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Question #40714

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 - px + q = 0$ , where $p = \lim_{x \to 0} \frac{\ln(1+4x)}{x}$ and $q = \lim_{x \to 0} \frac{1 - \cos(2\sqrt{2}x)}{x^2}$ . A sequence $\{t_r\}$ is defined for $r \in \mathbb{N}$ as $t_r = \sum_{j=0}^{r-1} \alpha^{r-1-j} \beta^j$ . If $n = t_3$ , find the sum of the coefficients of $x^{t_r}$ (for all $r \in \mathbb{N}$ ) in the binomial expansion of $(1+x)^n$ .

Options

A.
508
B.
99
C.
1
D.
4096

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

Quadratic Roots in Sequences|Coefficient of x^r in Expansion|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the 'Even Multiplicity' trap in sequences derived from quadratic roots.

Students often memorize the difference of powers formula $t_r = \frac{\alpha^r - \beta^r}{\alpha - \beta}$ for such geometric sums.

When $D=0$ ( $\alpha=\beta$ ), this formula yields an indeterminate $0/0$ form, causing students to either incorrectly assume $t_r=0$ or misapply Newton's sums as $\alpha^{r-1} + \beta^{r-1}$ .

Recognizing that the sum collapses to $r\alpha^{r-1}$ is crucial.

Furthermore, it tests the validity of the binomial index and dynamically selecting coefficients based on sequence growth, where terms beyond $n$ yield a coefficient of zero.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Quadratic Roots in Sequences
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free