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Question #41660

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ invertible matrix with real entries such that $A A^T = \begin{bmatrix} 5 & 2 & 3 \\ 2 & 6 & 2 \\ 3 & 2 & 5 \end{bmatrix}$ Let $B = \text{adj}(A)$ and denote the column vectors of $B$ by $\vec{b}_1, \vec{b}_2, \vec{b}_3$ . Let $P_1$ be the plane passing through the origin and containing the vectors $\vec{b}_1$ and $\vec{b}_2$ . Let $P_2$ be the plane passing through the origin and containing the vectors $\vec{b}_2$ and $\vec{b}_3$ . If $\theta$ is the acute angle between the planes $P_1$ and $P_2$ , then the value of $\cos \theta$ is:

Options

A.
$\frac{3}{5}$
B.
$\frac{4}{5}$
C.
$\frac{7}{13}$
D.
$\frac{17}{19}$

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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

Angle between two planes|Determinant as Geometric Area/Volume|Adjoint-Determinant Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the deep geometric interpretation of the adjoint matrix and its relationship with cross products.

The primary trap is computational: students may attempt to find matrix $A$ explicitly (which is not uniquely determined) or compute $B^T B = \text{adj}(A A^T)$ and use Lagrange's identity.

While the latter works, it is highly prone to arithmetic errors and time-consuming.

The conceptual shortcut requires realizing that the normals to the planes spanned by the columns of $B = \text{adj}(A)$ are exactly the rows of $\text{adj}(B)$ .

By the identity $\text{adj}(\text{adj}(A)) = |A|A$ , these normals are simply scaled versions of the rows of $A$ .

Thus, the angle between the planes is identical to the angle between the first and third rows of $A$ , which can be read directly from the given $A A^T$ matrix.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Angle between two planes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free