Date | November 2021 | Marks available | 2 | Reference code | 21N.3.AHL.TZ0.1 |
Level | Additional Higher Level | Paper | Paper 3 | Time zone | Time zone 0 |
Command term | Hence and Find | Question number | 1 | Adapted from | N/A |
Question
In this question you will explore some of the properties of special functions f and g and their relationship with the trigonometric functions, sine and cosine.
Functions f and g are defined as f(z)=ez+e-z2 and g(z)=ez-e-z2, where z∈ℂ.
Consider t and u, such that t, u∈ℝ.
Using eiu=cos , find expressions, in terms of and , for
The functions and are known as circular functions as the general point () defines points on the unit circle with equation .
The functions and are known as hyperbolic functions, as the general point ( ) defines points on a curve known as a hyperbola with equation . This hyperbola has two asymptotes.
Verify that satisfies the differential equation .
Show that .
.
.
Hence find, and simplify, an expression for .
Show that .
Sketch the graph of , stating the coordinates of any axis intercepts and the equation of each asymptote.
The hyperbola with equation can be rotated to coincide with the curve defined by .
Find the possible values of .
Markscheme
A1
A1
AG
[2 marks]
METHOD 1
substituting and M1
(M1)
A1
AG
METHOD 2
M1
M1A1
AG
Note: Accept combinations of METHODS 1 & 2 that meet at equivalent expressions.
[3 marks]
substituting into the expression for (M1)
obtaining (A1)
Note: The M1 can be awarded for the use of sine and cosine being odd and even respectively.
A1
[3 marks]
substituting and attempt to simplify (M1)
A1
[2 marks]
METHOD 1
substituting expressions found in part (c) (M1)
A1
METHOD 2
M1
A1
Note: Accept equivalent final answers that have been simplified removing all imaginary parts eg etc
[2 marks]
M1
A1
A1
Note: Award A1 for a value of 1 obtained from either LHS or RHS of given expression.
M1
(hence ) AG
Note: Award full marks for showing that .
[4 marks]
A1A1A1A1
Note: Award A1 for correct curves in the upper quadrants, A1 for correct curves in the lower quadrants, A1 for correct -intercepts of and (condone and ), A1 for and .
[4 marks]
attempt to rotate by in either direction (M1)
Note: Evidence of an attempt to relate to a sketch of would be sufficient for this (M1).
attempting to rotate a particular point, eg (M1)
rotates to (or similar) (A1)
hence A1A1
[5 marks]