cameroon gce A level June 2024 further mathematics 2
cameroon gce A level June 2024 further mathematics 2
(a) A similarity transformation (similitude) is defined by
z 1 = — 2i.
ii i)) Show Given that the object in the complex 2 in the diagram plane, thebelow invariant , point of the transformation is (0,^) –
5 .
copy it, and on the same axes sketch its image, z (3 marks)
( b) Show that the point with polar coordinates (r,0) is also the point with Cartesian coordinates
whose position vector is r — r cos6\ + r sin 0j.
Two points A and B lie on a plane which contains the origin 0. The position vector of A is
i) Find thei position + >/3j, vector while of theBpolar andcoordinates the polar coordinates (r, 8. Given that a sequence, Un , defined by
i) Find
ii) Given that v = n(u – 3), k R, prove that vfH.j = kun, and find \i and k.
iii) Find Vn in terms of nThe differential equation
d 2 y _ dy
—— + (> — + 9?/ = 4r;
dx2 dx
has general solution A c
i) Find p (x ).
ii)Given that y = 1, = 1 when x = 0 , find the values of A and B .
— ,\.r
-;i.r + Iixc wherep ( x ) is its particular integral
(3 marks)
(4 marks)
dx
Given that a group G has identity elemente and that a’ ] = a , a e G
i) Show that n~ = a
ii ) Construct the operation table for
// = ({1,5, 7,11} , X]9) , where xJ 2 is multiplication modulo 12.
2.
(2 marks)
(1 mark)
Hence or otherwise solve
iii) completely the equation x3 — x
iv) the system of congruences
5.r = 2( mocll 2)
x = 2( mod 7)
(2 marks)
(4 marks)
2
3. (a) Express/ ( x) x ^ — lin partial fractions. (3 marks)
( x + l)2 (x2 + l)
(b) The lines Lx and L.} are defined by
L
, : r = — 4i + 4 j + 7k + t ( 2i – j + k);
L2
: r = — 2i + j + 6k + s( 2i + j + k), t,s E M.
Find
i) the coordinates of the foot of the perpendicular from the point P (2, — 1,8) to L^
ii) the Cartesian equation of the plane containing and L2 .
(2 marks)
(3 marks)
4. A curve has equation y = 4\/x , x > 0.
i) Show that the length of the arc, S
,
of the curve for which 1 < x < 4 is given by
dx. (3 marks)
Using the substitution x = 4 sinh20 ,
ii) find S (leaving your answers in logarithmic form). (5 marks)
