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Question #40876

Difficulty: 8/10 | 🟢 Free

The Question

Let $g(t) = 4t^3 - 12t^2 + 8t$ and let $\alpha$ be the point of local maximum of the function $h(x) = \int_{0}^{x} g(t) dt$ . A plane $P$ makes intercepts $a, b,$ and $c$ on the $x, y,$ and $z$ axes respectively, where $a = f(-1)$ , $b = f(2)$ , and $c = f(3)$ , with $f(x) = \int_{\alpha}^{x} g(t) dt + 3$ . If the point $D(2k, k, -2k)$ is coplanar with the points where the plane $P$ meets the coordinate axes, then the value of $k$ is

Options

A.
1
B.
2
C.
3
D.
4

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

Collinearity and Coplanarity|Variable plane intercepts|Fundamental Theorem of Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the Fundamental Theorem of Calculus with 3D geometry, specifically targeting the evaluation of definite integrals with variable upper bounds.

The lethal trap lies in the lower bound $\alpha$ of the integral defining $f(x)$ .

Students often perform indefinite integration and append the constant $+3$ , ignoring the evaluation of the lower bound $t=1$ which contributes an additional $-1$ to the function.

If the lower bound is ignored (or if a local minimum $\alpha \in \{0, 2\}$ is incorrectly chosen), the function becomes $f(x) = x^4 - 4x^3 + 4x^2 + 3$ , leading to intercepts $12, 3, 12$ and a trap answer of $k=3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Collinearity and Coplanarity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free