Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Show that the lines \[\frac{5 - x}{- 4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} \text { and } \frac{x - 8}{7} = \frac{2y - 8}{2} = \frac{z - 5}{3}\] are coplanar.

Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.

Maximize Z = 5x + 3y
Subject to

\[3x + 5y \leq 15\]
\[5x + 2y \leq 10\]
\[ x, y \geq 0\]

Maximize Z = 9x + 3y
Subject to 

2x + 3y ≤ 13

3x + y ≤ 5

x, y ≥ 0

Minimize Z = 18x + 10y
Subject to 

\[4x + y \geq 20\]
\[2x + 3y \geq 30\]
\[ x, y \geq 0\]

Maximize Z = 50x + 30y
Subject to 

\[2x + y \leq 18\]
\[3x + 2y \leq 34\]
\[ x, y \geq 0\]

Maximize Z = 4x + 3y
subject to

\[3x + 4y \leq 24\]
\[8x + 6y \leq 48\]
\[ x \leq 5\]
\[ y \leq 6\]
\[ x, y \geq 0\]

Maximize Z = 15x + 10y
Subject to 

\[3x + 2y \leq 80\]
\[2x + 3y \leq 70\]
\[ x, y \geq 0\]

Maximize Z = 10x + 6y
Subject to

\[3x + y \leq 12\]
\[2x + 5y \leq 34\]
\[ x, y \geq 0\]

Maximize Z = 3x + 4y
Subject to

\[2x + 2y \leq 80\]
\[2x + 4y \leq 120\]

Maximize Z = 7x + 10y
Subject to 

\[x + y \leq 30000\]
\[ y \leq 12000\]
\[ x \geq 6000\]
\[ x \geq y\]
\[ x, y \geq 0\]

Minimize Z = 2x + 4y
Subject to 

\[x + y \geq 8\]
\[x + 4y \geq 12\]
\[x \geq 3, y \geq 2\]

Minimize Z = 5x + 3y
Subject to 

\[2x + y \geq 10\]
\[x + 3y \geq 15\]
\[ x \leq 10\]
\[ y \leq 8\]
\[ x, y \geq 0\]

Minimize Z = 30x + 20y
Subject to 

\[x + y \leq 8\]
\[ x + 4y \geq 12\]
\[5x + 8y = 20\]
\[ x, y \geq 0\]

Maximize Z = 4x + 3y
Subject to 

\[3x + 4y \leq 24\]
\[8x + 6y \leq 48\]
\[ x \leq 5\]
\[ y \leq 6\]
\[ x, y \geq 0\]

Minimize Z = x − 5y + 20
Subject to

\[x - y \geq 0\]
\[ - x + 2y \geq 2\]
\[ x \geq 3\]
\[ y \leq 4\]
\[ x, y \geq 0\]

Maximize Z = 3x + 5y
Subject to

\[x + 2y \leq 20\]
\[x + y \leq 15\]
\[ y \leq 5\]
\[ x, y \geq 0\]

Minimize Z = 3x₁ + 5x₂
Subject to

\[x_1 + 3 x_2 \geq 3\]
\[ x_1 + x_2 \geq 2\]
\[ x_1 , x_2 \geq 0\]

Maximize Z = 2x + 3y
Subject to

\[x + y \geq 1\]
\[10x + y \geq 5\]
\[x + 10y \geq 1\]
\[ x, y \geq 0\]

Maximize Z = −x₁ + 2x₂
Subject to

\[- x_1 + 3 x_2 \leq 10\]
\[ x_1 + x_2 \leq 6\]
\[ x_1 - x_2 \leq 2\]
\[ x_1 , x_2 \geq 0\]