Find the vector equation of the following diagram.

Find  \(\overrightarrow {BM} \).

Differentiate \(y = \dfrac{1}{{\sqrt {5x – 1} }}\).

Find \(a\) and \(b\),   if  \(\dfrac{d}{{dx}}\dfrac{{{x^2}}}{{{{\left( {2x + 3} \right)}^2}}} = \dfrac{{ax}}{{{{\left( {2x + 3} \right)}^b}}}\).

At an open-cut coal mine, a machine removes \( V\) cubic metres of coal in t hours, where \(V = 10t – \dfrac{{{t^2}}}{{10}}\).   At what time is the machine removing coal at the rate of 9 m³ per hour?

Find the maximum and minimum values of \(P = 400 + 100\sin \dfrac{{\pi t}}{3}\).

Find  \(p\) and \(q\),   if  \(\int {\left( {x + 2} \right)\sqrt {2{x^2} + 8x} } dx = \dfrac{1}{p}\sqrt {{{\left( {2{x^2} + 8x} \right)}^q}} + C\).

Find \(a\),   if  \(\int_0^3 {x\sqrt {x + 1} } dx = \dfrac{a}{{15}}\).

Find \(a\) and \(b\),   if  \(\int_3^6 {\dfrac{{4{x^3} – 3x + 2}}{x}} dx = a + {\log _e}b\).

Find the area of the region bounded by   \(y = {x^3} – 2{x^2}\) and \(y = {x^2} – 2x\),   correcting to one decimal place.