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Question #76208

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = 3\hat{i} + 3\hat{j} + 3\hat{k}$ and $\vec{b}(\theta) = \cos\theta \hat{i} + \sin\theta \hat{j} + \hat{k}$ , where $\theta \in [0, 2\pi]$ . Define the vector $\vec{v}(\theta) = (\vec{a} \times \vec{b}) - (\vec{b} \times \vec{a})$ . If $M$ is the maximum possible magnitude of $\vec{v}(\theta)$ , find the number of distinct solutions to the equation $\sin x + \cos x = \frac{M}{6\sqrt{6}}$ in the interval $[0, 4\pi]$ .

Options

A.
3
B.
4
C.
5
D.
6

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Vector Product Anti-Commutativity|Direction Cosines constraint|Auxiliary Angle Method

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 anti-commutativity of the cross product and the optimization of trigonometric expressions using the Auxiliary Angle Method.

A common trap is failing to recognize that $\vec{v}(\theta) = 2(\vec{a} \times \vec{b})$ , and subsequently misidentifying the maximum of the magnitude.

Students often assume the extremum of $f(t) = 5 - 2t - t^2$ occurs at the boundary values $t = \pm\sqrt{2}$ (derived from the amplitude of $\sin\theta+\cos\theta$ ), yielding a sub-optimal maximum of $3+2\sqrt{2}$ .

By forcing the substitution $t = \sin\theta+\cos\theta$ , the true maximum is revealed at the internal stationary point $t = -1$ .

The recursive step ensures complete failure if the maximum $M$ is incorrectly determined.

Quick Facts About This Question

Question Information

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