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Question #72334

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = l\hat{i} + m\hat{j} + n\hat{k}$ be a unit vector, where $l, m, n$ are its direction cosines. Let $\vec{b} = \hat{i} + \hat{j} + \hat{k}$ . Let $S$ be the set of all possible values of $\vec{a} \cdot \vec{b}$ that satisfy the inequality $\frac{|\vec{a} \times \vec{b}|^2 - 2}{\vec{a} \cdot \vec{b}} \ge 1$ If $M$ and $m$ are the supremum and infimum of the set $S$ respectively, then the value of $m^2 + \frac{1}{M}$ is:

Options

A.
$\frac{7+\sqrt{5}}{2}$
B.
$2+\sqrt{5}$
C.
$\frac{5+\sqrt{5}}{2}$
D.
$\frac{9+\sqrt{5}}{2}$

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

Lagrange Identity|Direction Cosines constraint|Cross Multiplication Trap

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

The question tests the application of Lagrange's identity alongside geometric constraints (direction cosines), but its primary discriminator is the 'Cross Multiplication Trap' in inequalities.

Students often blindly cross-multiply $\frac{1-x^2}{x} \ge 1$ to get $1-x^2 \ge x$ , implicitly assuming $x > 0$ .

This yields the incorrect interval $[\frac{-1-\sqrt{5}}{2}, \frac{-1+\sqrt{5}}{2}]$ , leading to a false infimum $m = \frac{-1-\sqrt{5}}{2}$ and the trap answer $2+\sqrt{5}$ .

A rigorous approach requires analyzing the sign of $x$ using the wavy curve method, revealing the valid negative interval bounded by the Cauchy-Schwarz minimum $-\sqrt{3}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Lagrange Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free