Cambridge International AS and A Level Mathematics Pure Mathematics 1

Sequences and series P1^ 3 10 The first three terms of a geometric sequence are 100, 90 and 81. (i) Write down the common ratio ...

Exercise 3B P1^ 3 16 A pendulum is set swinging. Its first oscillation is through an angle of 30°, and each succeeding oscillati ...

Sequences and series P1^ 3 22 The 1st term of an arithmetic progression is a and the common difference is d, where d ≠ 0. (i) Wr ...

Binomial expansions 95 P1^ 3 Binomial expansions A special type of series is produced when a binomial (i.e. two-part) expression ...

Sequences and series P1^ 3 ExamPlE 3.12 Write out the binomial expansion of (2a − 3 b)^5. SOlUTION The binomial coefficients for ...

Binomial expansions P1^ 3 and so the expansion is x^10 + 10 x^9 y + 45 x^8 y^2 + 120 x^7 y^3 + 210 x^6 y^4 + 252 x^5 y^5 + 210 x ...

Sequences and series 98 P1^ 3 aCTIvITy 3.1 The table shows an alternative way of laying out Pascal’s triangle. Column (r) 0 1 2 ...

Binomial expansions P1^ 3 SOlUTION (i) 5 0 5 05 0 120 1 120 1     = !( −! )!= × = (ii) 5 1 5 14 120 124  5    == × ! ...

Sequences and series P1^ 3 The expansion of (1 + x)n When deriving the result for n r     you found the binomial coefficie ...

Binomial expansions P1^ 3 ExamPlE 3.18 The first three terms in the expansion of ()ax+bx 6 where a  0, in descending powers of ...

Sequences and series 102 P1^ 3 Adding terms You have seen that each term in Pascal’s triangle is formed by adding the two above ...

Exercise 3C P1^ 3 ExERCISE 3C 1 Write out the following binomial expansions. (i) (x + 1)^4 (ii) (1 + x)^7 (iii) (x + 2)^5 (iv) ( ...

Sequences and series P1^ 3 10 (i) Find the first three terms, in descending powers of x, in the expansion of (^) ()x−x^2 5 . (ii ...

Key points P1^ 3 Cubes A cube is painted red. It is then cut up into a number of identical cubes, as in figure 3.5. How many of ...

Functions 106 P1^ 4 Functions Still glides the stream and shall forever glide;  The form remains, the function never dies. Willi ...

The language of functions 107 P1^ 4 Mappings In mathematics, many (but not all) mappings can be expressed using algebra. Here ar ...

Functions P1^ 4 ●?^ For each of the examples above: (i) decide whether the mapping is one-to-one, many-to-many, one-to-many or m ...

The language of functions P1^ 4 ExaMPlE 4.1  Sketch the graph of y = 3 x + 2 when the domain of x is (i) x ∈  (ii) x ∈ + (iii) ...

Functions P1^ 4 Figure 4.3 illustrates some different types of mapping. The graphs in (a) and (b) illustrate functions, those in ...

Exercise (^4) a 111 P1^ 4 2  For each of the following mappings: (a) write down a few examples of inputs and corresponding outpu ...