91Ó°ÊÓ

If the function \(f(x)=x^{4}+b x^{2}+8 x+1\) has a horizontal tangent and a point of inflection for the same value of \(x\), then the value of \(b\) is equal to \(\begin{array}{llll}\text { (a) }-1 & \text { (b) } 1 & \text { (c) } 6 & \text { (d) }-6\end{array}\)

A curve is represented parametrically by the equations \(x=t+e^{\text {at }}\) and \(y=-t+e^{a t}\) when \(t \in R\) and \(a>0 .\) If the curve touches the axis of \(x\) at the point \(A\), then the coordinates of the point \(A\) are (a) \((1,0)\) (b) \((1 / e, 0)\) (c) \((e, 0)\) (d) \((2 e, 0)\)

For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in \(\mathrm{km}\) ). If the speed of the water current is steady at \(\mathrm{C} \mathrm{km} / \mathrm{h}\), then the most economical speed of the steamer going against the current will be (a) \(1.25 \mathrm{C}\) (b) \(1.5 \mathrm{C}\) (c) \(1.75 \mathrm{C}\) (d) \(2 C\)

Statement I Shortest distance between \(|x|+|y|=2\) and \(x^{2}+y^{2}=16\) is \(4-\sqrt{2}\). Statement II Shortest distance between the two smooth curves lies along the common normal.

The equation \(|\ln m x|=p x\) where \(m\) is a positive constant has exactly three roots for (a) \(p<\frac{m}{e}\) (b) \(0