Question

Please answer the following
#10

Answer 1

(a+1) is the factor of the given equation by hit and trial method,
so , (a+1) =0
⇒a = -1

Now, putting the value in the given expression
(-1)²⁴⁵⁴-(-1)²⁴¹² +2 
=1-1 +2
=2

Answer 2

a⁵+a⁴+a³+a²+a+1=0
or, a⁵+a⁴+a³+a²+a=-1
or, a(a⁴+a³+a²+1)=-1
or, a⁴+a³+a²+a+1=-1/a
or, a⁴+a³+a²+a=-1/a-1
or, a(a³+a²+a+1)=-(1+a)/a
or, a³+a²+a+1=-(1+a)/a²
or, a³+a²+a=-(1+a)/a²-1
or, a(a²+a+1)=-{(1+a)/a²+1}
or, a(a²+a+1)=-(1+a+a²)/a²
or, a=-1/a²
or, a₃=-1
∴, $a^{2454}-a^{2412}+2$
=$( a^3)^{818} - (a^3)^{804} +2$
=$(-1)^{818} - (-1)^{804} +2$
=1-1+2
=2
∴, Answer: (4) 2