The standard international (SI) unit for molar conductivity, \( \Lambda_m \), is expressed as: \[ S \, m^2 \, \text{mol}^{-1} \] In this expression:
- \( S \) (Siemens) denotes the unit of electrical conductivity,
- \( m^2 \) indicates the area over which the conductivity is measured,
- \( \text{mol}^{-1} \) signifies per mole of the solution's electrolyte.
The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.
Solve the following L.P.P. by graphical method:
Maximize:
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]