Date | May 2021 | Marks available | 4 | Reference code | 21M.2.SL.TZ1.9 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 1 |
Command term | Hence and Find | Question number | 9 | Adapted from | N/A |
Question
Consider the function f defined by f(x)=90e-0.5x for x∈ℝ+.
The graph of f and the line y=x intersect at point P.
The line L has a gradient of -1 and is a tangent to the graph of f at the point Q.
The shaded region A is enclosed by the graph of f and the lines y=x and L.
Find the x-coordinate of P.
Find the exact coordinates of Q.
Show that the equation of L is y=-x+2 ln 45+2.
Find the x-coordinate of the point where L intersects the line y=x.
Hence, find the area of A.
The line L is tangent to the graphs of both f and the inverse function f-1.
Find the shaded area enclosed by the graphs of f and f-1 and the line L.
Markscheme
Attempt to find the point of intersection of the graph of f and the line y=x (M1)
x=5.56619…
=5.57 A1
[2 marks]
f'(x)=-45e-0.5x A1
attempt to set the gradient of f equal to -1 (M1)
-45e-0.5x=-1
Q has coordinates (2 ln 45, (accept () A1A1
Note: Award A1 for each value, even if the answer is not given as a coordinate pair.
Do not accept or as a final value for . Do not accept or as a final value for .
[4 marks]
attempt to substitute coordinates of (in any order) into an appropriate equation (M1)
OR A1
equation of is AG
[2 marks]
A1
[1 mark]
appropriate method to find the sum of two areas using integrals of the difference of two functions (M1)
Note: Allow absence of incorrect limits.
(A1)(A1)
Note: Award A1 for one correct integral expression including correct limits and integrand.
Award A1 for a second correct integral expression including correct limits and integrand.
A1
[4 marks]
by symmetry (M1)
A1
Note: Accept any answer that rounds to (but do not accept ).
[2 marks]