UPSC CDS 1 2023 Math Paper

Concept:A quadratic expression is a perfect square when its discriminant is zero.This means it has a repeated root.The discriminant formula for $ax^2+bx+c$ is $b^2-4ac$.Explanation:The given expression is $mx^2+mx+8x+9$.Combine the $x$ terms to get standard form: $mx^2+(m+8)x+9$.Here, $a=m$, $b=m+8$, and $c=9$.For a perfect square, set the discriminant to zero:$b^2-4ac=0$Substitute the values:$(m+8)^2-4\cdot m \cdot 9=0$Expand $(m+8)^2$ using the formula $(a+b)^2=a^2+2ab+b^2$:$m^2+16m+64-36m=0$Simplify:$m^2-20m+64=0$Factor the quadratic:$m^2-16m-4m+64=0$$m(m-16)-4(m-16)=0$$(m-4)(m-16)=0$Thus, $m-4=0$ or $m-16=0$.So $m=4$ or $m=16$.Answer:The values of $m$ are $4$ and $16$.This matches option D: 4, 16.