9. WMA11/01 (Edexcel) IAL P1 June 2021 Q9 Graphs of Trig Functions, Reciprocal Functions
Figure 4 shows a sketch of the curve with equation
The line l, shown in Figure 4, is an asymptote to
(a) State an equation for l.
(1)
A copy of Figure 4, labelled Diagram 1, is shown on the next page.
(b)
(i) On Diagram 1, sketch the curve with equation
stating the equation of the horizontal asymptote of this curve.
(ii) Hence, giving a reason, state the number of solutions of the equation
in the region
(4)
(c) State the number of solutions of the equation
(i)
(ii)
(2)
SOLUTION
a-
b- i-
The equation of a horizontal asymptote is x=1.
b-ii-
The number of solution is 5 because between - 2π ≤ x ≤ 2π there are 5 intersection between y-tanx and y=1/x+1.
c- i-
As there is only 1 solution between 0 and π so in between 0 to 40π there will be 40 solution.
40 solutions
ii- Let’s first analyze the number of solutions from 0π to -10π. From 0 to -2π, there are 3 solutions. Out of these 3 solutions, two of them are along the graph as it stretches parallel to the x-axis, whereas one solutions lies where the graph stretches parallel to y-axis. Now, further, from -2π to -10π, the graph stretches parallel to x-axis. Therefore, there will be 8 more solutions parallel to x-axis from -2π to -10π. Therefore, from 0π to -10π, there will be total 8+3=11 solutions.
Now, lets analyze for the range 0π to 5/2π, where it is equal to 2.5π. We already know, that from 0π to 2π, there are 2 solutions. And to know the number of solutions there would be between 2π to 2.5π, see the number of solutions between 0π and π/2. There is only 1 solution between 0π and π/2, so will be in between 2π and 5π/2. Thus, the total solutions between 0π and π/2 are 2+1=3 solutions.
Hence, the total number of solution between -10π and 5π/2 are 11+3=14.
14 solutions
