The equation cosec x = x 2 has a root, α , between 1 and 2 . The equation can be rearranged either as x = sin - 1 1 x 2 or x = 1 sin x . Write down two possible iterative formulae, one based on each given rearrangement.

Given Cosec x = x 2 ⇒ x 2 = 1 sin x ⇒ x = 1 sin x or ⇒ 1 x 2 = sin x ⇒ x = sin - 1 1 x 2 Therefore, the iterative formula is x n + 1 = 1 sin x n or x n + 1 = sin - 1 1 x n 2

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AS and A Level

IMPORTANT

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By sketching graphs of $y = e^{x}$ and $y = x + 6,$ determine the number of roots of the equation $e^{x} - x - 6 = 0 .$

Important Questions on Numerical Solutions of Equations

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6A>Q 10

Show graphically that the equation

2^{x} = x + 4

has exactly two roots.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6A>Q 11

By sketching two suitable graphs on the same diagram, show that the equation

\cot x = x^{2}

has one root between

0

and

\frac{π}{2}

radians.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6A>Q 12

By sketching a suitable pair of graphs, deduce the number of roots of the equation

x = \tan 2 x

for

0 < x < 2 π .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6B>Q 3

The equation $(x - 0.5) e^{3 x} = 1$ has a root $α .$

Show, by sketching the graph of $y = e^{3 x}$ and one other suitable graph, that $α$ is the only root of this equation.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6B>Q 4

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

x^{3} + 10 x = x + 5

has only one root that lies between

0

and

\underline{1} .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6B>Q 7

The equation $x^{4} - 1 - x = 0$ has a root, $α,$ between $x = 1$ and $x = 2 .$

Show that $α$ also satisfies the equation $x = \sqrt[4]{1 + x} .$

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6B>Q 7

The equation $x^{4} - 1 - x = 0$ has a root, $α,$ between $x = 1$ and $x = 2 .$

Show that $α$ also satisfies the equation $x = \sqrt[4]{1 + x} .$

Write down an iterative formula based on the above equation

Use your iterative formula, with a starting value of $x_{1} = 1.5,$ to find $α$ correct to $2$ decimal places. Give the result of each iteration to $4$ decimal places.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 6 - Numerical Solutions of Equations>EXERCISE 6B>Q 8

The equation $cosec x = x^{2}$ has a root, $α,$ between $1$ and $2 .$ The equation can be rearranged either as $x = \sin^{- 1} (\frac{1}{x^{2}})$ or $x = \sqrt{\frac{1}{\sin x}} .$

Write down two possible iterative formulae, one based on each given rearrangement.

By sketching graphs of y=ex and y=x+6, determine the number of roots of the equation ex-x-6=0.

Important Questions on Numerical Solutions of Equations

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By sketching graphs of $y = e^{x}$ and $y = x + 6,$ determine the number of roots of the equation $e^{x} - x - 6 = 0 .$