MEDIUM
AS and A Level
IMPORTANT
Earn 100

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

HARD
AS and A Level
IMPORTANT
Show, by calculation, that ex-x-6=0 has a root that lies between x=2.0 and x=2.1.
HARD
AS and A Level
IMPORTANT
Show, by calculation, that x+2e5x=1 has a root between x=0 and x=-0.2
MEDIUM
AS and A Level
IMPORTANT
Show, by sketching the graph of y=e5x and one other suitable graph, that this is the only root of this equation.
HARD
AS and A Level
IMPORTANT
Show by calculation that the equation cos-12x=1-x has a root α, where 0.4α0.5.
HARD
AS and A Level
IMPORTANT

The diagram shows the graph of y=cos-12x where -0.5x0.5.

Question Image

By drawing a suitable graph on a copy of the diagram, show that α is the only root of cos-12x=1-x for -0.5x0.5.

HARD
AS and A Level
IMPORTANT

The curve with equation y=sinx2x+3 where x<-π2 intersects the line y=1 at the point α, 1.

By sketching the graph of y=sinx for -2π<x<-π2 radians and one other suitable graph, deduce that this is the only point where the curve y=sinx2x+3 intersects the line y=1.

HARD
AS and A Level
IMPORTANT

The curve with equation y=sinx2x+3 where x<-π2 intersects the line y=1 at the point α, 1.

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

MEDIUM
AS and A Level
IMPORTANT
By sketching a suitable pair of graphs, show that the equation x3+4x=7x+4 has only one root for 0x5.