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Question #41758

Difficulty: 8/10 | 🟢 Free

The Question

A variable plane makes intercepts $a = \sin \theta$ , $b = \cos \theta$ , and $c = \sin 2\theta$ on the $x, y$ , and $z$ axes respectively, where $\theta \in [0, 2\pi)$ . If the plane passes through the point $\left(\frac{1}{4}, \frac{1}{4}, -\frac{1}{2}\right)$ , then the number of such distinct planes is:

Options

A.
0
B.
2
C.
3
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Variable plane intercepts|Parametric Form Manipulation|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to reject extraneous solutions that arise during parametric manipulation of 3D geometric equations.

The trap is set when the student converts the intercept form of the plane into a trigonometric equation.

Solving the quadratic in $u = \cos \theta + \sin \theta$ yields roots $u=1$ and $u=-1/2$ .

The root $u=1$ gives $\theta = 0, \pi/2$ , which mathematically satisfy the algebraic equation but geometrically force the intercepts to be zero.

A plane cannot have a zero intercept while maintaining the standard intercept form $x/a + y/b + z/c = 1$ (it would pass through the origin, making all unique intercepts undefined).

Students who blindly count the roots of the trigonometric equation will incorrectly answer 4, falling for the 'Extraneous Solution Rejection' trap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Variable plane intercepts
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free