Q.64 If |−2X + 9| = 3 then the possible value of |−X| − X2 would be:
- (A) 30
- (B) −30
- (C) −42
- (D) 42
The equation |−2X + 9| = 3 yields two solutions: X = 3 and X = 6, both satisfying the original equation. Evaluating |−X| − X² at these points gives -6 and -30, respectively, so -30 matches option (B). None of the other options arise from these valid solutions.
Solving the Equation
The absolute value equation |−2X + 9| = 3 splits into two cases:
- Case 1: −2X + 9 = 3 → −2X = −6 → X = 3
Verification:|−2(3) + 9| = |3| = 3 ✓ - Case 2: −2X + 9 = −3 → −2X = −12 → X = 6
Verification:|−2(6) + 9| = |−3| = 3 ✓
Evaluating Expression
| X Value | |−X| | X² | |−X| − X² |
|---|---|---|---|
| X = 3 | 3 | 9 | -6 |
| X = 6 | 6 | 36 | -30 ✓ |
Option Analysis
(A) 30
Positive value mismatches both −6 and −30; no solution yields this.
(B) −30 ✓
Matches exactly at X = 6.
(C) −42
Exceeds magnitudes; for example, |−X| − X² = −42 implies X² − |X| = 42, unsolved by X=3 or 6.
(D) 42
Positive, inconsistent with quadratic dominance for real X ≥ 0.
