Before You Begin

Question #41876

Difficulty: 8/10 | 🟢 Free

The Question

Let $n \ge 4$ be an integer such that the vectors $\vec{a} = \binom{n}{2} \hat{i} + \binom{n}{3} \hat{j} + \binom{n}{4} \hat{k}$ , $\vec{b} = \hat{i} + \hat{j} + \hat{k}$ , and $\vec{c} = \hat{i} + 2\hat{j} + 2\hat{k}$ are coplanar. Let $R = (\sqrt{2}+1)^n$ , $I = [R]$ , and $f = \{R\}$ , where $[\cdot]$ and $\{\cdot\}$ denote the greatest integer and fractional part functions respectively. Consider the vectors $\vec{p} = \frac{1 - (-1)^I}{2} \hat{i} + (1-f)\hat{j} + \lambda \hat{k}$ and $\vec{q} = \hat{i} - \hat{j} + f \hat{k}$ . If the angle between $\vec{p}$ and $\vec{q}$ is obtuse, find the maximum integer value of $\lambda$ .

Options

A.
476
B.
477
C.
478
D.
479

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

Integral Part and Fractional Part|Linear Independence vs Dependence|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of linear algebra and binomial theorem properties.

The first trap is recognizing that the determinant condition for coplanarity simplifies to a basic binomial equality, yielding $n=7$ .

The 'Lethal Collision' occurs when evaluating the dot product for the obtuse angle condition, which forces students to resolve the fractional part $f$ of $(\sqrt{2}+1)^7$ .

A common mistake is assuming $R + F$ is an integer (which is only true for even powers), leading to $f = 1 - F$ and an incorrect bound for $\lambda$ .

By correctly identifying that $R - F$ is an even integer for odd powers, students find $f = F$ and $I = 478$ , which nullifies the $\hat{i}$ component of $\vec{p}$ and elegantly transforms the inequality into $\lambda < I + f - 1$ , yielding the exact maximum integer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Integral Part and Fractional Part
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free