UPSC CDS 2 2022 Math Paper

Concept:The given expression simplifies to $x^2$ by factoring and algebraic expansion.Explanation:Step 1: Write the numerator as $x^2(x^4 - 24x^2 + 144)$.Step 2: Expand $(x - 2\sqrt{3})^2 = x^2 + 12 - 4\sqrt{3}x$.Step 3: Multiply the two denominator factors: $(x^2 + 4\sqrt{3}x + 12)(x^2 + 12 - 4\sqrt{3}x)$.Step 4: This is of the form $(A + B)(A - B)$ with $A = x^2 + 12$ and $B = 4\sqrt{3}x$.Step 5: The product becomes $(x^2 + 12)^2 - (4\sqrt{3}x)^2 = (x^4 + 24x^2 + 144) - (48x^2)$.Step 6: Simplify: $x^4 + 24x^2 + 144 - 48x^2 = x^4 - 24x^2 + 144$.Step 7: The denominator now equals $x^4 - 24x^2 + 144$.Step 8: Cancel the common factor $(x^4 - 24x^2 + 144)$ from numerator and denominator.Step 9: The final result is $x^2$.Answer:$x^2$