The curve with equation y = sin x 2 x + 3 where x < - π 2 intersects the line y = 1 at the point α , 1 . By sketching the graph of y = sin x for - 2 π < x < - π 2 radians and one other suitable graph, deduce that this is the only point where the curve y = sin x 2 x + 3 intersects the line y = 1 .

Given that: The curve with equation y = sin x 2 x + 3 where x < - π 2 intersects the line y = 1 at the point α , 1 . The graph of y = sin x for - 2 π < x < - π 2 radians and y = 2 x + 3 One point of intersection for - 2 π < x < - π 2 So, only one root of 1 = sin x 2 x + 3 on this domain. Also, should x be less than - 2 π or should x be greater than - π 2 , the line and the curve will not intersect again and so this is the only point of intersection of y = sin x 2 x + 3 and y = 1 .

The curve with equation y = sin x 2 x + 3 where x < - π 2 intersects the line y = 1 at the point α , 1 . Using calculations, show that - 2 < α < - 1 . 9

Given that: The curve with equation y = sin x 2 x + 3 where x < - π 2 intersects the line y = 1 at the point α , 1 . Let f x = sin x 2 x + 3 - 1 = 0 Then, f - 2 = sin - 2 2 - 2 + 3 - 1 = - 0 . 09070 f - 1 . 9 = sin - 1 . 9 2 - 1 . 9 + 3 - 1 = 0 . 1828 Change of sign indicate the presence of root Therefore, - 2 < α < - 1 . 9 Hence, proved

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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

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 .

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

The diagram shows the graph of $y = \cos^{- 1} 2 x$ where $- 0.5 ⩽ x ⩽ 0.5 .$

Question Image

By drawing a suitable graph on a copy of the diagram, show that $α$ is the only root of $\cos^{- 1} 2 x = 1 - x$ for $- 0.5 ⩽ x ⩽ 0.5 .$

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 8

The curve with equation $y = \frac{\sin x}{2 x + 3}$ where $x < - \frac{π}{2}$ intersects the line $y = 1$ at the point $(α, 1) .$

By sketching the graph of $y = \sin x$ for $- 2 π < x < - \frac{π}{2}$ radians and one other suitable graph, deduce that this is the only point where the curve $y = \frac{\sin x}{2 x + 3}$ intersects the line $y = 1 .$

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 8

The curve with equation $y = \frac{\sin x}{2 x + 3}$ where $x < - \frac{π}{2}$ intersects the line $y = 1$ at the point $(α, 1) .$

Using calculations, show that $- 2 < α < - 1.9$

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 9

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

x^{3} + 4 x = 7 x + 4

has only one root for

0 ⩽ x ⩽ 5 .

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