A level north west regional mock gce 2022 further mathematics 3

A level north west regional mock gce 2022 further mathematics 3

A level north west regional mock gce 2022 further mathematics 3

Given the differential equation ( x + dy+ y ~ 2x= 0 ,
(i) Show that (.v+1) – +3 = 0
(tx ax’
(2 marks)
1
Given, also, that y=- when A- = 1
(ii) Show that the solution of the differential equation as series of ascending powers of ( x-1), up to
| 3 ]
and including the term in (*-l.)3, is j+ “(A’~ l) 4- “(-v- l)2
= 0.8
2/,1^’U
1
\ (*-D3 (4 marks)
(2 marks)
16
(iii) Find the value of y when *
Hence.
yn< i – y.
and a step length of 0.2, find the value of y when
(4 marks)
(3 marks)
(iv) Using tlie approximation
x = 1.4
(v) Estimate J ydx using Simps
dx ) «- i
on’s rule with 3 ordinates.
(work throughout with 4 decimal places)
2. (a) A particle moves with a constant angular velocity (0 round the curve with polar equation
r = o(l + 3cos0), <« > 0
Find, in terms of a,co and 9,the maximum speed of the particles.
(b) At time t, the radial and transverse components of the velocity of a particle are 6r and 2yr,
respectively. Given that the particle is initially at the point with polar coordinates ( 4,0 ),find the polar
equation of its path.
(6 marks)
(6 marks)
3. A smooth sphere A of mass 2m moving with velocity (3i T 4f )ms~1 impinges obliquely on a smooth
sphere B of mass 3m moving with velocity (2i — j)ms~ x.At the instant of collision, the line of centres
of A and B is parallel to the vector i .
2
Given that the coefficient of restitution between A and B is
— ,
(i) Find the velocities of A and B after collision.
(ii) Show that if A is deflected by the impact through an angle 9,then 11 ‘ tan 8 — 2
3
(7 marks)
(4 marks)
4. (a) A particle is performing simple harmonic motion between two points which are 8 metres apart.
When the particle is 2 metres from the centres from the centre of oscillations, its speed is y/3 mj’1,
Find :
(i) The period of motion of the particle. (4 marks)
(ii) The acceleration of the particle when it is -8 metres from the centre. (2 marks)
3
(b) A light elastic string of natural length 2a is fastened at one end to a fixed point. It hangs vertically
and carries at its other end, a particle of mass 3m. In the position of equilibrium, the length of the
. 8a
string is —.
Find :
(i) The modulus of elasticity of the string
(ii) The period of small vertical oscillations of the particle when it
equilibrium position and then released.

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