Before You Begin

Question #41731

Difficulty: 8/10 | 🟢 Free

The Question

Let a line $L$ in 3D space be given by the parametric equations $x = 3t + 1$ , $y = 2 - 3t$ , and $z = 4t$ , where $t \in \mathbb{R}$ . A function $f: (0, 3) \to \mathbb{R}$ is defined as $f(t) = [x(t)] + [y(t)] + [z(t)]$ where $[ \cdot ]$ denotes the greatest integer function. Let $m$ be the number of points in the interval $(0, 3)$ where $f(t)$ is discontinuous, and $n$ be the number of points in $(0, 3)$ where $\lim_{\tau \to t} f(\tau)$ does not exist. Find the value of $m + n$ .

Options

A.
28
B.
34
C.
26
D.
30

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

Parametric form pitfalls|Parametric Form Manipulation|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the subtle distinction between limits and continuity when dealing with parametric lines passed through step functions.

The primary trap lies in the composition $[3t] + [-3t]$ .

Students often correctly identify that the left-hand and right-hand limits of this sum are both $-1$ at integer values of $3t$ , but incorrectly conclude that the function is continuous there.

They fail to realize that the actual function value jumps to $0$ , creating a removable discontinuity.

Furthermore, the problem forces students to carefully separate the set of points where the limit does not exist (driven solely by $[4t]$ ) from the set of points where the function is discontinuous (the union of discontinuities from both components), punishing those who conflate the two concepts.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Parametric form pitfalls
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free