Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

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Important Questions on Linear Programming

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Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

Let the numbers

2, b, c

be in an A.P. and

A = [\begin{matrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2} \end{matrix}]

. If

d e t (A) \in [2,16],

then

c

lies in the interval:

HARD

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If

A = [\begin{matrix} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{matrix}]

, then for all

θ \in (\frac{3 π}{4}, \frac{5 π}{4})

,

\det (A)

lies in the interval :

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If

S = \{x \in [0, 2 π] : |\begin{matrix} 0 & \cos x & - \sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{matrix}| = 0\},

then

\sum_{x \in S} \tan (\frac{π}{3} + x)

is equal to:

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If the system of linear equations

x - 4 y + 7 z = g

;

3 y - 5 z = h

;

- 2 x + 5 y - 9 z = k

is consistent, then:

HARD

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If

A = [\begin{array}{l} 1 & 1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{array}]

. Find

A^{- 1}

. Hence, solve the system of equations

x + y + z = 6, x + 2 z = 7, 3 x + y + z = 12

.

EASY

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

The sum of the real roots of the equation

|\begin{matrix} x & - 6 & - 1 \\ 2 & - 3 x & x - 3 \\ - 3 & 2 x & x + 2 \end{matrix}| = 0,

is equal to:

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If

|\begin{matrix} x^{2} + x & x + 1 & x - 2 \\ 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 \\ x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix}| = a x - 12

, then

a

is equal to:

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

Show that the lines

\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}

and

\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}

are coplanar.

HARD

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If

A = [\begin{array}{ccc} 1 & 2 & - 3 \\ 3 & 2 & - 2 \\ 2 & - 1 & 1 \end{array}]

. Find

A^{- 1}

and use it to solve the following system of equations

x + 2 y - 3 z = 6, 3 x + 2 y - 2 z = 3, 2 x - y + z = 2

.

EASY

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

Let

d \in R,

and

A = [\begin{matrix} - 2 & 4 + d & (\sin θ) - 2 \\ 1 & (\sin θ) + 2 & d \\ 5 & (2 \sin θ) - d & (- \sin θ) + 2 + 2 d \end{matrix}],

θ \in [0, 2 π] .

If the minimum value of

d e t (A)

is

8,

then a value of

d

is:

EASY

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

There are two values of

a

which makes determinant,

Δ = |\begin{matrix} 1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2 a \end{matrix}| = 86

, then sum of these number is

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If the system of linear equations

x - 2 y + k z = 1

2 x + y + z = 2

3 x - y - k z = 3

has a solution

(x, y, z), z \neq 0,

then

(x, y)

lies on the straight line whose equation is:

EASY

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

The inverse of matrix is

[\begin{matrix} 2 & 5 & 0 \\ 0 & 1 & 1 \\ - 1 & 0 & 3 \end{matrix}]

is

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If

Δ_{1} = |\begin{matrix} x & s i n θ & c o s θ \\ - s i n θ & - x & 1 \\ c o s θ & 1 & x \end{matrix}|

and

Δ_{2} = |\begin{matrix} x & s i n 2 θ & c o s 2 θ \\ - s i n 2 θ & - x & 1 \\ c o s 2 θ & 1 & x \end{matrix}|,

x \neq 0;

then for all

θ \in (0, \frac{π}{2})

:

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If

A = [\begin{array}{l} 1 & 2 \\ 4 & 2 \end{array}]

then show that

| 2 A | = 4 | A |

.

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

Describe an inverted pyramid of biomass with the help of an example.

EASY

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

Which of the following ecological pyramids is generally inverted?

HARD

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

Let

A = [\begin{matrix} 2 & b & 1 \\ b & b^{2} + 1 & b \\ 1 & b & 2 \end{matrix}],

where

b > 0

. Then the minimum value of

\frac{\det (A)}{b}

is:

EASY

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

Let $λ$ be a real number for which the system of linear equations

$x + y + z = 6,$

$4 x + λ y - λ z = λ - 2$ and

$3 x + 2 y - 4 z = - 5$

has infinitely many solutions. Then $λ$ is a root of the quadratic equation:

MEDIUM

Mathematics>Algebra>Linear Programming>Mathematical Formulation of a Linear Programming Problem

If $a, b, c$ are non - zero real numbers and if the system of equations

$(a - 1) x = y + z$

$(b - 1) y = x + z$

$(c - 1) z = x + y$

has a non - trivial solution, then $a b + b c + c a$ equals :