Question 14:
--- 14(a) ---
14(a) | Forms general force
equation (at least three
terms) with at least two
terms correct (allow
equivalent notation for
derivatives – condone a
and v). | AO3.1b | M1 | 20 
9m −x −mg −6mx = mx
 9 
x+6x+9x =10
λ2 +6λ+9 =0
λ= −3 ( twice )
CF:
x = Ae−3t + Bte−3t
PI:
10
x =
9
General Solution:
10
x= Ae−3t +Bte−3t +
9
−10
x=0,t =0⇒ A=
9
x =−3Ae−3t +Be−3t −3Bte−3t
0=−3A+B
30
B =−
9
10 10 10
x = − e−3t − te−3t +
9 3 9
Obtains fully correct
general force equation &
cancels down into 2nd order
DE form (allow equivalent
notation for derivatives). | AO1.1b | A1
Obtains correct solution to
their Auxiliary Equation | AO1.1a | M1
Obtains their correct RHS
of Complementary Function | AO1.1b | A1F
Obtains their correct (non-
zero) Particular Integral | AO1.1b | B1F
Obtains correct RHS of
General Solution
(ft their CF & non-zero PI,
but must have two
unknowns) | AO2.2a | A1F
Uses x=0 when t =0 to
obtain correctA | AO1.1b | B1
Sets their correct x =0
when t =0 | AO3.3 | M1
Obtains correct B – can
be unsimplified. | AO1.1b | A1
Obtains correct final
equation – can be
unsimplified. | AO2.1 | R1
--- 14(b) ---
14(b) | States critical damping
because the Auxiliary
Equation has equal roots
(or equivalent) | AO1.2 | B1 | Critical damping, because the
Auxiliary Equation has equal
roots
Total | 11
Q | Marking Instructions | AO | Marks | Typical Solution