UPSC CDS II 2025 Mathematics Paper

Concept:If (x + 2) is a factor of a polynomial, then substituting x = –2 must give zero (Factor Theorem).Explanation:Evaluate each polynomial at x = –2 and check which result is zero.Option A: $(-2)^5 - 4(-2)^4 - 3(-2)^3 + 8(-2)^2 - 14(-2) + 12$ = -32 - 64 + 24 + 32 + 28 + 12 = 0.Option B: $(-2)^5 + 4(-2)^4 - 3(-2)^3 + 8(-2)^2 - 14(-2) + 12$ = -32 + 64 + 24 + 32 + 28 + 12 = 128 (not zero).Option C: $(-2)^5 - 4(-2)^4 + 3(-2)^3 + 8(-2)^2 - 14(-2) + 12$ = -32 - 64 - 24 + 32 + 28 + 12 = -48 (not zero).Option D: $(-2)^5 - 4(-2)^4 - 3(-2)^3 + 8(-2)^2 + 14(-2) + 12$ = -32 - 64 + 24 + 32 - 28 + 12 = -56 (not zero).Only Option A gives zero.Answer:Option A: $x^{5} - 4x^{4} - 3x^{3} + 8x^{2} - 14x + 12$