NIMCET 2010 — Mathematics PYQ

1. Simplify the value of $X$:

Given $X=-5+2\sqrt{-4}$.

Since $\sqrt{-4}=\sqrt{4}\cdot \sqrt{-1}=2i$, we have:

2. Form a quadratic equation from $X$:

To avoid calculating high powers of a complex number directly, we create a quadratic factor:

Squaring both sides:

Since $i^2=-1$:

3. Divide the polynomial by the quadratic factor:

We now divide the given polynomial $P(X)=X^4+9X^3+35X^2-X+4$ by $(X^2+10X+41)$ using long division:

• Step 1: $(X^4+9X^3+35X^2)÷(X^2+10X+41)$ gives $X^2$.

  • Subtract $X^2(X^2+10X+41)=X^4+10X^3+41X^2$.

  • Remainder: $-X^3-6X^2-X+4$.

• Step 2: $(-X^3-6X^2-X)÷(X^2+10X+41)$ gives $-X$.

  • Subtract $-X(X^2+10X+41)=-X^3-10X^2-41X$.

  • Remainder: $4X^2+40X+4$.

• Step 3: $(4X^2+40X+4)÷(X^2+10X+41)$ gives $+4$.

  • Subtract $4(X^2+10X+41)=4X^2+40X+164$.

  • Final Remainder: $4-164=-160$.

4. Conclusion:

The polynomial can be written as:

Substituting the value of $X$ where $X^2+10X+41=0$:

Correct Option:

(b) $-160$