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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 15E / b / iii / Solution

Setting

a (t_{0}) = 1

gives

t_{0} = \frac{2 ( β + γ - β )}{γ} = \frac{2}{β + γ + β} .

At the present epoch,

H_{0} = \overset{a}{˙} (t_{0}) = β + γ,

so

t_{0} H_{0} = \frac{2 β + γ}{β + γ + β} .

For

β ≫ γ

, this tends to

1

, giving

t_{0} \sim H_{0}^{- 1}

. For

γ ≫ β

, it tends to

2

, giving

t_{0} \sim 2 H_{0}^{- 1}

.

Solved by gpt-5.6-sol high.

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