The two ends of a cricket pitch are denoted by \(A\), \(B\) and are at a distance \(l\) apart. The bowler bowls from \(A\), the ball leaving his hand at a height \(a\) from the ground at an angle \(\theta\) above the horizontal. The ball bounces at a point which divides \(AB\) in the ratio \(1 : \alpha\), and then hits the stumps at \(B\) at a height \(b\). The ground is assumed to be smooth and the coefficient of restitution between the ball and the ground is \(e\), where \(e > \sqrt{b/a}\). Show that, if \(\alpha > 0\), one value of \(\alpha\) lies between \(0\) and \(e\) while the other lies between \(e\) and \(2e\).
A heavy particle is attached at one end of a long string. The string is wound round a rough circular cylinder of radius \(a\) whose axis is horizontal, and the weight hangs freely at a height \(c\) below the axis of the cylinder. The particle is given a horizontal velocity \(u\), in the direction away from and perpendicular to the vertical plane through the axis of the cylinder. If \(a/c\) is small, and if \(n\) is an integer such that \(0 \leq 2n \leq \frac{u^2 - 5gc}{3\pi ga} < 2n + 1,\) show that the string first slackens after rotating through an angle of approximately \((2n + 1)\pi\).
Discuss the reasoning in the following statements:
Assuming that the length of the circumference of a circle lies between the total lengths of side of the inscribed and circumscribed regular polygons of \(n\) sides, derive upper and lower bounds for \(\pi\). Show that when \(n = 8\) these bounds reduce to \[4\sqrt{(2-\sqrt{2})} < \pi < 8(\sqrt{2}-1).\] How would you continue to evaluate the corresponding bounds for \(n = 16, 32, 64, \ldots\), using no operation more advanced than the taking of a square root?
Evaluate the integrals \[\int_0^u \tan^{-1}x\,dx; \quad \int_0^v \sqrt{x(a-x)}\,dx \quad (0 \leq v \leq a).\] If \[I_n = \int (ax^2 + 2bx + c)^n\,dx,\] find a relation between \(I_n\) and \(I_{n-1}\), and comment on special cases.
Consider a complex variable \(z = x + iy\), and show that in the \((x, y)\) plane the two sets of equations \[\text{Re}(z^2) = \text{const.}, \quad \text{Im}(z^2) = \text{const.}\] describe two families of mutually orthogonal hyperbolae, also that \[\text{Re}(z^{-1}) = \text{const.}, \quad \text{Im}(z^{-1}) = \text{const.}\] describe families of mutually orthogonal circles. (By \(\text{Re}(\omega)\) and \(\text{Im}(\omega)\) are meant the real and imaginary parts of the complex variable \(\omega\).)
Find all the real roots of the two following equations in \(x\): \[\cos(x\sin x) = \frac{1}{2};\] \[\cos 2x + 2\cos a\cos x - 2\cos 2a = 1.\]
A rectangle lies wholly in the region \[0 < x < a,\] \[0 \leq y \leq \frac{\beta}{x^\gamma} \quad (\alpha, \beta, \gamma > 0),\] and has two of its sides along the coordinate axes. Determine the rectangle of this type which has greatest area, paying attention to the relative values of \(a\), \(\beta\) and \(\gamma\).
Sketch the curves \[x^n + y^n = 1\] for \(n = -1, 1, 2, 3, 4\). Also, sketch the curves \(y = f(x)\), \(y = f'(x)\), for a function \(f(x)\) which obeys \[f'(0) < 0, \quad f''(x) > 0;\] \[\frac{f(x)}{x} \to 1 \text{ as } x \to +\infty\] in the range \(x > 0\).
Prove the expansion \[f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x) + \ldots + \frac{h^{n-1}}{(n-1)!}f^{(n-1)}(x) + \frac{1}{(n-1)!}\int_0^h f^{(n)}(x+y)(h-y)^{n-1}\,dy\] by integrating the remainder term by parts. Assuming that the remainder term can be written \(h^n f^{(n)}(x + \theta h)/n!\), show that if \(f^{(n+1)}(x+y)\) is continuous in \(y\) at \(y = 0\) then \(\theta \to 1/(n+1)\) as \(h \to 0\).