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Question #41870

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be the number of distinct permutations of the letters of the word "ALGEBRA" such that all the vowels are adjacent to each other. A function $F(x)$ is defined as $F(x) = \int_{0}^{\frac{N}{60} x} |t - x^2| \, dt$ The value of $F'(4)$ is equal to

Options

A.
112
B.
48
C.
256
D.
144

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

String Method (Adjacency)|Newton-Leibniz Formula with variable limits|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the application of the generalized Newton-Leibniz rule when the integrand itself is a function of the variable of differentiation, specifically involving a non-differentiable modulus function.

A common trap is blindly applying the standard Leibniz rule $\frac{d}{dx} \int_a^{h(x)} f(t) dt = f(h(x))h'(x)$ , which yields 48.

This ignores the fact that the integrand $|t - x^2|$ depends on $x$ .

To solve correctly, students must first resolve the modulus by splitting the integral at its critical point $t = x^2$ , which falls inside the variable limits $[0, 6x]$ determined by the String Method permutation count.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	String Method (Adjacency)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free