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Question #41203

Difficulty: 8/10 | 🟢 Free

The Question

A variable plane $P$ has intercepts $a(t), b(t),$ and $c(t)$ on the $x, y,$ and $z$ axes respectively, where $a(t) = t^3 + k$ , $b(t) = \frac{15}{2}t^2$ , and $c(t) = t^3 + 36t$ for a parameter $t \in [0, 4]$ . Consider the system of planes given by the equations: $P_1: x + y + z = 1$ $P_2: x + 2y + 3z = 1$ $P_3: a(t)x + b(t)y + c(t)z = 1$ If the coefficient matrix of this system is non-singular for all $t \in [0, 4]$ , what is the complete set of values for the real constant $k$ ?

Options

A.
$(-\infty, -32) \cup (0, \infty)$
B.
$(-\infty, -28) \cup (-27, \infty)$
C.
$(-\infty, -32) \cup (-27, \infty)$
D.
$(-\infty, -28) \cup (0, \infty)$

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

Singular vs Non-Singular Matrices|Geometry Constraint Modeling|Variable plane intercepts|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of linear algebra (determinant conditions for non-singular matrices) and differential calculus (global optimization on a closed interval).

The primary trap lies in finding the extrema of the determinant function.

Students frequently find the critical points ( $t=2, 3$ ) and incorrectly assume the local extrema ( $28$ and $27$ ) define the bounding range, leading to the trap option $(-\infty, -28) \cup (-27, \infty)$ .

A robust understanding requires evaluating the boundary conditions ( $t=0, 4$ ) to find the true global extrema ( $0$ and $32$ ), ensuring the parameter $k$ strictly avoids the entire global range of the function.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Singular vs Non-Singular Matrices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free