## Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. |

## 1. | A-Level |

(a) Using a suitable substitution, or otherwise, find

$$ \int \frac{x}{(3x^2 - 5)^2} dx$$(b) Solve the differential equation below giving your answer in the form \(y = f(x)\). It is given that given that y = \( \frac{1}{2} \) when x = 0.

$$ \frac{dy}{dx} = \frac{2xy^3}{(3x^2 - 5)^2}$$## 2. | A-Level |

(a) Express the following fraction in partial fractions.

$$ \frac{1}{F(5-3F)} $$The popularity of a student rock group is measured during their first year of gigs. The number of fans is modelled by the differential equation:

$$ \frac{dF}{dt} = \frac{F}{15} (5-3F) \quad 0 \le t \le 12 $$where F, in hundreds, is the number of fans and t is the time measured in months since the band began performing regularly.

(b) Given that there were 100 fans when the measurements began, determine the time taken, in months, for the number of fans to increase by 50%.

(c) Show that:

$$ F= \frac{A}{B+C^{-\frac{t}{3}}} $$where A, B and C are integers to be found.

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