Answer:
a) factor 
b) factor 
c) factor 
d) factor 
Explanation:
For an oscillating spring-mass system, the time period is expressed as:


where:
represents the frequency of oscillation
signifies the mass linked to the spring
is the spring's stiffness constant
a) If the mass is doubled:
- New mass,

Thus, the new time period:




this leads to factor
as per the question.
b) When the stiffness constant is quadrupled, holding other factors constant:
New stiffness constant, 
Thus, the new time period:

this results in factor
as required.
c) When both mass and stiffness constant are quadrupled:
New stiffness, 
New mass, 
Thus, the new time period:

which leads to factor
as stated in the question.
d) If amplitude is quadrupled, the time period remains unaffected because T does not depend on amplitude as demonstrated by the equation.
Thus, factor 