5}-3 x^{3}+x^{2}-4=0$ has one positive root.

(a)

Verify by calculation that this root lies between 1 and 2 .

[ 2 ]

(b)

Show that the equation can be rearranged in the form

\left.x=\sqrt[3]{( } 3 x+\frac{4}{x^{2}}-1\right)

[ 1 ]

(c)

Use an iterative formula based on this rearrangement to determine the positive root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

(a)

By sketching suitable graphs, show that the equation

4 x^{2}-1=\cot x

has only one root in the interval $0<x<\frac{1}{2} \pi$ .

[ 2 ]

(b)

Verify by calculation that this root lies between 0.6 and 1 .

[ 2 ]

(c)

Use the iterative formula

x_{n+1}=\frac{1}{2} \sqrt{ }\left(1+\cot x_{n}\right)

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

[ 3 ]

[Maximum number: 2]

A curve has parametric equations

x=t^{2}+3 t+1, \quad y=t^{4}+1

The point P on the curve has parameter p. It is given that the gradient of the curve at P is 4 .

(a)

Verify by calculation that the value of p lies between 1.8 and 2.0.

[ 2 ]

(b)

Use an iterative formula based on the equation in part (i) to find the value of p correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(a)

By sketching a suitable pair of graphs, show that the equation $\sec x=2-\frac{1}{2} x$ has exactly one root in the interval $0 \leqslant x<\frac{1}{2} \pi$ .

[ 2 ]

(b)

Verify by calculation that this root lies between 0.8 and 1 .

[ 2 ]

(c)

Use the iterative formula $x_{n+1}=\cos ^{-1}\left(\frac{2}{4-x_{n}}\right)$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

(a)

The sequence of values given by the iterative formula

x_{n+1}=\pi-\sin ^{-1}\left(\frac{1}{\mathrm{e}^{-\frac{1}{2} x_{n}}+1}\right),

with initial value $x_{1}=2$ , converges to one of these roots.
Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

[Maximum number: 5]

The curve with equation

y=5 \mathrm{e}^{2 x}-8 x^{2}-20

crosses the x-axis at only one point. This point has coordinates (p, 0).

(a)

Show by calculation that 0.75<p<0.85.

[ 2 ]

(b)

Use an iterative formula based on the equation in part (i) to find the value of p correct to 5 significant figures. Give the result of each iteration to 7 significant figures.

[ 3 ]

[Maximum number: 7]

The equation $x=\frac{10}{\mathrm{e}^{2 x}-1}$ has one positive real root, denoted by $\alpha$ .

(a)

Show that $\alpha$ lies between x=1 and x=2.

[ 2 ]

(b)

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

x_{n+1}=\frac{1}{2} \ln \left(1+\frac{10}{x_{n}}\right)

converges, then it converges to $\alpha$ .

[ 2 ]

(c)

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

[ 3 ]

[Maximum number: 3]

The diagram shows a semicircle A C B with centre O and radius r. The tangent at C meets A B produced at T. The angle B O C is x radians. The area of the shaded region is equal to the area of the semicircle.

(a)

Use the iterative formula $x_{n+1}=\tan ^{-1}\left(x_{n}+\pi\right)$ to determine x correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]

[Maximum number: 3]

The diagram shows the curve $y=\frac{\sin x}{x}$ for $0<x \leqslant 2 \pi$ , and its minimum point M.

(a)

The iterative formula

x_{n+1}=\tan ^{-1}\left(x_{n}\right)+\pi

can be used to determine the x-coordinate of M. Use this formula to determine the x-coordinate of M correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[ 3 ]