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Question #66439

Difficulty: 8/10 | 🟢 Free

The Question

Let $t_n$ be a non-constant geometric progression of complex numbers with $t_1 = 1$ . If the system of linear equations in $x, y, z$ : $t_1 x + t_2 y + t_3 z = 0$ $t_1 x + i t_2 y - t_3 z = 0$ $t_1 x + t_3 y + t_5 z = 0$ has non-trivial solutions, find the value of $\sum_{n=1}^{40} n S_n$ , where $S_n = \sum_{k=1}^n t_k$ and $i = \sqrt{-1}$ .

Options

A.
$390 + 410i$
B.
$410 + 390i$
C.
$390 - 410i$
D.
$400 + 400i$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Cramer's Rule Constraints|Selection of GP Terms|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 application of Cramer's Rule for non-trivial solutions ( $D=0$ ) combined with complex number algebra.

The primary trap lies in solving the resulting complex quadratic equation $r^2(i-1) + 2r - (1+i) = 0$ .

Students often fail to correctly separate real and imaginary components, leading to incorrect roots for the common ratio $r$ .

Furthermore, once $r=i$ is found, the secondary cyclic trap requires recognizing the periodicity of $S_n$ (period 4) to evaluate the arithmetic-complex-geometric series.

Ignoring this periodicity leads to massive calculation errors.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Cramer's Rule Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free