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Question #75811

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be an $n \times n$ matrix ( $n \ge 2$ ) with $\det(A) = c^2 > 0$ . Let $D(t) = \det(k t A)$ where $k$ is a non-zero scalar and $t > 0$ is a parameter. The normal to the rectangular hyperbola $xy = c^2$ at the point $P\left(ct, \frac{c}{t}\right)$ intersects the y-axis at $Y_0$ . Which of the following correctly expresses $Y_0$ in terms of the derivative $D'(t)$ ?

Options

A.
$\frac{c}{t} - \frac{t^{4-n} D'(t)}{n k^n c}$
B.
$\frac{c}{t} - \frac{t^3 D'(t)}{k c}$
C.
$\frac{c}{t} - \frac{t^3 D'(t)}{k^n c}$
D.
$\frac{c}{t} - \frac{t^{4-n} D'(t)}{n k c}$

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

Scalar Multiplication Geometric Scaling|Rectangular Hyperbola|Differentiation of Determinants|Elementary Row Operation Invariance

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the multilinear property of determinants and its integration with coordinate geometry.

A common trap (Concept 215 error) is treating the determinant as a linear operator, incorrectly assuming $\det(ktA) = kt\det(A)$ or failing to scale both $k$ and $t$ by the matrix order $n$ .

This leads to a dimensional mismatch when substituting the derivative back into the geometric equation of the normal's y-intercept.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Scalar Multiplication Geometric Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free