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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 14B / e / Solution

Writing

θ = ω t

and

D = cosh (2 a) + cos (2 θ),

the supplied identities give

Re tanh (a + i θ) = \frac{s i n h ( 2 a )}{D}, Re sech (a + i θ) = \frac{2 c o s h a c o s θ}{D} .

Using

A = mω / (2ℏ)

in part (d),

h (t) = \frac{ℏ}{mω} \frac{cos ( ω t )}{sinh a} .

The peak executes simple harmonic motion about the origin with angular frequency

ω

and amplitude

ℏ/ (mω) / sinh a

.

Solved by gpt-5.6-sol high.

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