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Question #41639

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_n$ be a family of planes given by the equation $x + y + z = d_n$ , where $d_n$ is the coefficient of $t^2$ in the polynomial $(1+t)^{n+3} - (1+t)^{n+1}$ . A line $L$ is given by the symmetric equations $\frac{x - 2023}{1} = \frac{y - 2024}{1} = \frac{z - 2025}{1}$ . Let $Q_n(x_n, y_n, z_n)$ be the unique point of intersection of the plane $P_n$ and the line $L$ . Find the value of $\sum_{n=1}^{8} (x_n + y_n + z_n)$ .

Options

A.
96
B.
83
C.
54
D.
100

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

Parametric form pitfalls|Variable plane intercepts|The Telescopic Series Trap|Binomial Coefficient Summation Properties

Why This Question Is Tricky

Common Mistakes

This question tests students on two distinct JEE Advanced traps.

First, the 'Red Herring Trap': students often waste time parameterizing the line $L$ and solving for the intersection point, failing to realize that the sum of the coordinates is directly given by the plane's equation.

Second, the 'Telescopic Series Index Shift Trap': the summation $\sum (A_{n+2} - A_n)$ has an index shift of 2, meaning two terms survive at both the upper and lower bounds ( $A_{10} + A_9 - A_2 - A_1$ ).

Students blindly applying the standard $f(n) - f(n-1)$ telescopic property will incorrectly cancel terms, leading to distractors like 54 (computing $A_{10} - A_1$ ) or 100 (forgetting the lower bounds).

The value 83 serves as a secondary distractor for arithmetic miscalculations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Parametric form pitfalls
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free