List of practice Questions

Solve the LPP graphically: Minimize $Z = 5x + 10y$ subject to $x + 2y \le 120$, $x + y \ge 60$, $x, y \ge 0$.

Solve the following linear programming problem graphically:
Maximize \[ Z = 8000x + 12000y \] Subject to the constraints
\[ 3x + 4y \le 60 \] \[ x + 3y \le 30 \] \[ x \ge 0,\; y \ge 0 \]

For the feasible region shown below, the non-trivial constraints of the linear programming problem are

In a linear programming problem, the linear function which has to be maximized or minimized is called

Let $A$ be a square matrix. If $A^2 = A$, then the matrix $A$ is called:

For the linear programming problem: \[ {Maximize} \quad Z = 2x_1 + 4x_2 + 4x_3 - 3x_4 \] subject to \[ \alpha x_1 + x_2 + x_3 = 4, \quad x_1 + \beta x_2 + x_4 = 8, \quad x_1, x_2, x_3, x_4 \geq 0, \] consider the following two statements:
S1: If $ \alpha = 2 $ and $ \beta = 1 $, then $ (x_1, x_2)^T $ forms an optimal basis.
S2: If $ \alpha = 1 $ and $ \beta = 4 $, then $ (x_3, x_2)^T $ forms an optimal basis. Then, which one of the following is correct?

Consider the following regions: \[ S_1 = \{(x_1, x_2) \in \mathbb{R}^2 : 2x_1 + x_2 \leq 4, \quad x_1 + 2x_2 \leq 5, \quad x_1, x_2 \geq 0\} \] \[ S_2 = \{(x_1, x_2) \in \mathbb{R}^2 : 2x_1 - x_2 \leq 5, \quad x_1 + 2x_2 \leq 5, \quad x_1, x_2 \geq 0\} \] Then, which of the following is/are TRUE?

Let $ M_2(\mathbb{R}) $ be the vector space (over $ \mathbb{R} $) of all $ 2 \times 2 $ matrices with entries in $ \mathbb{R} $.
Consider the linear transformation $ T: M_2(\mathbb{R}) \to M_2(\mathbb{R}) $ defined by $ T(X) = AXB $, where

\[ A = \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 6 & 5 \\ -2 & -1 \end{bmatrix}. \]

If $ P $ is the matrix representation of $ T $ with respect to the standard basis of $ M_2(\mathbb{R}) $, then which of the following is/are TRUE?

Consider the linear programming problem (LPP):

\[ \text{Maximize } Z = 3x_1 + 5x_2 \]

Subject to:
\[ x_1 + x_3 = 4, \]
\[ 2x_2 + x_4 = 12, \]
\[ 3x_1 + 2x_2 + x_5 = 18, \]
\[ x_1, x_2, x_3, x_4, x_5 \geq 0. \]

Given that $ x_B = (x_3, x_2, x_1)^T $ forms the optimal basis of the LPP with basis matrix $ B $ and respective $ B^{-1} $:

\[ B^{-1} = \begin{bmatrix} \alpha & \beta & -\beta \\ 0 & \gamma & 0 \\ 0 & -\beta & \beta \end{bmatrix} \]

If $ (p, q, r) $ is the optimal solution of the dual of the LPP, then which one of the following is/are TRUE?

Solve the following linear programming problem graphically:
Maximize $z = 50x + 30y$
Subject to:
$2x + y \leq 18$
$3x + 2y \leq 34$
$x \geq 0$, $y \geq 0$

There are two types of fertilizers $F_1$ and $F_2$. $F_1$ consists of 10% nitrogen and 6% phosphoric acid. $F_2$ consists of 5% nitrogen and 10% phosphoric acid.
After testing the soil conditions, a farmer finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop.
If $F_1$ costs ₹ 6 per kg and $F_2$ costs ₹ 5 per kg, how much of each type of fertilizer should be used so that the cost is minimum.
Formulate a linear programming problem.

In a LPP, the maximum value of $z = 3x + 4y$ subject to the constraints $x + y \leq 40$, $x + 2y \leq 60$, $x, y \geq 0$ is

The maximum value of $ Z = 3x - y $ subject to the constraints \[ x + y \leq 8, \quad x \geq 0, \quad y \geq 0 \text{ is}. \]

The maximum value of $ Z = 2x + y $ subject to the constraints \[ x + y \leq 35, \quad x \geq 0, \quad y \geq 0 \] is.

Solve the linear programming problem graphically. Maximize: $ z = 3x + 5y $ Subject to: \[ x + 4y \leq 24, 3x + y \leq 21, x + y \leq 9, x \geq 0, y \geq 0 \] Also, find the maximum value of $ z $.

Let $ T, S : P_4(\mathbb{R}) \to P_4(\mathbb{R}) $ be the linear transformations defined by \[ T(p(x)) = xp'(x), \quad S(p(x)) = (x + 1)p'(x) \] for all $ p(x) \in P_4(\mathbb{R}) $. Then, the nullity of the composition $ S \circ T $ is ................

Consider the real vector space $ \mathbb{R}^3 $. Let $ T : \mathbb{R}^3 \to \mathbb{R} $ be a linear transformation such that \[ T(1, 1, 1) = 0, \quad T(1, -1, 1) = 0, \quad T(0, 0, 1) = 16. \] Then, the value of $ T \left( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \right) $ is equal to ............... (rounded off to two decimal places).

Let $ f_1, f_2, f_3 $ be nonzero linear transformations from $ \mathbb{R}^4 $ to $ \mathbb{R} $ and \[ \ker(f_1) \subset \ker(f_2) \cap \ker(f_3). \] Let $ T : \mathbb{R}^4 \to \mathbb{R}^3 $ be the linear transformation defined by \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad \text{for all } v \in \mathbb{R}^4. \] Then, the nullity of $ T $ is equal to:

Define $ T : \mathbb{R}^3 \to \mathbb{R}^3 $ by \[ T(x, y, z) = (x + z, 2x + 3y + 5z, 2y + 2z), \quad \text{for all } (x, y, z) \in \mathbb{R}^3 \] Then, which one of the following is TRUE?

Minimize \[ Z = 2x + y, \] subject to \[ 5x + 10y \le 50, \quad x + y \ge 1, \quad y \le 4, \quad x \ge 0, \quad y \ge 0. \]

Maximize \[ Z = 20x + 3y, \] subject to the constraints: \[ 3x + 2y \leq 10, \quad x \geq 0, \quad y \geq 0. \]

The minimum value of $ Z = 3x + 5y $ subject to the constraints: \[ x + y \leq 2, \quad x \geq 0, \quad y \geq 0 \] is:

The maximum value of $ Z = 3x + 2y $ subject to the constraints: \[ 3x + y \leq 15, \quad x \geq 0, \quad y \geq 0 \] is:

Maximize $ Z = 3x + 9y $ subject to the constraints: \[ x + 3y \leq 60, x + y \geq 10, x \leq y, x \geq 0, y \geq 0. \]

Minimize $ Z = 3x + 5y $ subject to the constraints: \[ x + 3y \geq 3, x + y \geq 2, x \geq 0, y \geq 0. \]