A level south west regional mock gce 2022 further mathematics 2
A level south west regional mock gce 2022 further mathematics 2
1. The variables x and y are functions of t and satisfy the differential equations —dx + x = 0.
t 2x -y and
a) Show that~atL + 2 3at^+ x — 0
b) Find the general solution for x, of the differential equation in (a) above, and deduce from
(1mark)
dx
— + 2x = y the general solution for . (4 marks)
c) Hence or otherwise, find x and y in terms of t, given that x = 1and = 0, when t = 0. (2 marks)
2. a) Show that the set M of all matrices of the form (J ,n e TL , forms an Abelian group under
multiplication of matrices. (Assume associativity )
b) Given that (x) =
(4 marks)
x3
(i) expressf (x ) in partial fractions (3 marks)
(x2+2)3′
(ii) hence, show thatf*f (x )dx =^ (3 marks)
3. a) Prove by mathematical induction that V n G 2+, £?=i(r2 +l)r! = n(.n +!)•
b) (i) Find the gcd of 54 and 21and express it in the form d = 54x + 21y.
(ii) Hence, solve the linear congruence 54x = 12(moof21)
4i a). Solve forx and y the equations coshx = 3sinhy and 2sinhx = 5 — 6coshy, expressing your
answers in logarithmic form..
b) Given that 7″2 =
(4 marks)
225
is the equation of an ellipse, find
25-16sin26
)
(i) the Cartesian equation of the ellipse
(ii) the eccentricity of the ellipse
(iii) the coordinates of the foci of the ellipse
(iv) the equations of the directrices of the ellipse
(v) the equations of the asymptotes of the ellipse
