Before You Begin

Question #41643

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_1, S_2,$ and $S_3$ be the sets of all real roots of the following equations, respectively:

(i) $\frac{d}{dx} \left( \int_{x}^{x^2} \frac{dt}{\sqrt{1-t^2}} \right) = 0$

(ii) $\frac{d}{dx} \left( \int_{x}^{x^2} \frac{dt}{\sqrt{t^2-1}} \right) = 0$

(iii) $\frac{d}{dx} \left( \int_{x}^{x^2} \frac{dt}{t\sqrt{t^2-1}} \right) = 0$

Let $P(\alpha, \beta, \gamma)$ be a point in 3D space where $\alpha = \sum_{x \in S_1} \frac{1}{x^2}$ , $\beta = \sum_{x \in S_2} x^2$ , and $\gamma = \sum_{x \in S_3} x^2$ . (If any set is empty, its corresponding sum is $0$ ).

If $\Pi$ is the plane whose $x, y,$ and $z$ intercepts are $3, -6,$ and $3$ respectively, then the perpendicular distance from the point $P$ to the plane $\Pi$ is:

Options

A.
2
B.
$\frac{17}{9}$
C.
4
D.
$\frac{5}{3}$

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

Distance from point to plane|Newton-Leibniz Formula with variable limits|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question tests the Newton-Leibniz formula while embedding a severe 'Function-Definition' trap regarding the domain of variable-limit integrals.

Students mechanically differentiate the integrals and solve $f'(x) = 0$ , finding roots like $x = 1/\sqrt{3}$ for the second equation and $x = \pm\sqrt{3}$ for the third.

However, for an integral to be real-valued, the entire interval between its limits must lie within the integrand's domain.

For the second and third integrals, the domain requires $[x, x^2] \subset (1, \infty)$ , meaning $x > 1$ .

This completely invalidates the root $1/\sqrt{3}$ (making $S_2$ empty) and rejects $-\sqrt{3}$ for $S_3$ .

Failing to perform this rigorous domain analysis leads to incorrect coordinates for point $P$ and traps the student into calculating the distance from an imaginary or invalid point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Distance from point to plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free