Find the most general solution of the system of equations \begin{align} 3x + 2y + z &= 7,\\ x + y + az &= 3,\\ (2a + 1)x + 4y + 5z &= 11, \end{align} where \(a\) is a given real number. Examine carefully any exceptional cases.
Let \(Z\), \(W\) be points with rectangular cartesian coordinates \((x, y)\), \((u, v)\) respectively, and suppose that the complex numbers \(z = x + iy\), \(w = u + iv\) are related by the equation \[ w = \frac{1-z}{1+z}. \] Show that, as \(Z\) varies on a general straight line \(l\) in the \((x, y)\) plane, \(W\) describes a circle in the \((u, v)\) plane. Identify geometrically those lines \(l\) which are exceptional in this respect. Let \(l_1\), \(l_2\), \(\ldots\), \(l_n\) be concurrent (non-exceptional) lines in the \((x, y)\) plane. Show that the corresponding circles \(C_1\), \(C_2\), \(\ldots\), \(C_n\) belong to a coaxial system. What condition must be satisfied by \(l_1\), \(l_2\) in order that \(C_1\), \(C_2\) should be orthogonal circles?
Prove that the geometric mean of a finite set of positive real numbers does not exceed their arithmetic mean. Prove that \(k = \frac{2\sqrt{2}}{3}\) is the smallest constant which has the following property: if \(a\), \(b\) are real numbers such that \(a \geq 2b > 0\), then \[ \sqrt{(ab)} \leq k\left(\frac{a+b}{2}\right). \] Show that, if \(a_1\), \(\ldots\), \(a_n\), \(b_1\), \(\ldots\), \(b_n\) are real numbers such that \[ a_i \geq 2b_i > 0 \quad (i = 1, \ldots, n), \] then \[ (a_1a_2\ldots a_nb_1b_2\ldots b_n)^{\frac{1}{2n}} \leq \frac{2\sqrt{2}}{3}\left(\frac{a_1 + a_2 + \ldots + a_n + b_1 + b_2 + \ldots + b_n}{2n}\right). \]
If \(f\), \(g\) are real-valued functions of a real variable, let \(f*g\) denote the function whose value at \(x\) is \(f(g(x))\). (i) Show that there is exactly one function \(u\) such that, for every \(f\), \[ f*u = u*f = f. \] (ii) Find the form of the most general function \(v\) such that, for every \(f\), \[ v*f = v. \] (iii) Show that a function \(w\) satisfies the condition \[ w*f + f \] if and only if, the equation \(w(t) = t\) has no solutions. Give an example of such a function \(w\).
Let \(x\) be a real number such that \(0 < x < 1\). Find all the maxima and minima of the function \[ f(x) = xx - \cos x. \] Show how to determine the number of distinct positive roots of the equation \(\cos x = xx\). Show that this number is even if, and only if, \[ \frac{x\sin^{-1}x + \sqrt{(1-x^2)}}{2\pi x} \] is an integer (the value of \(\sin^{-1}x\) being chosen between \(0\) and \(\frac{1}{2}\pi\)).
A man observes that the summit of a nearby hill is in a direction \(x\) radians east of north, and at an inclination \(\theta\) above the horizontal. He then walks due north, down a slope of uniform inclination \(\tan^{-1}k\) below the horizontal, a distance \(x\) yards (measured along the slope), and finds that the direction and inclination of the summit are now (respectively) \(\beta\) east of north, \(\phi\) above the horizontal. Show that \[ k\sin(\beta-\alpha) = \sin\alpha\tan\phi - \sin\beta\tan\theta. \] Calculate the height of the summit above the man's initial position.
(i) Evaluate the sum \[ \sum_{r=1}^{n} \sin^3 rx. \] (ii) By using an algebraic relation between \(\tan x\) and \(\tan 2x\), or otherwise, evaluate the sum \[ \sum_{r=0}^{n} 2^r\tan\left(\frac{y}{2^r}\right)\tan^3\left(\frac{y}{2^{r+1}}\right). \] Show that \[ \sum_{r=0}^{\infty} 2^r\tan\left(\frac{y}{2^r}\right)\tan^3\left(\frac{y}{2^{r+1}}\right) = \tan y - y. \]
Suppose that the function \(f(x)\) has derivatives of all orders. Show by induction that \[ \frac{d^n}{dx^n}\{f(\frac{1}{2}x^2)\} = \sum_{r=0}^{[\frac{1}{2}n]} a(n,r)x^{n-2r}f^{(n-r)}(\frac{1}{2}x^2), \] where \([\frac{1}{2}n]\) denotes the greatest integer not exceeding \(\frac{1}{2}n\), and the constants \(a(n,r)\) satisfy \begin{align} a(n,0) &= 1 \quad (n = 0, 1, 2, \ldots),\\ a(2r,r) &= a(2r-1,r-1) \quad (r = 1, 2, \ldots),\\ a(n+1,r) &= a(n,r) + (n-2r+2)a(n,r-1) \quad (n = 2r, 2r+1, \ldots; r = 1, 2, \ldots). \end{align}
(i) Integrate the function \[ \frac{1}{1+\sqrt{(1+e^x)}}. \] (ii) Show that the definite integrals \[ \int_0^1 \frac{\sin^{\frac{1}{2}}\pi x}{\sqrt{(x-x^2)}} dx, \quad \int_0^1 \frac{\cos^{\frac{1}{2}}\pi x}{\sqrt{(x-x^2)}} dx \] are equal. Hence, or otherwise, evaluate these integrals.
Show that, if \[ I_n = \int_0^1 x^n\sqrt{(1+x)} dx \quad (n = 0, 1, 2, \ldots), \] then \[ 0 < I_n < \frac{\sqrt{2}}{n+1}. \] Obtain a reduction formula for \(I_n\). Hence, or otherwise, show that \[ I_n > \frac{\sqrt{2}}{n+\frac{3}{2}}. \]