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Question #70522

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x, t) = \begin{pmatrix} t & x & x \\ x & t & x \\ x & x & t \end{pmatrix}$ and $N(x, t) = x M(x, t)$ . Let $f(x) = \int_{0}^{x} \frac{|N(x, t)|}{(x-t)^2} dt$ for $x > 0$ . Let $k = \frac{2}{25} f'(2)$ . If $P$ is a $3 \times 3$ matrix such that $|P| = k$ , find the value of $\log_2 \left( |\text{adj}(\text{adj}(2P))| \right)$ .

Options

A.
28
B.
24
C.
32
D.
20

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

Leibniz Rule for Differentiation under Integral|Determinant Scaling Property|The Zero-Factor Cancellation Trap

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

The question tests the 'Zero-Factor Cancellation Trap' in the context of the Leibniz rule and matrix singularities.

If a student attempts to differentiate the determinant $|N(x,t)|$ using Jacobi's formula or applies the Leibniz rule without first cancelling the $(x-t)^2$ factor, they encounter a singularity at the upper limit $t=x$ where the matrix becomes non-invertible (yielding an indeterminate $0/0$ boundary term).

By simplifying the integrand algebraically before applying calculus operations, the removable singularity is resolved.

The problem further layers determinant scaling properties ( $|cA| = c^n|A|$ ) and nested adjoint determinants to ensure comprehensive mastery.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Leibniz Rule for Differentiation under Integral
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free