WMA11 October 2021

4. WMA11/01​​ Edexcel​​ IAL P1 Oct 2021,​​ Q4​​ (Trignometric Functions – Graphs)

Figure 2 shows a sketch of the curve with equation y = f(x), where​​ 

fx=cos⁡2xo       0≤ x≤ k 

The point Q and the point​​ R(k, 0) lie on the curve and are shown in Figure 2.​​ 

(a) State​​ 

• the coordinates of Q,​​ 

• the value of k.​​ 

(3)

(b) Given that there are exactly two solutions to the equation​​ 

cos 2xo = p         in the region    0≤ x≤ k 

find the range of possible values for p.​​ 

(2)

SOLUTION​​ 

a-​​ 

i-​​ 

Always remember,​​ the graph of y=cos(2x) is a stretch of the graph y=cos(x) by the scale factor of ½ in the horizontal direction. Therefore, the x –cordinate of Q is half of the value of x-cordinate of the point ​​ in the same position on the graoh y=cos(x). So the x-cordinate woould be half of 180.

90,-1

ii-​​ 

Since 1.25 cycle of cos x graph, and also stretch by the scale factor of 1/2, the value of k is 1.25(360/2)=225

225

b-​​ 

SInce there are only two solutions, this means that y=p and y=cos 2x intersect at only at two points. We​​ need to find the range of p for which line y=p and y=cos 2x cuts each other at only 2 points.​​ 

This is only possible for p=I and ​​ when the line cuts the curve below the x-axis.The figure below shows some of the lines drawn in the range of p where the line​​ and curve cuts at two points only.​​ 

p=1  &  -1<p<0