Linear Programming - Class 12

Let $R$ be the feasible region in the first quadrant of the $xy$-plane defined by the system of inequalities: $x \le 5$ $y \le 6$ $x^2 y - 8xy + 16y \le x^3 - 8x^2 + 16x$ Find the maximum value of the objective function $Z = 4x + 5y$ over the region $R$.

Let a line $L$ lie in the plane $P: 2x + y + 2z = 12$. The projection of the line $L$ onto the $yz$-plane is the line given by the equation $y - 2z = 4$. Let $S$ be the set of all points $(x,y,z)$ on $L$ that satisfy the non-negativity conditions $x \ge 0$, $y \ge 0$, and $z \ge 0$. If $M$ and $m$ are the absolute maximum and absolute minimum values of the function $f(x,y,z) = x^2 + y^2 + z^2 - 6z - 22$ on the set $S$, respectively, find the value of $M + m$.

Let $S$ be the region in the $xy$-plane defined by the inequalities $x \ge 0$, $y \ge 0$, $x + y \le 4$, and $2x + y \le 6$. Let $f(\theta) = \max_{(x,y) \in S} (x \cos \theta + y \sin \theta)$ for $\theta \in \left[0, \frac{\pi}{2}\right]$. Find the value of the integral $\int_0^{\pi/2} f(\theta) d\theta$.

Let $\alpha$ and $\beta$ be the coefficients of $x^5$ and $x^3$ respectively in the binomial expansion of $(1-x^2)^4 (1-x)^3$.

Consider the determinants: $\Delta_1(x,y) = \begin{vmatrix} 1 & 1 & 1 \\ x & y & \alpha \\ x^3 & y^3 & \alpha^3 \end{vmatrix} \quad \text{and} \quad \Delta_2(x,y) = \begin{vmatrix} 1 & 1 & 1 \\ x & y & \beta \\ x^2 & y^2 & \beta^2 \end{vmatrix}$

Let $L_1(x,y) = 0$ be the line obtained from the linear factor of $\Delta_1(x,y)$ that has a negative slope. Let $L_2(x,y) = 0$ and $L_3(x,y) = 0$ be the lines obtained from the linear factors of $\Delta_2(x,y)$ that have zero and infinite slopes, respectively. Assume the leading coefficients of $x$ and $y$ in the expressions for $L_1, L_2,$ and $L_3$ are taken as positive.

Let $R$ be the feasible region in the first quadrant ($x \ge 0, y \ge 0$) satisfying the constraints $L_1(x,y) \le 0$, $L_2(x,y) \le 0$, and $L_3(x,y) \le 0$. Find the maximum value of the objective function $Z = 5x + 7y$ on the region $R$.

Let $x, y,$ and $z$ be the side lengths of a non-degenerate triangle such that $z = 1$ and the perimeter of the triangle is $3$. Consider the function $f(x, y) = x^3 + y^3 - 3xy$. If $M$ and $m$ are the supremum and infimum of $f(x, y)$ respectively, then the value of $4(M - m)$ is:

Let $R$ be the feasible region in the $xy$-plane defined by $x \ge 0, y \ge 0$ and the determinant constraints: $\begin{vmatrix} [x] & [y] \\ 1 & 2 \end{vmatrix} \le 202, \quad \begin{vmatrix} [x] & -[y] \\ 2 & 1 \end{vmatrix} \le 202$ where $[ \cdot ]$ denotes the greatest integer function. If $M$ is the supremum of the objective function $Z = 4x + 3y$ for $(x,y) \in R$, find the value of $M$.

Let $S = \left\{ (x, y) \in \mathbb{R}^2 : 0 \le x \le 5, 0 \le y \le 5, \frac{x - 3}{y - 2} \ge 1 \right\}$. Find the supremum of the function $f(x, y) = x^2 + y^2 - 10x - 10y + 50$ over the set $S$.

Let $f: [0, 3] \to \mathbb{R}$ be defined as $f(x) = \int_0^x (4t + 1 - 6\lfloor t \rfloor) dt$ where $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$. Find the absolute maximum value of $f(x)$ on the interval $[0, 3]$.

Let $S$ be the set of all vectors $\vec{v} = l\hat{i} + m\hat{j} + n\hat{k}$ where $l, m, n$ are the direction cosines of a ray originating from the origin $O$. The ray intersects the plane $P_1: 3x + 4y + 12z = 68$ at point $A$ and the plane $P_2: x + 2y + 2z = 6$ at point $B$. If $OA = 5$ and $OB = 3$, then the number of elements in $S$ is:

Consider the linear programming problem: Maximize $Z = \left( \lim_{x \to 0} \frac{\tan x - \sin x}{x^3} \right) x + \left( \lim_{x \to 0} \frac{\ln(\cos(\sqrt{2}x))}{x^2} \right) y$ subject to the constraints: $x - 2y \le \lim_{x \to 0} \frac{e^{x^2} - \cos(\sqrt{2}x)}{x^2}$ $x + y \ge \lim_{x \to 0} \frac{e^{3x} - 1}{\sin x}$ $x \ge 0, y \ge 0$ Which of the following statements is true regarding the optimal solution?

Let $\vec{a} = t\hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + t\hat{j} + \hat{k}$, and $\vec{c} = \hat{i} + \hat{j} + t\hat{k}$ be three linearly dependent vectors, where $t$ is a real number such that $t < 0$. Consider the following Linear Programming Problem:

Maximize $Z = 4x + 3y$ Subject to the constraints: $\vec{r} \cdot (\hat{i} + \hat{j}) \ge 1$ $\vec{r} \cdot (t\hat{i} + \hat{j}) \le 4$ $x \ge 0, y \ge 0$ where $\vec{r} = x\hat{i} + y\hat{j}$.

The maximum value of $Z$ is:

Let $S$ be the feasible region in the $xy$-plane defined by the inequalities: $y - |2x - 4| \ge 0$ $2x - y + 2 \ge 0$ Find the maximum value of the objective function $Z = 5x - 2y$ on the region $S$.

A factory produces two liquids, $X$ and $Y$. The daily production constraints are $x + y \ge 6$, $x \le 6$, and $y \le 6$, where $x, y \ge 0$ are the quantities in kiloliters. The daily operating cost (in thousands of rupees) is modeled by the objective function $C(x,y) = 4|x - y| + x + y$. What is the minimum possible daily operating cost?

Let $M = \begin{pmatrix} 5 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 5 \end{pmatrix}$ and $J = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$. A function $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = \det(M + xJ)$. Find the local maximum value of $f(x)$.

Let $S$ be the set of all points $P(x,y,z)$ in $\mathbb{R}^3$ such that the distance of $P$ from the origin $O$ is $2\sqrt{3}$, and the sum of the direction cosines of the position vector $\vec{OP}$ is $1$. Let $\vec{v} = 3\hat{i} + 3\hat{j}$. If $M$ and $m$ are the absolute maximum and minimum values of the dot product $\vec{OP} \cdot \vec{v}$ for all $P \in S$, then the value of $M^2 + m^2$ is equal to:

Let $R$ be the region in the $xy$-plane consisting of all points $(x, y)$ that satisfy the conditions: $x^2 + y^2 \le 36$ $\sum_{k=2}^{7} \frac{2x - 2k - 1}{(x-k)^2 (x-k-1)^2} \ge 0$ Let $M$ be the supremum of the objective function $Z = x + y$ over the region $R$. Find the value of $M$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{vmatrix} \ln(1+x^2) & |x^2-3| & 1 \\ |x^2-3| & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix}$ Let $S$ be the set of all points of local extrema (both local maxima and local minima) of the function $f(x)$. Find the value of $\sum_{x \in S} x^2$.

Let the feasible region $R$ in the $xy$-plane be defined by the system of inequalities: $x \ge 0, \quad y \ge 0$ $x - y \le 2$ $x + \sqrt{3}y \le 6$ Let $S$ be the set of all values $\theta \in [0, 2\pi)$ such that the objective function $Z = x \cos \theta + y \sin \theta$ attains its maximum value on $R$ at infinitely many points. Find the value of $\sum_{\theta \in S} \theta$.

Consider the linear programming problem: Minimize $Z = 3x + y$ subject to the constraints: $y - 3|x - 4| \ge 3$ $3x + y \ge 12$ $x + y \le 13$ $2x - y \le 10$ $x \ge 0, y \ge 0$ Let $S$ be the set of all points $(x,y)$ in the feasible region at which $Z$ attains its minimum value. If $S$ is a line segment of length $L$, find the value of $L^2$.

Let $S$ be the region in the complex plane consisting of all $z = x+iy$ satisfying the following system of inequalities: $|z - 2| \le |z + 2|$ $|z - 2i| \le |z + 2i|$ $|z| \ge |z - 2 - 2i|$ $|z - 4| \le |z - 8|$ $|z - 4i| \le |z - 8i|$ $|z - 3 - 3i| \le |z - 5 - 5i|$ Let $f(z) = \text{Re}((1-i)z)$. If $M$ is the maximum value of $f(z)$ for $z \in S$, and $E$ is the set of all points $z \in S$ such that $f(z) = M$, find the length of the line segment representing $E$.

Consider the linear programming problem: Maximize $Z = c_1 x + c_2 y$ Subject to the constraints: $x \ge 0, y \ge 0$ $x + \lambda y \le L$ $3x + 2y \le 15$ where $c_1 = \frac{2}{\binom{24}{12}} \sum_{r=0}^{12} \binom{12}{r}^2$, $c_2 = \frac{2}{\binom{24}{12}} \sum_{r=0}^{12} r \binom{12}{r}^2$, and $L = \lim_{n \to \infty} \sum_{k=1}^n \frac{40k^3}{n^4}$. If the problem has multiple optimal solutions, what is the maximum value of $Z$?

Consider the optimization problem:

Maximize $Z = 2[x] + 3y$

subject to the constraints: $2x + 3y \le 18$ $x \ge 0, y \ge 0$

where $[x]$ denotes the greatest integer less than or equal to $x$. Let $S$ be the set of all optimal solutions $(x, y)$ that maximize $Z$. Which of the following statements is true?

A discrete random variable $X$ takes values in $\{1, 2\}$ with the probability mass function: $P(X=1) = \frac{1}{4} + \frac{1}{2\pi} \arcsin(\sin \alpha)$ $P(X=2) = \frac{3}{4} - \frac{1}{2\pi} \arccos(\cos \alpha)$ where $\alpha \in [0, 2\pi]$. Let $S$ be the set of all values of $\alpha$ for which this defines a valid probability mass function. For a fixed $\alpha \in S$, consider the linear programming problem: Maximize $Z = P(X=1) x + P(X=2) y$ subject to: $2x + y \le 6$ $x + 3y \le 9$ $x \ge 0, y \ge 0$ Let $M(\alpha)$ be the maximum value of $Z$ for a given $\alpha$. Find the minimum value of $M(\alpha)$ over all $\alpha \in S$.

Let $z = x + iy$ be a complex number satisfying $x \ge 0$, $y \ge 0$, and the inequality $|z - 2 - 4i| + |iz + 12 + 4i| \le 10$. Find the minimum value of the objective function $f(z) = |z - 2 - 4i|^2 + |iz + 12 + 4i|^2$.

Let $R$ be the feasible region in the $xy$-plane defined by the system of inequalities: $x + y \le 10, \quad x \le 7, \quad y \le 8, \quad 2x + 3y \le 30, \quad x \ge 0, \quad y \ge 0$ Consider the objective function $Z = 20 - |x - 2y + 5| - |2x + y - 10|$. If $M$ is the global maximum value of $Z$ over the region $R$, and it is attained at the point $(x_0, y_0)$, then the value of $M + x_0 + y_0$ is

Let $E_1: x^2 - a x + 8 = 0$ and $E_2: x^2 - b x + 32 = 0$ be two quadratic equations with real coefficients having a positive common root $r$. Let the non-common roots of $E_1$ and $E_2$ be $\alpha$ and $\beta$ respectively. It is given that $\alpha, r, \beta$ form a geometric progression in that order.

Consider the linear programming problem: Maximize $Z = (a+b)x + (ab)y$ subject to the constraints: $\alpha x + r y \le a$ $r x + \beta y \le b$ $x \ge 0, y \ge 0$

Find the maximum value of $Z$.

Let $A = \{ (x,y) \in \mathbb{R}^2 : 3x + 4y \le 25, \; 4x + 3y \le 25, \; x + y \le 7, \; 0 \le x \le 6, \; 0 \le y \le 6 \}$. Let $f: A \to B$ be a surjective function defined by $f(x,y) = 7x + 7y + 5$. Which of the following intervals represents the set $B$?

Let $c$ be the number of positive integers $t$ satisfying $\frac{(t-1)(t-5)}{t-3} \le 0$, and $d$ be the number of positive integers $t$ satisfying $\frac{t^2 - 7t + 12}{t - 2} \le 0$. Let $S$ be the region in the $xy$-plane defined by the inequalities $x \ge 0$, $y \ge 0$, $[x] + [y] \le 4$, and $2[x] - [y] \ge 1$, where $[ \cdot ]$ denotes the greatest integer function. If $M$ is the supremum of the objective function $Z = cx + dy$ over the region $S$, find the value of $M$.

Let $S$ be the feasible region defined by the system of inequalities: $x \ge 0, \quad y \ge 0$ $x + y \le 14$ $24 \le 3x + 4y \le 24 + 5d$ where $d > 0$ is the perpendicular distance between the parallel lines $3x + 4y = 24$ and $3x + 4y = 24 + 5d$. If the maximum value of the objective function $Z = 4x + 5y$ on the region $S$ is $62$, find the area of the region $S$.

Let $X$ be a discrete random variable uniformly distributed over the set $\{3, 7, 11, 15\}$. Let $\mu$ and $\sigma^2$ denote the mean and variance of $X$, respectively.

Consider the following linear programming problem: Maximize $Z = \sigma^2 x + \mu y$ subject to the constraints: $x + y \le 34$ $2x + y \le 39$ $5x + 2y \le 88$ $9x + 2y \le 144$ $x \ge 0, y \ge 0$

The maximum value of $Z$ is:

Let $\hat{n} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})$. A plane $\Pi$ in $\mathbb{R}^3$ is defined by the equation $\vec{r} \cdot \hat{n} = 2\sqrt{3}$. A feasible region $S$ on $\Pi$ is bounded by the non-negativity conditions $\vec{r} \cdot \hat{i} \ge 0$, $\vec{r} \cdot \hat{j} \ge 0$, $\vec{r} \cdot \hat{k} \ge 0$, and the following projection constraints: $\vec{r} \cdot (\hat{i} + 2\hat{j}) \le 8$ $\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) \le 14$ Determine the maximum value of the objective function $Z = \vec{r} \cdot (10\hat{i} + 24\hat{j} + 8\hat{k})$ for any position vector $\vec{r} \in S$.

Let $R$ be the feasible region in the first quadrant bounded by the $y$-axis, the line $y = 2$, and the curve $y = \int_0^x |t^2 - 1| dt$. If $A$ is the area of the region $R$, find the maximum value of the objective function $Z = 4A x - 5y$ over the region $R$.

Let $S$ be the set of all real values of $\lambda$ for which the linear programming problem:

Maximize $Z = 5x + 2y + 3z$

subject to the equality constraints: $4y - 2z = \lambda^2 - 2\lambda$ $-2x + 3z = 1 - 3\lambda$ $3x - 9y = 2$ $x, y, z \ge 0$

has a non-empty feasible region. The sum of all elements in $S$ is:

Let $S_k = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 \le 25, \; x^2 - y^2 \ge k, \; x \ge 0, \; y \ge 0 \}$. Let $M(k)$ be the maximum value of the objective function $Z = 3x + 4y$ subject to $(x,y) \in S_k$. Find the sum of all real values of $k$ for which $M(k) = 24$.

Let $S$ be the set of all ordered pairs $(x, y)$ of real numbers satisfying the system of equations: $\binom{8}{[x]} + \binom{8}{[y]} = 84$ $[x + y] = [x] + [y]$ $x - y = [x] - [y] - \frac{1}{5}$ $2x + 3y = 2[x] + 3[y] + \frac{41}{10}$ where $[t]$ denotes the greatest integer less than or equal to $t$. Find the number of elements in $S$.

Let $A$ be the origin and $B$ be the point representing the complex number $3+4i$ on the Argand plane. Let $C$ be a variable point representing the complex number $w$ such that $|w - (8+6i)| \le 2$. Let $M$ and $m$ be the maximum and minimum possible areas of $\triangle ABC$, respectively.

A factory produces quantities $x$ and $y$ (where $x, y \in \mathbb{R}$) of two goods, X and Y. The profit per unit of X is $M$ and per unit of Y is $m$. The production constraints are:

1. The total quantity produced cannot exceed 10 units.
2. The quantity of X produced must be at least the quantity of Y.
3. At least 2 units of X and 1 unit of Y must be produced.

What is the maximum possible profit?

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(t) = t^3 - 6t^2 + 9t$. Let $S$ be the set of all points $(x, y)$ in the Cartesian plane satisfying the following system of constraints:

1. $x \ge 0$
2. $y - x \ge 1$
3. $2x + y \le 8$
4. $f(t)$ is strictly increasing on the interval $[x, y]$.

Define the objective function $Z(x, y) = \frac{f(y) - f(x)}{y - x}$. Find the maximum value of $Z(x, y)$ subject to $(x, y) \in S$.

Let $R$ be the region in 3D space consisting of all points $P(x,y,z)$ that satisfy the following conditions:

1. $P$ lies in the octant $x \le 0$, $y \le 0$, $z \ge 0$.
2. The distance of $P$ from each of the coordinate planes is at most 2.
3. The distance of $P$ from the plane $x + 2y + 2z + 1 = 0$ is at most 1.

If the volume of the region $R$ is $V$, find the value of $3V$.

A manufacturing plant produces three types of components, with daily production quantities denoted by the column vector $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$. The production constraints are modeled by the matrix equation $(A + I)X = B$, where $I$ is the $3 \times 3$ identity matrix, $B = \begin{pmatrix} 2 \\ 5 \\ 7 \end{pmatrix}$, and $A$ is a $3 \times 3$ matrix such that $A^2 = I$. If it is given that $A B = \begin{pmatrix} -2 \\ 21 \\ 19 \end{pmatrix}$, and the profit function is $P = 10x + 15y + 20z$, what is the maximum possible daily profit?

Let the feasible region $S$ be defined by the inequality $|x| + |y| \le 4$. A matrix $M$ is defined as $M = \begin{pmatrix} |x| - 2 & |y| - 2 \\ |y| - 2 & |x| - 2 \end{pmatrix}$ If $Z = \det(M)$, find the maximum value of the objective function $Z$ for $(x,y) \in S$.

Let $Z(x,y)$ be the objective function defined by the line integral: $Z(x,y) = \int_{(0,0)}^{(x,y)} \left( (v e^{u^2 v^2} + 2u)du + (u e^{u^2 v^2} + 2v)dv \right)$ where the integral is taken over any piecewise smooth curve connecting $(0,0)$ to $(x,y)$. Consider the linear programming problem: Maximize $Z(x,y)$ subject to the constraints: $2x + y \ge 4$ $x - y \le 1$ $x \ge 0, y \ge 0$ Which of the following represents the maximum value of $Z(x,y)$?

Let $R$ be the feasible region in the first quadrant ($x \ge 0, y \ge 0$) satisfying the following constraints:

1. $\left( \int_{-\pi/4}^{\pi/4} \ln\left(t + \sqrt{1+t^2}\right) dt \right) x + x + y \le 4$
2. $x \sin \alpha + y \cos \alpha \le 5$ for all $\alpha \in [0, 2\pi]$

Consider the objective function $Z(x,y) = \int_{0}^{x^2 y} \frac{4-t}{1+t^2} dt$. The maximum value of $Z(x,y)$ over the region $R$ is:

Let $S$ be the line segment defined by the equation $x + y = 10$ for $5 \le x \le 8$. Given two points $A(4,4)$ and $B(2,6)$, find the minimum value of $PA + PB$ as the point $P$ varies over the segment $S$.

Let $X$ be a discrete random variable taking values $-1, 0,$ and $1$ with probabilities $p_1, p_2,$ and $p_3$ respectively. It is given that $p_2 \ge \frac{1}{5}$ and $|E[X]| \le \frac{2}{5}$, where $E[X]$ denotes the expected value of $X$. The maximum possible variance of $X$ is

Let $A = \begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 11 & 24 \\ -2 & 0 \end{pmatrix}$ be two matrices. A feasible region $R$ in the $xy$-plane is defined by the system of inequalities: $x \ge 0$, $y \ge 0$, and $x + 214y \le 1$. The region $R$ is transformed into a new region $R'$ by the linear transformation $T(\mathbf{v}) = M\mathbf{v}$, where the transformation matrix is given by $M = (A+B)^2 - (A^2+B^2)$. What is the area of the region $R'$?

Let $T(t)$ be a triangle in the $xy$-plane with vertices $A(t, 0)$, $B(0, t-1)$, and $C(|t|, |t-1|)$. If $S(t)$ denotes the area of the triangle $T(t)$, find the value of $12 \int_{-1}^{2} S(t) dt$

Let $R$ be the feasible region in $\mathbb{R}^3$ defined by the following system of inequalities: $x \ge 0, y \ge 0, z \ge 0$ $2x + y + 3z \le 12$ $x + 3y + 2z \le 12$ $3x + 4y + 5z \ge 24$ If $L$ is the maximum distance between any two points in $R$, find the value of $49 L^2$.

Let $R$ be the feasible region in the $xy$-plane defined by the inequalities $x \ge 0$, $y \ge 0$, and $x + y \le 25$. Find the maximum value of the objective function $Z(x,y) = 2x + y - 3|x-4|^{2/3}$ over the region $R$.

Let $\vec{F}(x) = \int (2x \hat{i} + 2x \hat{j}) dx$. The constant of integration is $\vec{K} = C \hat{i} - C \hat{j}$, where $C$ is an integer. Let $\vec{V}_1 = \vec{F}(1)$ and $\vec{V}_2 = \vec{F}(2)$. Consider the Linear Programming Problem: Maximize $Z = x + y$ Subject to: $\vec{r} \cdot \vec{V}_1 \ge 12$ $\vec{r} \cdot \vec{V}_2 \le 18$ $x \ge 0, y \ge 0$ where $\vec{r} = x \hat{i} + y \hat{j}$. Let $S$ be the set of all integers $C \in [-10, 10]$ for which the feasible region is non-empty. Find the number of elements in $S$.