A solid right circular cone of semi-vertical angle \(\alpha\) has its apex and the circumference of its base lying in the surface of a sphere of radius \(R\). Show that if \(\alpha\) is varied for fixed \(R\), the total surface area of the cone is a maximum for $$\sin\alpha = (1+\sqrt{17})/8.$$
Show that $$\int_0^{\frac{1}{4}\pi} f(\cos\theta)d\theta = \int_0^{\frac{1}{4}\pi} f(\sin\theta)d\theta,$$ and hence that $$\int_0^{\frac{1}{4}\pi} \ln\sin x dx = -\frac{1}{8}\pi\ln 2.$$ $$[\ln x = \log_e x.]$$
Let \(I_n\) be defined as $$I_n = \int_{-1}^1 (x^2 + 1)^n dx,$$ where \(n\) is not necessarily a positive integer. Obtain a relationship between \(I_n\) and \(I_{n-1}\) and hence evaluate \(I_{-\frac{1}{2}}\) without further integration. Evaluate also \(I_{-2}\).
Show that the triangles in the complex plane with vertices \(z_1, z_2, z_3\) and \(z_1', z_2', z_3'\) respectively are similar if $$\begin{vmatrix} z_1 & z_1' & 1 \\ z_2 & z_2' & 1 \\ z_3 & z_3' & 1 \end{vmatrix} = 0.$$ Discuss whether the converse of this result is true.
For \(n > 2\), prove by induction that $$(1-a_1)(1-a_2)\ldots(1-a_n) > 1-(a_1+a_2+\ldots+a_n),$$ where \(a_1, a_2, \ldots, a_n\) are positive numbers less than unity. Expanding \((1+x/n)^n\) by the binomial theorem, show that, for \(n\) a positive integer greater than 2, and \(x\) positive, $$S_n - \frac{x^2}{2n}S_{n-2} < \left(1+\frac{x}{n}\right)^n < S_n,$$ where $$S_n = \sum_{r=0}^n x^r/r!.$$ Given that, as \(n \to \infty\), \(S_n\) approaches a finite limit (dependent on \(x\)), show that \((1+x/n)^n\) approaches the same limit.
Let the sequence \((x_n)\) of positive numbers be defined by $$(1) \quad x_1 = 6, \quad \text{and} \quad (2) \quad x_{n+1} = \sqrt{8x_n - 15}.$$ Show that \(5 < x_{n+1} < x_n\) for all \(n\), and that \(x_n \to 5\) as \(n \to \infty\). Discuss what happens when (1) is replaced by \(x_1 = 4\).
A boy standing at the corner \(B\) of a rectangular pool \(ABCD\) with \(AB = 2\)m, \(AD = 4\)m has a boat in the corner \(A\) at the end of a string of length 2m. He walks slowly towards \(C\) along \(BC\) keeping the string taut. Locate the boy on \(BC\) when the boat is 1m from \(BC\). $$[\int \operatorname{cosec}\theta d\theta = \ln \tan \frac{1}{2}\theta.]$$
Find a solution of \(d^2y/dx^2 = y\) for which \(y = 0\) when \(x = l\), and \(y = a\) when \(x = 0\). Assuming a particular integral of the form \(x(A\cosh x + B\sinh x)\), or otherwise, solve $$\frac{d^2y}{dx^2} = y + 2\cosh(l-x),$$ given that \(y = 0\) when \(x = l\), and \(y = a\) when \(x = 0\).
A family of parabolas is given by the equation $$(x-at)^2 = 4a(y-at^2), \quad (1)$$ where \(a\) is a positive constant and \(t\) is a real-valued parameter. Show that the number of members of the family passing through a given point \((x_0, y_0)\) is 0, 1 or 2 according as \(x_0^2\) is greater than, equal to, or less than \(5ay_0\). Show that, for each fixed value of \(t\), the function \(y\) of the variable \(x\) defined by the equation (1) satisfies the differential equation $$5x^2\left(\frac{dy}{dx}\right)^2 - 4ax\left(\frac{dy}{dx}\right) + x^2 - ay = 0. \quad (2)$$ Deduce a solution of (2) which cannot be obtained by giving any fixed value to \(t\) in (1). How many solutions of (2) are there for which \(dy/dx\) is everywhere continuous and \(y = 0\) when \(x = 0\)?
Write down the expansions of \(e^x\) and \((1-x)^{-1}\) as power series in \(x\). Show that, for \(0 < a < \frac{1}{2}\), $$\int_0^a \frac{e^x-1}{x}dx < a + \frac{1}{4}a^2(1-\frac{1}{8}a)^{-1}.$$ Show also that $$1.80 < \int_0^1 \frac{e^x-1}{x}dx < 1.83.$$