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Question #40886

Difficulty: 8/10 | 🟢 Free

The Question

Let $L(t)$ be a variable line passing through the point $A(0, 2, 1)$ and having direction vector $\vec{d} = (6+5t-t^2)\hat{i} + \hat{j} + (t^2-6t+8)\hat{k}$ , where $t \in \mathbb{R}$ . Let $P(t)$ be the point of intersection of the line $L(t)$ with the $xy$ -plane. If $S$ is the set of all integer values of $t$ for which the algebraic projection of the vector $\vec{AP(t)}$ on the $x$ -axis is non-positive, then find the sum of the elements of $S$ .

Options

A.
11
B.
17
C.
20
D.
12

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

Projection vs. Coordinate|Parametric Form Manipulation|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of 3D parametric line equations with rational inequalities.

The core trap lies in the 'Wavy Curve' method (Method of Intervals).

When students project the vector onto the x-axis, they obtain a rational function of the parameter 't'.

A common mistake is to cross-multiply or ignore the domain constraints of the denominator, leading to the inclusion of the critical points t=2 and t=4 (phantom solutions).

Geometrically, at these values, the line is parallel to the xy-plane and never intersects it, making the projection undefined.

Failing to exclude these points yields an incorrect sum of 17, while completely ignoring the denominator's effect on the sign yields 20.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Projection vs. Coordinate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free