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

MEDIUM

AS and A Level

IMPORTANT

Earn 100

The curve $y = x^{3} + 5 x - 1$ cuts the $x$ -axis at the point $(α, 0) .$

By sketching the graph of $y = x^{3}$ and one other suitable graph, deduce that this is the only point where the curve $y = x^{3} + 5 x - 1$ cuts the $x$ -axis.

Important Questions on Numerical Solutions of Equations

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 3

The curve $y = x^{3} + 5 x - 1$ cuts the $x$ -axis at the point $(α, 0) .$

Using calculations, show that $0.1 < α < 0.5 .$

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 4

On the same axes, sketch the graphs of

y = l n (x + 1)

and

y = 3 x - 4 .

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 4

Hence deduce the number of roots of the equation

l n (x + 1) - 3 x + 4 = 0 .

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 5

By sketching graphs of

y = e^{x}

and

y = x + 6,

determine the number of roots of the equation

e^{x} - x - 6 = 0 .

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 5

Show, by calculation, that

e^{x} - x - 6 = 0

has a root that lies between

x = 2.0

and

x = 2.1 .

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 6

Show, by calculation, that

(x + 2) e^{5 x} = 1

has a root between

x = 0

and

x = - 0.2

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 6

Show, by sketching the graph of

y = e^{5 x}

and one other suitable graph, that this is the only root of this equation.

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 7

Show by calculation that the equation

\cos^{- 1} 2 x = 1 - x

has a root

α,

where

0.4 ⩽ α ⩽ 0.5 .