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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.



(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.

$$ \frac{dy}{dx} = \frac{2xy^3}{(3x^2 - 5)^2}$$



(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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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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