Before You Begin

Question #41038

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $3 \times 3$ matrices of the form $B = \begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{pmatrix}$ , where $a, b, c, d \in \{1, 2, 3, 4, 5, 6\}$ . A matrix $B$ is chosen at random from $S$ . Let $P$ be the probability that $B$ is non-singular.

Consider the matrix $A(x) = \begin{pmatrix} 0 & x^2 - x & \sin(\pi x) \\ x - x^2 & 0 & e^{x-1} - 1 \\ -\sin(\pi x) & 1 - e^{x-1} & 0 \end{pmatrix}$ and let $f(x) = \det(B A(x) B^T)$ .

Find the value of $648 P + f'(2024)$ .

Options

A.
605
B.
648
C.
562
D.
691

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

Multiplication Theorem|Complementary Counting|Skew-Symmetric Determinant|Differentiation of Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the Multiplication Theorem of Determinants and Complementary Counting, while embedding a classic JEE trap: differentiating a determinant of a skew-symmetric matrix.

Students often attempt to apply Jacobi's formula or differentiate the determinant column-by-column, failing to recognize that a skew-symmetric matrix of odd order has a determinant identically equal to zero.

Identifying this 'kill-switch' immediately reduces $f'(2024)$ to 0.

The second phase requires complementary counting to find the probability of a matrix being non-singular by calculating the number of solutions to $ad = bc$ for dice rolls, rewarding systematic frequency grouping over brute-force counting.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Multiplication Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free