Find the range of values of \(x\) for which \(\ln(x^2-3)\) is defined. Leave your answer in surd form.[2]
Given that \(y=\ln[e^x(x^2-3)]\), show that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) can be expressed in the form of \(\dfrac{(x+a)(x+b)}{x^2-3}\).[3]
It is given that \(y=x^3+px^2+qx+10\) where \(p\) and \(q\) are integers. The only values of \(x\) for which \(y\) is a decreasing function of \(x\) are those values for which \(3<x<7\). Find the value of \(p\) and of \(q\).[4]
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Answer:(a)(i) \(x<-\sqrt3\) or \(x>\sqrt3\); (ii) \(\dfrac{(x+3)(x-1)}{x^2-3}\). (b) \(p=-15,\ q=63\).