Consider the curve with equation \(f(x) = p{x^2} + qx\) , where p and q are constants. The point \({\text{A}}(1{\text{, }}3)\) lies on the curve. The tangent to the curve at A has gradient \(8\). Find the value of p and of q .

substituting \(x = 1\) , \(y = 3\) into \(f(x)\)     (M1)

\(3 = p + q\)     A1

finding derivative     (M1)

\(f'(x) = 2px + q\)     A1

correct substitution, \(2p + q = 8\)     A1

\(p = 5\) , \(q = - 2\)     A1A1     N2N2

[7 marks]

A good number of candidates were able to obtain an equation by substituting the point \(1{\text{, }}3\) into the function’s equation. Not as many knew how to find the other equation by using the derivative. Some candidates thought they needed to find the equation of the tangent line rather than recognising that the information about the tangent provided the gradient of the function at the point. While they were usually able to find this equation correctly, it was irrelevant to the question asked.