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Question #41790

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be a $3 \\times 3$ real matrix such that $N^3 = O$ and $N^2 \\neq O$ . Let $I$ be the $3 \\times 3$ identity matrix. Define the functions $f(x) = \\det(x^2 I + x N + N^2)$ and $h(x) = \\det(I + x N N^T)$ for $x \\in \\mathbb{R}$ . It is given that $\\text{Tr}(N N^T) = 13$ .\n\nA company manufactures two products, $P_1$ and $P_2$ . The profit per unit of $P_1$ is $c_1 = f'(1)$ and the profit per unit of $P_2$ is $c_2 = h'(0)$ . The production of $x$ units of $P_1$ and $y$ units of $P_2$ ( $x, y \\ge 0$ ) is subject to the constraints:\n $x + y \\le 20$ \n $12x + 26y \\le 312$ \nLet $M$ be the maximum total profit, and $k$ be the number of distinct production plans $(x,y)$ with integer coordinates that yield this maximum profit. Find the value of $M + k$ .

Options

A.
157
B.
158
C.
159
D.
160

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

Word Problem Modeling|Multiple Optimal Solutions|Differentiation of Determinants|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to connect linear algebra with linear programming.

The first trap involves evaluating the derivative of determinants involving nilpotent matrices; students must realize that $(xI-N)(x^2I+xN+N^2)=x^3I$ to find $f(x)$ , and use Jacobi's formula or characteristic polynomial properties for $h'(0)$ .

The primary LPP trap is 'Multiple Optimal Solutions'.

The objective function $Z = 6x + 13y$ is perfectly parallel to the constraint $12x + 26y \\le 312$ .

Students who only check the vertices of the feasible region will find the maximum value $M=156$ but will likely assume the solution is unique (finding $k=1$ ), leading to the distractor 157.

They must solve the Diophantine equation $6x+13y=156$ and filter the roots through the secondary constraint $x+y \\le 20$ to correctly identify $k=2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Word Problem Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free