Before You Begin

Question #41037

Difficulty: 8/10 | 🟢 Free

The Question

A system undergoes three sequential state transitions from state 0 to state 3. For a parameter $t \in (0, 1)$ , the probability of transitioning from state 0 to state 1 is $t$ , from state 1 to state 2 is $\sqrt{t}$ , and from state 2 to state 3 is $t^{3/2}$ . Let $P(t)$ be the probability that the system successfully reaches state 3 from state 0.

Three points in 3D space have position vectors: $\vec{a} = (\cos^{-1}t) \hat{i} + t \hat{j}$ $\vec{b} = (\cos^{-1}t) \hat{j} + \hat{k}$ $\vec{c} = \hat{i} + t \hat{k}$ Let $V(t)$ be the volume of the tetrahedron formed by the origin and the points with position vectors $\vec{a}, \vec{b}, \vec{c}$ .

Find the value of $\lim_{t \to 1^-} \frac{dV}{dP}$ .

Options

A.
$-\frac{1}{18}$
B.
$\frac{1}{18}$
C.
$-\frac{1}{3}$
D.
$\frac{1}{3}$

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

Multiplication Theorem|Determinant as Geometric Area/Volume|Parametric Derivative Endpoint Trap|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question integrates the Multiplication Theorem of probability to establish a parametric relationship, combined with the geometric interpretation of a determinant as the volume of a tetrahedron.

The primary trap lies in the parametric derivative at the boundary $t \to 1^-$ .

Students often carelessly evaluate the term $\frac{\cos^{-1}t}{\sqrt{1-t^2}}$ as $0 \times \infty = 0$ without applying L'Hopital's rule or trigonometric limits, leading to an incorrect derivative of $1/6$ and a final answer of $1/18$ .

Additionally, forgetting the $1/6$ factor in the tetrahedron volume formula leads to the trap answers $\pm 1/3$ .

This rigorously tests limit evaluation at endpoint singularities within a multi-concept framework.

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