Edexcel A Level Further Maths Paper 4A: Further Pure Mathematics 2 June 2019 question paper 9FM0/4A

A complex number z = x + i y is represented by the point P in an Argand diagram.

Given that

show that the locus of P has equation

Hence find the maximum value of | z |

The matrix A is given by

Show that 2 is a repeated eigenvalue of A and find the other eigenvalue.

Hence find three non-parallel eigenvectors of A.

Find a matrix P such that PAP is a diagonal matrix.

The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus k times the number of visits received in the month before that, where k is a positive constant.

Given that V_n is the number of visits to the website in month n,

write down a general recurrence relation for V_{n+ 2} in terms of V_{n+ 1}, V_n and k.

For a particular website you are given that

• k

• In month 1, there were 65 visits to the website.

• In month 2, there were 71 visits to the website.

Show that

This model predicts that the number of visits to this website will exceed one million for the first time in month N.

Find the value of N.

Use Fermat’s Little Theorem to find the least positive residue of 6 modulo 13

Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if

there are no restrictions where they sit in the row,

Alan, Enid, Felix and Graham sit together,

Brenda sits at one end of the row and Graham sits at the other end of the row,

Charles and Devindra do not sit together.

Prove that, for n ≥ 2

Hence show that

A binary operation * is defined on positive real numbers by

a * b = a + b + ab

Prove that the operation * is associative.

The set G = {1, 2, 3, 4, 5, 6} forms a group under the operation of multiplication modulo 7

Show that G is cyclic.

The set H = {1, 5, 7, 11, 13, 17} forms a group under the operation of multiplication modulo 18

List all the subgroups of H.

Describe an isomorphism between G and H.

A transformation from the z-plane to the w-plane is given by

Show that the circle C with equation | z + i | = 1 in the z-plane is mapped to a circle D in the w-plane, giving a Cartesian equation for D.

Sketch C and D on Argand diagrams.

Figure 1 shows the vertical cross section of a child’s spinning top. The point A is vertically above the point B and the height of the spinning top is 5 cm.

The line CD is perpendicular to AB such that CD is the maximum width of the spinning top.

The spinning top is modelled as the solid of revolution created when part of the curve with polar equation

LV URWDWHG WKURXJK UDGLDQV DERXW WKH LQLWLDO OLQH

Show that, according to the model, the surface area of the spinning top is

where k is a constant to be determined.