A cube stands on a horizontal surface, and supports a second cube of equal size which is balanced on a vertex in such a way that its corresponding diagonal is vertical, and would if continued pass through the centre of the lower cube. The sun shines vertically overhead. Show that the upper cube can be rotated about its vertical diameter so that the lower cube will lie entirely in its shadow.
\(ABCD\) is a parallelogram, and \(E\) a point not necessarily in the plane of \(ABCD\). Show that \(a^2 + b^2 + g^2 + h^2 = b^2 + d^2 + e^2\), these being the lengths shown in the figure, and find a relation involving only \(a, b, c, d, e, f\). (You may use vector geometry.)
If \(a\) and \(b\) are real positive constants, show that the equation $$\pm\sqrt{\left(\frac{x}{a}\right)} \pm \sqrt{\left(\frac{y}{b}\right)} = 1$$ represents a conic section, and by considering the behaviour of the curve for large values of \(x\) and \(y\), that this conic section is a parabola. Find the direction of the axis of this parabola, and sketch the curve, indicating the sections of it which correspond to the various possible combinations of the \(\pm\) signs.
The number \(a_{11} + a_{22} + a_{33}\) is called the trace of the matrix $$\mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}.$$ If \(\mathbf{A}\) and \(\mathbf{B}\) are two \(3 \times 3\) matrices, show that the traces of the matrices \(\mathbf{AB}\) and \(\mathbf{BA}\) are equal. If the matrix \(\mathbf{AB}\) represents a rotation through an angle \(\phi\) about the directed axis \(U\) and \(\mathbf{A}\) represents a rotation interchanging the axes \(U\) and \(V\), explain why \(\mathbf{BA}\) represents a rotation through the angle \(\phi\) about \(V\). Given that the matrix $$\mathbf{M} = \begin{pmatrix} \cos\phi & \sin\phi & 0 \\ -\sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ represents a rotation through the angle \(\phi\) about the \(z\)-axis, and that the matrix \(\mathbf{C}\) represents a rotation about some axis, find a formula for the angle of rotation in terms of the trace of \(\mathbf{C}\).
If \(x = c + \frac{1}{4}\cos^8\theta\), \(y = (1-x)\cot\theta\), where \(c\) is a positive constant and \(\theta\) is a variable parameter, find a relation of the form \(y^2 = f(x, c)\). Sketch the graphs of this relation for the cases (i) \(c > 1\), (ii) \(\frac{1}{4} < c < 1\). What value of \(c\) makes the graph a circle?
Show that the function \(y = \sin^2(m\sin^{-1}x)\) satisfies the differential equation \[(1-x^2)y'' = xy' + 2m^2(1-2y).\] Show that, at \(x = 0\), \[y^{(n+2)} = (n^2 - 4m^2)y^{(n)} \quad (n \geq 1)\] and derive the MacLaurin series for \(y\).
Let \(f\) be the function of two real variables defined by \[f(x, y) = x^2 + xy + y^4.\] Find the range of \(f\) when the domain of \(f\) is:
Give a definition of an integral as the limit of a sum. By considering \[\sum_{n=0}^{N-1} (aq^n)^p(aq^{n+1} - aq^n),\] where \(p\) is a positive integer, \(q^N = b/a\) and \(b > a > 0\), show that \[\int_a^b x^p dx = \frac{b^{p+1} - a^{p+1}}{p+1}.\]
By considering \(\int_0^1 [1 + (\alpha-1)x]^n dx\), or otherwise, show that \[\int_0^1 x^k(1-x)^{n-k} dx = \frac{k!(n-k)!}{(n+1)!}.\] Deduce the value of \[\int_0^{\pi/2} \sin^{2n+1}\theta \cos^{2n+1}\theta d\theta.\]
(i) Find a first-order differential equation satisfied by each member of the family \(F\) of curves \[y = c\exp(x^2) \quad (-\infty < c < \infty).\] Write down the differential equation satisfied by any curve which is orthogonal to every member of \(F\) and hence find the set of orthogonal trajectories to \(F\). (ii) Verify that \(y = x^2/4a\) is a solution of the differential equation \[y = x\frac{dy}{dx} - a\left(\frac{dy}{dx}\right)^2.\] Explain why each tangent to the curve \(y = x^2/4a\) is also a solution of the differential equation.