Binomial expansion

Find the first three terms in the expansion of (2 + x)^-3 in ascending powers of x.

Determine the range of values of x for which the expansion in part (b) is valid, giving a

reason for your answer.

Expand (3 - 2x)^-2 in ascending powers of x up to and including the term in x².

State the set of values of x for which this expansion is valid.

When is expanded in ascending powers of x, the coefficient of x is zero.

Determine the value of the constant a.

Find the first three terms in the expansion of (4 + 3x in ascending powers of x.

State the range of values of x for which the expansion in part (a) is valid.

Determine the possible values of the constant a.

Find the first four terms in the expansion of (3 + 2x)⁵ in ascending powers of x.

Hence determine the coefficient of y³ in the expansion of (3 + 2y + 4y²)⁵.

Find the first three terms in the expansion of (8 - 3x in ascending powers of x.

State the range of values of x for which the expansion in part (a) is valid.

Find the coefficient of x² in the expansion of .

The diagram shows triangle ABC in which angle A is 60° and the lengths of AB and AC are (4 + h) cm and (4 - h) cm respectively.

Show that the length of BC is p cm where

p² = 16 + 3h².

Hence show that, when h is small, p ≈ 4 + λ h² + μ h⁴, where λ and μ are rational numbers

whose values are to be determined.

In this question you should assume that - 1 < x < 1.

For the binomial expansion of (1 - x)^-2

find and simplify the first four terms,

write down the term in xⁿ.

Write down the sum to infinity of the series 1 + x + x² + x³ + ….

Hence or otherwise find and simplify an expression for 2 + 3x + 4x² + 5x³ + … in the form where a and b are constants to be determined.

In this question you must show detailed reasoning.

The probability that Paul’s train to work is late on any day is 0.15, independently of other days.

The number of days on which Paul’s train to work is late during a 450-day period is denoted by the random variable Y. Find a value of a such that P(Y > a) ≈

The number of days on which Paul’s train to work is late during a 50-day period is modelled by the random variable X.

Find the values of r for which P (X = r) ⩽ P (X = r + 1).

Hence find the most likely number of days on which the train will be late during a 50-day period.