Choose scalar quantities.

Find the vector equation of the following diagram.

Find the positive value of \(k\),   correcting to two significant figures,   if the length of  \( \begin{pmatrix} k \\ 2 \end{pmatrix} \)  is 5.

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

Find  \(a\),  if  \(y = 2{x^3} + a{x^2} – 24x + 1\)  has a maximum at  \(x = – 4\).

A particle  \(P\)   moves in a straight line with a velocity function of  \(s\left( t \right) = {t^3} – 6{t^2} + 1\) metres,  where  \(t \ge 0\),  \(t\)   in seconds.    Find the total distance travelled for the first 5 seconds.

Find the maximum \(y\)-value given   \(y = – 2{x^2} + 8x + 10\)  for  \(3 \le x \le 5\).

Find  \(\int {\dfrac{{4{x^4} + 3{x^3}}}{x}dx} \).

If a curve has a stationary point  \(\left( {-1,-3} \right)\)  and a gradient of  \(4x + 4\),   find the value of   \(y\)  when  \(x = 1\).

Find the area between the  \(x\)-axis  and  \(y = x – 2\)   from  \(x = 1\) to \(x = 3\).