@acer sorry my bad, I meant to write Bi>0. do you think that this assumption may lead to invalid results too?

@acer Bi=1. is a transition point, So dividing by Bi-1 would change the final result but I will try.

what about using (assume(Bi>1), do you think it's more correct than just dividing by (Bi-1)? I tried this and replaced Bi=1 after computing the discriminant and did get the imaginary and real components.

by the way, thank you for your help, I really appreciate it.

@acer is it possible to isolate the expression (Bi-1) or remove it from both imaginary and real components of the determinant?

@dharr for the other values of Bi, the determinant is different from zero but for the case Bi=1 it doesn't work.

The only thing I noticed is that the 3 first rows of the matrix misses values in the 5th and 6 the column.

thank you @tomleslie .it's the second times you help me, I really appreciate.

but I'm concerned about the values of wi they are seems too small, conventionaly in the literature it's range between [0,1]. but in this case it's unrealistically small!!!  

 @tomleslie thanks, you helped me a lot .  it's exactly what i was looking for.

and it did work for the other expressions too.

thank you agais

@radaar @dharr  thank you so much for your help guys, a really appreciate. 

thank you very much @tomleslie , I appreciate your help , I was trying to find a way to find the value for wich this set of equation doesn't not have a value ,your method is much easier.

did you have any idea why , there is a failure for i=3???

@Carl Love  i tried to solve the set of equations eqns with a different method , without a loop point by point for different value of R . and i found thar for the case before (first one i showed you ) where i took pi/3 or smaller value

i don't get a solution for the R=2 , maple give me the same answer as before.

 Error, invalid input: subs received fsolve({-0.3994442905e1...

thank you @Carl Love  for your reply

so here is the value of eqns

EQ1:=-0.3658176771e158*R^5*k^6+0.1197625584e156*R^7*k^6-0.2307441257e157*R^6*k^6+
0.2373683399e149*k^10*R^6-0.4604984942e150*k^8*R^9+0.5397454183e141*k^10*R^5-0.9681081748e151*k^10*R^7+
0.1055536715e152*k^10*R^8+0.7077378706e159*R^4*k^6-0.2632724142e160*R^2*k^2-0.3832933894e159*k^6*R^3+
0.1254499089e155*k^8*R^4+0.7705342889e156*k^8*R^5+0.4189228678e154*k^8*R^7+0.2157017583e152*k^8*R^8-
0.1110433302e157*k^8*R^6-0.1897793285e159*R^5*k^4+0.2795058995e160*R^3*k^2-0.1283566031e160*k^4*R^2+
0.5996226867e159*k^4*R^3+0.1586468565e160*R^4*k^4+0.5637311169e155*R^7*wr^6-0.8291507503e150*R^9*wr^8+
0.7211072458e159*R^3*wr^2-0.4696292176e158*R^5*wr^4+0.2303210148e159*R^2*k*wr+
0.1153376262e159*R^5*k^5*wr+0.5962564806e157*R^6*k^3*wr^3+0.1307059178e157*R^7*k^2*wr^4+
0.1722336636e157*R^7*k^4*wr^2-0.2047974585e157*R^7*k^3*wr^3-0.7294860071e156*R^7*k^5*wr+
0.8214745295e157*R^6*k^5*wr-0.1059726315e158*R^6*k^4*wr^2-0.4277124681e156*R^7*k*wr^5-
0.1317710426e159*R^5*k^4*wr^2-0.1315979188e157*R^6*k^2*wr^4+0.6507547922e158*k^3*R^5*wr^3-
0.9512005739e159*R^4*k^5*wr+0.2245188019e152*k^2*R^8*wr^6-0.2037523291e157*k^7*R^5*wr+
0.2544724043e157*R^6*k^5*wr^3-0.1324018988e153*k^3*R^8*wr^5+0.3666875159e153*k^6*R^8*wr^2-
0.4840355193e153*k^5*R^8*wr^3+0.3483577352e153*k^4*R^8*wr^4+0.4502364820e155*R^7*k^4*wr^4+
0.5780346230e155*R^7*k^6*wr^2-0.6927498035e155*R^7*k^5*wr^3-0.2468865010e155*R^7*k^7*wr-
0.1417906765e153*R^8*k^7*wr+0.3850099151e157*k^7*R^6*wr-0.4789549199e157*R^6*k^6*wr^2-
0.1509711711e155*R^7*k^3*wr^5+0.2093106269e159*R^3*k^5*wr+0.5019126789e155*R^6*k*wr^5-
0.1174377962e158*R^5*k^2*wr^4-0.9776216355e157*R^4*k^3*wr^3-0.8291506292e150*k*R^8*wr^7+
0.2045567038e154*R^7*k^2*wr^6+0.7535691845e151*R^9*k*wr^7+0.1959077906e154*R^6*k^3*wr^5-
0.2886958381e152*k^2*R^9*wr^6+0.6052482292e152*k^3*R^9*wr^5-0.2432511261e152*k^6*R^9*wr^2+
0.5633489560e152*k^5*R^9*wr^3-0.7532479920e152*k^4*R^9*wr^4+0.5414451228e151*k^7*R^9*wr+
0.7948362383e151*k^4*R^8*wr^6-0.2695096756e141*k^9*R^5*wr+0.1810724765e150*k^7*R^6*wr^3-
0.5560322553e152*k^5*R^8*wr^5+0.1742863886e153*k^8*R^8*wr^2-0.2248251357e153*R^8*k^7*wr^3+
0.1561623558e153*k^6*R^8*wr^4-0.4585372774e152*R^7*k^6*wr^4-0.1050103036e153*k^8*R^7*wr^2+
0.1006564542e153*R^7*k^7*wr^3+0.5193168841e152*k^9*R^7*wr-0.6851720501e152*k^9*R^8*wr+
0.1329441588e150*k^9*R^6*wr-0.4333530826e150*k^8*R^6*wr^2+0.7948361223e151*R^7*k^5*wr^5-
0.4241471530e156*R^5*k^5*wr^3-0.6487699529e154*k^7*R^4*wr-0.4912359992e156*R^6*k^4*wr^4+
0.1671701197e157*R^5*k^6*wr^2+0.3251512287e159*R^4*k^4*wr^2-0.2895955940e160*R^3*k*wr+
0.5691329808e159*k^3*R^5*wr-0.6106530531e159*R^5*k^2*wr^2+0.2805756161e159*R^5*k*wr^3+
0.2252707208e159*k^3*R^2*wr-0.2175920319e160*R^4*k^3*wr+0.7783586793e159*R^4*k^2*wr^2-
0.3596902417e158*R^4*k*wr^3-0.1381534162e160*R^3*k^3*wr+0.5890704144e159*R^3*k^2*wr^2:




EQ2:=0.9944277974e152*k^9*R^5+0.7051551189e151*k^9*R^8+0.5376138054e154*k^9*R^6+0.5031295700e153*R^8*k^7+
0.1401428815e160*R^4*k^3+0.3677843691e159*R^2*k^5-0.6487714504e154*k^9*R^7-0.3109455243e158*k^7*R^4-
0.1302056540e155*k^7*R^3-0.5588713615e141*k^9*R^4-0.8271328158e157*R^6*k^5-0.1597777164e160*R^2*k+
0.5382020885e159*k^5*R^4-0.1359023474e155*R^7*k^7-0.1914676454e160*R^3*k^5-0.4062170891e160*R^3*k^3-
0.794066920e158*k^3*R^2+0.4933165952e158*k^7*R^5-0.7273796239e156*k^7*R^6+0.1068159898e159*k^5*R^5+
0.2657232919e157*R^6*wr^5+0.5961249821e159*R^2*wr-0.2955900422e159*R^4*wr^3-0.4079394413e153*R^8*wr^7+
0.3414082691e154*k*R^8*wr^6+0.2179911892e155*R^8*k^3*wr^4+0.1366986665e155*R^8*k^5*wr^2+
0.4732243206e158*R^6*k^2*wr^3+0.2228082156e157*R^5*k*wr^4+0.3614336273e158*R^6*k^4*wr+
0.1522393478e160*R^4*k*wr^2+0.1543513617e153*k^7*R^8*wr^2+0.2295277755e153*k^5*R^8*wr^4+
0.5383648276e155*k^6*R^7*wr^3+0.3249361247e155*k^8*R^7*wr-0.2296637021e155*R^8*k^4*wr^3-
0.4194825142e154*R^8*k^6*wr-0.1181670691e155*R^8*k^2*wr^5-0.1762948993e159*R^3*k*wr^2-
0.5982225503e158*R^6*k^3*wr^2-0.1801905024e158*R^6*k*wr^4-0.2564678017e160*R^4*k^2*wr-
0.1731896242e155*k^6*R^6*wr^3-0.1240235794e153*k^4*R^8*wr^5-0.2470933955e153*k^6*R^8*wr^3-
0.2282537590e155*k^5*R^7*wr^4-0.6073545976e155*k^7*R^7*wr^2-0.5157959628e152*k^8*R^8*wr-
0.2151213452e155*k^8*R^6*wr-0.2120181539e158*k^4*R^5*wr^3-0.1275241440e159*R^3*k^3*wr^2-
0.3916123799e153*R^7*k*wr^6-0.4392250447e151*k^2*R^8*wr^7-0.4392249806e151*k^3*R^7*wr^6-
0.1656169872e158*k^5*R^4*wr^2-0.1717265504e159*R^4*k^2*wr^3-0.2748228814e153*k^8*R^5*wr+
0.7568341634e159*k^3*R^4*wr^2+0.4430597297e157*k^4*R^6*wr^3+0.3198928829e157*k^6*R^6*wr-
0.5412849994e155*k^3*R^7*wr^4-0.1313203543e156*R^7*k^5*wr^2-0.5478310765e158*R^5*k^2*wr^3-
0.2591739233e159*R^5*k^4*wr-0.1158828050e159*k^6*R^5*wr-0.1755138554e157*k^3*R^6*wr^4+
0.8686806621e158*k^5*R^5*wr^2-0.5417633870e157*k^5*R^6*wr^2+0.1032425127e155*k^2*R^7*wr^5+
0.2766178950e160*k^2*R^3*wr+0.2031346932e159*k^3*R^5*wr^2-0.1101688848e160*k^4*R^4*wr+
0.1202413246e156*k^4*R^7*wr^3+0.6887637726e155*k^6*R^7*wr+0.2990262050e155*k^7*R^6*wr^2+
0.3727248780e154*k^4*R^7*wr^5+0.4816850327e158*k^6*R^4*wr+0.5830536329e159*k^2*R^2*wr+
0.2700127319e156*R^6*k^2*wr^5+0.2395382093e156*k^3*R^5*wr^4+0.3539940642e154*k^5*R^6*wr^4+
0.1132352362e153*k^7*R^5*wr^2+0.1384788790e160*R^3*k^4*wr+0.3615542175e152*k^3*R^8*wr^6:

thank you for your help @Carl Love  i did us you said 

Failures:= table():
 i from 1 by 1 to 101 do
R:=(i-1):
S:=fsolve(eqns,{k, wr},{wr=0..10,k=0..10}):
if eval(S,1)::specfunc(fsolve) then
Failures[i]:= eval(S,1)
else
 w(i):= eval(k, S)
end if
end do:
eval(Failures);

but i get this error messaage

Error, invalid input: eval received (k = k, wr = wr), which is not valid for its 2nd argument, eqns

ps: sorry if i did something wrong,but I'm a beginner with maple .