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Question #76393

Difficulty: 8/10 | 🟢 Free

The Question

Let $l$ and $m$ be two distinct real numbers that satisfy the equation $x^2 - \left(S_k + i\left(\sum_{r=1}^k r - 15\right)\right)x + \left(P_k + i(k^2 - 25)\right) = 0$ where $k \in \mathbb{Z}^+$ , $S_k = \frac{2}{k}$ , and $P_k = \frac{1-k}{k^2}$ . If $l, m, n$ are the direction cosines of a line $L$ in 3D space, find the value of $625(l^4 + m^4 + n^4)$ .

Options

A.
281
B.
269
C.
312
D.
225

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Direction Cosines constraint|Quadratic Roots in Sequences|Algebraic Manipulation of i

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the student's ability to reconcile complex polynomials with real geometric constraints.

The primary trap is ignoring the fact that a linear equation (representing the imaginary part) having two distinct real roots implies its coefficients must identically vanish.

Students often incorrectly attempt to apply the quadratic formula directly to the complex coefficients or assume the roots are complex conjugates, missing the geometric constraint that direction cosines must be strictly real.

By forcing the imaginary components to zero, the sequence parameter $k$ is uniquely determined, unlocking the sum-of-squares constraint for the direction cosines.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Direction Cosines constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free