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Question #67916

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} x & \sqrt{1-x^2} \\ \sqrt{3} & 1 \end{pmatrix}$ and $B(x) = \begin{pmatrix} 39 & x+1 \\ 208 & 4x^2 \end{pmatrix}$ be two matrices defined for real values of $x$ . Let $S$ be the set of all real numbers $x$ for which the matrix product $M(x) = A(x)B(x)$ is singular. Then the value of $36 \sum_{x \in S} x^2$ is equal to:

Options

A.
43
B.
70
C.
187
D.
214

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

Inverse Existence Criterion|Determinant Product Rule|Extraneous Solution Rejection|Domain Constraint Intersections

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

This question tests the Inverse Existence Criterion ( $\det(M) \neq 0$ ) combined with two critical JEE Advanced traps: Extraneous Solution Rejection and Domain Constraint Intersections.

Students must apply the Determinant Product Rule ( $\det(AB) = \det(A)\det(B)$ ) and solve for the roots.

The first trap occurs when solving $\det(A)=0$ , where squaring introduces an extraneous negative root ( $x = -\sqrt{3}/2$ ) that fails the non-negativity condition of the principal square root.

The second trap lies in $\det(B)=0$ , which yields a root ( $x=2$ ) that must be rejected because it falls outside the implicit domain of matrix $A$ ( $x \in [-1, 1]$ ).

The distractors perfectly map to failing one or both of these checks.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Inverse Existence Criterion
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free