what should be the length of aluminum and a brass rods at STP. so that brass rod is longer than the aluminum rod by 4 cm at all conditions of temperature

Answer 1

Aluminum's thermal coefficient of expansion is about 25% higher than that of brass. See
http://hyperphysics.phy-astr.gsu.edu/hbase/tables/thexp.html
for verification.

Let L1' = Al rod length at 0 C, in cm
Let L2' = Brass rod length at 0 C, in cm
T = rod temperature in deg. C
a = thermal expansion coefficient of brass.

L1' - L2' = 4
At any temperature T,
L1' + 1.25*a*L1*(T-0) - L2' - a*L2(T-0)
= L1' - L2' -(0.25*a*L1*T - L2*a*T)
= 4 at any T, if 0.25 L1 = L2