cameroon gce A level June 2023 further mathematics 3
cameroon gce A level June 2023 further mathematics 3
Aiforce ) FindFthe ,—ofwork magnitude doneVby \.^)Fn, in *,ADi Ndisplacing , acts ; 03 alongthe a*particle oline aiffrom ^ [firh the point wherejA Ja)2=>jim04kto)3Jm the c3 point
where a = 2. ? P.HT/.Ifl Vi’/iUT*4(6 marks)
The forces F
2 and F3 act at points with position vectors r2 and r3 respectively, where
1 ‘ M
F i ‘ > < » 2 = (“i + j)N > r2 = .(j + k)nJ.
F, = (-i 4- 4k)N, r3 = (i + j + k)iri
The system of forces Fx , F2 and F3 is equivalent to a single force, ll „ acting through the
origin O together with a couple G .
Show that
i * ” ‘ )V
tji!:i.r/ *:->11 j j« j ; -jf
» J‘J) . . j
I
1 icm.1!
ji
t it
VilO’
(2 marks)
(6 marks)
ii) R- (4i-3J- 4k)N / . 2 3 ^ >’ j f >! * d, I
iii) the magnitude of the couple G is 3>/l3 Nrn . t: } f :>. o
2. (a) The coordinates of points on the curve y = f ( x ) are ( x,ij ) where x and y are given by
the table below:
x 0 5 10 15 20 25 30
y 0 9 14 18 22 23 19
• \
Use Simpson’s rule with 6 strips to estimate the area enclosed by the curve y = f ( x) the x —
axis and the ordinates at x = 0 and x = 30.
(b) Given that
s . (4 marks)
d v .
„2
— f x — y = 0 and that y = 0 when x = 1, • ! *. * . ?
d x
find the first three non ~ zero terms in the Taylor series expansion of y at x =T. (6 marks)
3. A particle P, of mass 2 kg, moves in straight horizontal line with simple harmonic motion with
centre O. When P is at a point distant 1.5 metres from O, its speed is 4 metres per second and
its acceleration is —6ms-2
Find
the period of motion.
ii) the amplitude of the motion.
iii) the maximum speed of P.
iv) the maximum force acting on P.
i) (4 marks)
(3 marks)
(2 marks)
(3 marks)
4. A car of mass 1500kg starts from rest and moves along a straight level road surface with its
engine working at the rate of 6000 v , where v is the velocity of the car at time t . Given that
the total resistance to the motion of this car when traveling at speed v ms 1 is
GOOOv
where u is a constant,
u
show that
i) u is the maximum speed of the car along this road,
ii) the equation of motion of this car is
4 1 d v
(3 marks)
• *IY iw’\ (2 marks)
u u — v d t
Hence, find
iii) the time taken for the speed of the car to be raised from 0 to 4 u .
iv) the distance travelled by the car in this time.
(4 marks)
(3 marks
