Evaluate \(\mathop {\lim }\limits_{x \to \infty } \dfrac{{{x^2} + 3x – 1}}{{{x^2} – 2x – 3}}\).

Differentiate \(f\left( x \right) = \sqrt[3]{{{x^4}}}\) .

Find the positive angle \(\theta \),  correcting to the nearest degree, that makes the normal and the \(x\)-axis where the gradient of the normal to  \(y = {x^3} – 3{x^2} + 7\)  at  \(x = 1\).

Find  \(x\)  for which  \(f\left( x \right) = \dfrac{{2x}}{{x – 3}}\)  is a decreasing function.

Find the turning point and the nature of the turning point of  \(y = \sqrt {1 – {x^2}} \).

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

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

Find  \(p\)  and  \(q\),   if   \(\int {{{\left( {\dfrac{{2x}}{3} + 1} \right)}^3}} dx = \dfrac{1}{p}{\left( {2x + 3} \right)^q} + C\).

If a curve has a stationary point  \(\left( {1,5} \right)\)  and a gradient of  \(y’ = 8x + p\), where \(p\)  is a constant,   find the value of  \(p\),  and  \(y\)  when  \(x = 2\).

Find the value of  \(k\)    if   \(\int_{ – 1}^2 {\left( {3{x^2} + 4x + k} \right)dx} = 30\).