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Question #41160

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \cos^{-1}(\cos x) + 3\left[\frac{x}{\pi}\right]\sin x$ on the interval $[0, 2\pi]$ , where $[ \cdot ]$ denotes the greatest integer function. Let $k = 1$ if Lagrange's Mean Value Theorem is applicable to $f(x)$ on $[0, 2\pi]$ , and $k = 0$ otherwise. Let $S$ be the set of all points $c \in (0, 2\pi)$ such that the tangent to the curve at $x = c$ is parallel to the chord joining the endpoints of the interval. Find the value of $k + \sum_{c \in S} c$ .

Options

A.
$2\pi - \arccos\left(\frac{1}{3}\right)$
B.
$1 + 2\pi - \arccos\left(\frac{1}{3}\right)$
C.
$\pi + \arccos\left(\frac{1}{3}\right)$
D.
$2\pi - \arccos\left(\frac{1}{4}\right)$

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

The Quadrant Adjustment Logic|LMVT Tangent Parallelism|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of LMVT on a function containing a step function and an inverse trigonometric term.

The primary trap is the 'Step Function Discontinuity' in the derivative: students often assume that since $f(x)$ is continuous at $x=\pi$ , LMVT is applicable ( $k=1$ ).

However, the step function causes an abrupt change in the coefficient of $\sin x$ , making it non-differentiable.

Furthermore, finding the points parallel to the chord requires the 'Quadrant Adjustment Logic'.

If a student blindly differentiates $\cos^{-1}(\cos x)$ as $1$ in the interval $(\pi, 2\pi)$ , they will incorrectly solve $1 + 3\cos c = 0 \implies c = \pi + \arccos(1/3)$ , falling for a carefully placed distractor.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Quadrant Adjustment Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free