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Question #40500

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3$ be the roots of the equation $z^3 - (2+i)z^2 + (3+2i)z - 3i = 0$ . A line $L$ passes through the point $A(1, 0, 0)$ and has direction cosines $(l, m, n)$ which are proportional to $(\text{Re}(z_1), \text{Re}(z_2), \text{Re}(z_3))$ . A plane $P_1$ has a normal vector $\vec{n}_1$ whose direction ratios are proportional to $(\text{Im}(z_1), \text{Im}(z_2), \text{Im}(z_3))$ . If the line $L$ intersects the plane $P_2: x + y + z = 1 + \sqrt{2}$ at a point $B$ , and $\theta$ is the acute angle between the planes $P_1$ and $P_2$ , then the value of $15 (AB)^2 \cos^2 \theta$ is:

Options

A.
1
B.
2
C.
3
D.
5

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

Vieta’s Formulas in Complex Domain|Line-Plane intersection logic|Direction Cosines constraint|Angle between two planes

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of complex polynomial roots with 3D vector geometry, specifically targeting the normalization constraint of direction cosines and the phase-dependence of dot products.

A common trap is attempting to use Vieta's formulas to find the sum of squares of real/imaginary parts without finding the roots, which is algebraically intractable here because $\sum |z_i|^2$ cannot be isolated from symmetric sums alone.

This forces the student to identify the root $z=i$ .

Furthermore, students often fail to normalize the direction ratios $(0, 1, 1)$ into direction cosines $(0, 1/\sqrt{2}, 1/\sqrt{2})$ , leading to an incorrect parameter $r$ and distance $AB$ .

Finally, the angle calculation requires preserving the signs of the imaginary parts; ignoring the phase (signs) alters the dot product $\vec{n}_1 \cdot \vec{n}_2$ , destroying the invariant nature of the intersection.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Vieta’s Formulas in Complex Domain
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free