MAH-CET 2026 — Mathematics PYQ

Step 1: Understand the nature of the modulus function

The modulus function $|3-x|$ represents an absolute value, which denotes distance on a number line. By definition, the absolute value of any real expression is always non-negative. That means:

$$|3-x|\ge 0$$

Step 2: Analyze based on the value of $a$

• Case 1: When a < 0 (Negative value)

  An absolute value can never be equal to a negative number. For example, $|3-x|=-5$ has no real solution.

  Therefore, if a < 0, there is no solution, which is represented by the empty set:

  $$\varphi$$

• Case 2: When $a\ge 0$ (Non-negative value)

  If $a$ is greater than or equal to zero, we can remove the modulus sign by considering both positive and negative possibilities:

  $$3-x=\pm a$$

  Let's solve for $x$ in both scenarios:

  1. Taking the positive sign ($+a$):

     $$3-x=a⟹x=3-a$$

  2. Taking the negative sign ($-a$):

     $$3-x=-a⟹x=3+a$$

  Thus, when $a\ge 0$, the solution set contains the elements:

  $$\{3-a,3+a\}$$

Step 3: Combine both cases

Bringing both cases together, the complete solution set is $\{3-a,3+a\}$ for $a\ge 0$ and $\varphi$ for a < 0.

Correct Answer:

(a) $\{3-a,3+a\}$ for $a\ge 0$ and $\varphi$ for a < 0