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Question #40756

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(\vec{a})$ and $B(\vec{b})$ be two distinct points such that $\vec{a}$ and $\vec{b}$ are non-collinear vectors. A point $P(\vec{p})$ divides the line segment $AB$ strictly internally. If $\vec{p} = \lambda \vec{a} + \mu \vec{b}$ , where $\lambda = \log_2(4\sin x)$ and $\mu = \log_2(2\cos x)$ , find the sum of all possible values of $x$ in the interval $[0, 2\pi]$ .

Options

A.
$\frac{\pi}{12}$
B.
$\frac{\pi}{2}$
C.
$3\pi$
D.
$\frac{5\pi}{12}$

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

Internal vs External Division|Scalar Multiplication Geometric Scaling|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of vector geometry and transcendental equations.

The primary trap lies in the compound function boundaries.

Students typically solve the logarithmic equation to find $x = \pi/12$ and $5\pi/12$ , and often select their sum ( $\pi/2$ ).

However, they fail to verify the geometric constraint of strict internal division, which mandates that the scalar coefficients $\lambda$ and $\mu$ must be strictly positive.

The root $5\pi/12$ yields a negative value for $\mu$ (since $\cos 75^\circ < 0.5$ ), corresponding to external division, and must be rejected.

Furthermore, proving $\sin 15^\circ > 1/4$ requires a neat algebraic deduction without a calculator, adding a layer of analytical rigor typical of JEE Advanced.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Internal vs External Division
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free