Partial fractions

Given that

find the value of each of the constants A, B, C and D.

The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation

The plant cell has an initial volume of 20 microlitres.

Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.

Show that

where A, B and k are constants to be found.

The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell.

Find the value of L, giving a reason for your answer.

Express f (x) in partial fractions in terms of k.

Hence find the exact value of k for which

When chemical A and chemical B are mixed, oxygen is produced. A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time. The total volume of oxygen produced, Vm³, t hours after the chemicals were mixed, is modelled by the differential equation

where k is a constant. Given that exactly 2 hours after the chemicals were mixed, a total volume of 3 m³ of oxygen had been produced,

solve the differential equation to show that

The scientist noticed that

• there was a time delay between the chemicals being mixed and oxygen being produced

• there was a limit to the total volume of oxygen produced

Deduce from the model

the time delay giving your answer in minutes,

the limit giving your answer in m³

Given that f(x) can be expressed in the form

where A, B and C are constants

find the value of B and the value of C

show that A = 0

Use binomial expansions to show that, in ascending powers of x

where p, q and r are simplified fractions to be found.

Find the range of values of x for which this expansion is valid.

Given that f(x) can be expressed in the form

where A, B and C are constants

find the value of B and the value of C

show that A = 0

Use binomial expansions to show that, in ascending powers of x

where p, q and r are simplified fractions to be found.

Find the range of values of x for which this expansion is valid.

Given that

find the values of the constants A, B and C

Hence, using algebraic integration, find the exact value of

giving your answer in the form a + bln 2 where a and b are integers to be found.

Explain why you can deduce that q = 4

Figure 4 shows a sketch of part of the curve C. The region R, shown shaded in Figure 4, is bounded by the curve C, the x-axis and the line with equation x = 3

Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found.

Find the values of the constants A, B and C.

Prove that f (x) is a decreasing function.

Find, in its simplest form,

Hence find

Use integration by parts to find

giving your answer in the form a + b ln 2, where a and b are constants to be found.

Given that y 0, find

Use the substitution x = 4sin² to show that

Hence use integration to find

giving your answer in the form a + b, where a and b are exact constants.

A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation

where P is the population in thousands, and t is the time measured in years since the start of the study. Given that P = 3 when t = 0,

solve this differential equation to show that

find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures.

The finite region R is bounded by the curve C, the x- axis, the line with equation x = 1 and the line with equation x = 4 This region is shown shaded in Figure 2 The region R is rotated through 360 ° about the x- axis.

Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form a + b ln c, where a, b and c are constants.

Express in partial fractions

Find the values of the constants A, B and C.

where a and b are constants.

Find the values of the constants A, B and C.

Find the particular solution of the differential equation

Find the values of the constants A, B and C.

term in x². Give each coefficient as a simplified fraction.

₂

Given that f (x) can be expressed in the form

find the values of B and C and show that A = 0.

Hence, or otherwise, find the series expansion of f (x), in ascending powers of x, up to and including the term in x². Simplify each term.

Find the percentage error made in using the series expansion in part (b) to estimate the value of f (0.2). Give your answer to 2 significant figures.

Hence obtain the solution of

for which y = 0 at x =x giving your answer in the form sec² x = g(y).