Evaluate \(\begin{vmatrix} 2 &-1 &-2 \\0 & 2 & -1\\3 & -5& 0 \end{vmatrix}.\)

Show that 10P₃ = 9 P₃ + 3. 9P₂

Convert the parabola y²=4x+4y into standard form.

Evaluate \(\cot\left(\frac{-15\pi}{4}\right)\)

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

Show that the functions f(x) = 5x - \(\left| x \right| \) is continuous at x = 0

Differentiate \({ x }^{ \frac { 2 }{ 3 } }\) from first principles

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_x(1, -1)

A fruit grower can use two types of fertilizers in his garden, brand P and brand Q. The amounts (in Kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs atIeast 240 kgs of phosphoric acid, at least 270 kg of potash and atmost 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, formulate the above as mathematical LPP.

Construct a network diagram for the following situtation A < B; B < D, E; C