RRB Group D 27 Nov 2025 Shift 3 Solved Paper

Concept:The given equation $x^{2}+1=2x$ can be used to find $x^{4}-1$ either by algebraic manipulation or by solving for $x$.Explanation:Start with $x^{2}+1=2x$.Rearrange to get $x^{2}=2x-1$.Square both sides: $(x^{2})^{2} = (2x-1)^{2}$.This gives $x^{4} = 4x^{2} - 4x + 1$.Now substitute $x^{2}=2x-1$ into the expression for $x^{4}$:$x^{4} = 4(2x-1) - 4x + 1$.Simplify: $4(2x-1)=8x-4$, so $x^{4} = 8x-4 -4x +1 = 4x -3$.Then $x^{4} -1 = (4x-3)-1 = 4x-4$.Alternatively, notice that the original equation $x^{2}+1=2x$ can be written as $x^{2}-2x+1=0$, which is $(x-1)^{2}=0$.Thus $x=1$ is the only solution.Substituting $x=1$ gives $x^{4}-1 = 1^{4}-1 = 0$.Both methods show that $x^{4}-1$ equals $0$.Answer:Option B. 0.