A regular dodecahedron is bounded by twelve regular pentagons. Find to the nearest degree the obtuse angle between two adjacent faces.
The normals at the points \(A\), \(B\), \(C\) of a parabola meet in a point \(P\), and \(H\) is the orthocentre of the triangle \(ABC\). Prove that
The parametric vector equation of a line \(l\) through the origin in three-dimensional Euclidean space is \(\mathbf{r} = t\mathbf{k},\) where \(\mathbf{k}\) is a constant unit vector and \(t\) denotes distance measured along \(l\) from the origin. A point \(P\) has position vector \(\mathbf{s}\). Find the position vector of the reflection of \(P\) in \(l\), i.e. of the point \(Q\) such that \(PQ\) is bisected at right angles by \(l\). If \(\mathbf{r} = t\mathbf{k}_i\) (\(i = 1, 2\)) are two distinct lines through the origin, and \(S_i\) (\(i = 1, 2\)) are the operations of reflection with respect to these lines, prove that \(S_1 S_2 = S_2 S_1\) if and only if the two lines are perpendicular.
A point with rectangular Cartesian coordinates \((x_1, x_2)\) in the Euclidean plane is represented by the \(1 \times 2\) matrix or row-vector \(\mathbf{x} = (x_1 \; x_2)\). Interpret the \(1 \times 1\) matrix \(\mathbf{x}\mathbf{x}'\), where \(\mathbf{x}'\) is the transpose of \(\mathbf{x}\). \(T(\mathbf{a}, \mathbf{d})\) denotes the transformation of the plane which sends the point \(\mathbf{x}\) into the point \(\mathbf{x}\mathbf{a} + \mathbf{d}\), where \(\mathbf{a}\) is a non-singular \(2 \times 2\) matrix and \(\mathbf{d}\) is a row-vector. Prove that the set of all such transformations forms a group \(G\). What is (i) the identity element of \(G\), (ii) the inverse of \(T(\mathbf{a}, \mathbf{d})\)? Find a necessary and sufficient condition that the distance between any two points should be equal to the distance between their transforms by \(T(\mathbf{a}, \mathbf{d})\), and prove that such distance-preserving transformations form a subgroup of \(G\).
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 \(x_1 = 4\).