cameroon gce A level June 2023 further mathematics 2
cameroon gce A level June 2023 further mathematics 2
a) Use the substitution toy == v, where v
x2 — + xi) = 2 (a-2?/2 + 1) .
da: v ‘
(2 marks)
5. > f Is;/ M rOl ‘ iV lOflT/. M
into a differential equation involving v and x only.
Hence, find the general solution of this differential equation in terms of x and y.
b) Given that /loos 4®/+v B,sin 4a: is a particular solution of the differential equation ^
– ~ 4
(3 marks)
‘
V :/!i: ! VJ > I
?
<}— d 2 y + 9j/ = 14 sin 4rr #•* rji .. jt ‘t <•1! <ni
i . … dx~ v 1*1« vro i ” / « ‘JUO f ;
(2 marks)
(3 marks)
Find the values of the real constants Aand B.
Hence, find the general solution of the differential equation.
2. Given two vectors
* •
a = x i — 2j + k
b = i + to ~ k,
a
x
, ye Z ,
and that axb = i + 4j+ 5k.
i) Calculate the values of the real constants, x and y . (3 marks)
ii) Show that a and b are linearly independent. (2 marks)
iii) Find the Cartesian equation of the plane that contains a and b and passes through the point with
position vector j + k. (2 marks)
3. Prove that
1+ 4a:
1 — 4a:/
Hence, or otherwise, solve the equation tanh-1 4a: — In 2 = 0.
tanh 1 4a: = ] In
• ^
* < 71 – (3 marks) (3 marks) 4. Given that 3a:2 + 10a: -f- 9 /(*) > x ^ —2 ,
( a: + 2)4
Use the substitution u = x + 2, or otherwise, to express f ( x ) in partial fractions.
Hence, show that
(4 marks)
o 25
x \ a x — — (4 marks)
-l 24
5. a) Show that the set of matrices of the form
x y )
, where x ^ 0, and x, y R
-y X
forms a group under matrix multiplication. ( Assume Associativity).
Go on to the next page
)
(5 marks)
