Before You Begin

Question #41404

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all twice continuously differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(0) = 3$ and $\int_0^x \begin{vmatrix} f(t) & \sin t \\ f''(t) & -\sin t \end{vmatrix} dt = f(x) \cos x - 3 - \sin x (f'(x))^3$ for all $x \in \mathbb{R}$ . Find the value of $\sum_{f \in S} f\left(\frac{\pi}{2}\right)$ .

Options

A.
6
B.
9
C.
12
D.
3

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Linearity and Partitioning|Differentiation of Determinants|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

This question tests the differentiation of determinants and the 'Zero-Factor Cancellation Trap'.

Students must recognize that the integrand is the derivative of a determinant, applying the linearity property of determinant differentiation.

The critical trap occurs when simplifying the resulting differential equation to $\sin x [ (f'(x))^3 - f'(x) ] = 0$ .

A common mistake is to divide by $f'(x)$ (assuming it is non-zero), which incorrectly eliminates the constant function solution $f(x)=3$ .

Furthermore, students must use the continuity of $f'$ (implied by twice differentiability) to deduce that $f'$ must be a constant function across the entire domain, preventing piecewise combinations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Linearity and Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free