Q.38 Which of the following point(s) lies (lie) on the plane 2x + 3y + z = 6?
(A) (0, 0, 6)
(B) (0, 2, 0)
(C) (1, 1, 1)
(D) (3, 0, 0)
All four points (A, B, C, D) lie on the plane 2x + 3y + z = 6. This multiple-choice question tests the substitution method for verifying points on a plane equation.
Verification Method
A point (x₀, y₀, z₀) lies on the plane 2x + 3y + z = 6 if substituting these coordinates equals 6. This direct substitution confirms plane membership.
Option Analysis
Each option satisfies the equation exactly:
| Option | Point | Calculation | Result |
|---|---|---|---|
| (A) | (0, 0, 6) | 2(0) + 3(0) + 6 = 6 | Lies on plane |
| (B) | (0, 2, 0) | 2(0) + 3(2) + 0 = 6 | Lies on plane |
| (C) | (1, 1, 1) | 2(1) + 3(1) + 1 = 6 | Lies on plane |
| (D) | (3, 0, 0) | 2(3) + 3(0) + 0 = 6 | Lies on plane |
Plane Intercepts
The plane intersects axes at x=3, y=2, z=6, matching options (D), (B), (A) as intercept points. Option (C) lies interior to these intercepts.
