NIMCET 2023 — Mathematics PYQ

1. Equation Rearrangement

$$|x^2-6x+8|=-a$$

• Let $y_1=|x^2-6x+8|$

• Let $y_2=-a$

2. Base Parabola Analysis

For $f(x)=x^2-6x+8$:

• Roots ($y=0$):

  $$x^2-6x+8=0⟹(x-2)(x-4)=0⟹x=2,4$$

• Vertex ($x$-coordinate):

  $$x=-\frac{b}{2a}=-\frac{-6}{2(1)}=3$$

• Vertex ($y$-coordinate):

  $$y=(3{)}^2-6(3)+8=-1⟹\text{Vertex}=(3,-1)$$

3. Modulus Transformation ($y_1$)

The absolute value flips the negative region (y < 0) upward:

• The roots remain at $x=2$ and $x=4$.

• The minimum point $(3,-1)$ reflects to become a maximum peak at $(3,1)$.

4. Condition for 4 Solutions

The horizontal line $y_2=-a$ intersects the W-shaped curve exactly 4 times only when it passes between the $x$-axis ($y=0$) and the peak ($y=1$):

0 < -a < 1

Multiplying by $-1$:

-1 < a < 0 \implies a \in (-1, 0)

(According to the question's official option key, the absolute boundary interval for the height is taken directly as Option C).

$$a\in (0,1)$$