Discuss the solutions of the equations: \begin{align*} x+y+3z &= 4, \\ x+2y+4z &= 5, \\ x-y+az &= b, \end{align*} in the cases (i) \(a=b=1\), (ii) \(a=1\) and \(b=2\), (iii) \(a=b=2\).
Prove that for real values of \(x\) the rational function \[ \frac{5x^2 - 18x - 35}{8(x^2 - 1)} \] takes all real values except those between 1 and 4. Draw a rough graph of the function.
If \(u_0=2\), \(u_1=2\cos\theta\), and \[ u_n = u_1 u_{n-1} - u_{n-2}, \quad (n>1) \] prove that \(u_n=2\cos n\theta\). By successive applications of the relation express \(u_5\) as a polynomial in \(u_1\), say \(f(u_1)\). Then by considering the equation \(f(u_1)=1\) shew that \(2\cos\frac{\pi}{15}\) is a root of the quartic equation \[ x^4 + x^3 - 4x^2 - 4x + 1 = 0. \] What are the other three roots?
The points \(A\) and \(B\), at a distance \(a\) apart on a horizontal plane, are in line with the base \(C\) of a tower. The elevations of the top of the tower from \(A\) and \(B\) are observed to be \(\alpha\) and \(\beta\). Find the distance \(BC\). If the observations of elevation are uncertain by \(\pm\epsilon\), shew that the maximum possible percentage error in the calculated value of \(BC\) is \[ \frac{100\epsilon\sin(\beta+\alpha)}{\sin\alpha\cos\beta\tan(\beta-\alpha)}. \]
If \[ S_n(\theta) = \sum_{r=1}^n \cos^r\theta \sin r\theta \] prove (by induction or otherwise) that, if \(0<\theta<\pi\), \[ S_n(\theta) = \cot\theta(1-\cos^n\theta \cos n\theta). \] Prove also that \(S_n(0) = 0\). If \(n\) is fixed, what is the limit of \(S_n(\theta)\) as \(\theta\to 0\)? Shew that the series converges as \(n\to\infty\) for all values of \(\theta\), but that its sum is not continuous.
Water is poured gently into a bowl having the form of a surface of revolution with its axis vertical. If the rate of increase of volume is \(w\) shew that the rate of increase of depth is \(w/a\), and the rate of increase of the area of the wetted surface is \(2\pi rw/a\), where \(a\) denotes the area of the water-surface, and \(r\) the intercept between the meridian curve and the axis on a normal to the curve through a point on the water-surface. If the meridian curve is \[ y = \frac{ax^2}{a^2+x^2}, \] and the axis of rotation is \(x=0\), find \(r\) when \(a=\pi a^2\).
State (without proof) Rolle's theorem, and deduce that there is a number \(\xi\) between \(a\) and \(b\) such that \[ f(b) - f(a) = (b-a)f'(\xi), \quad (1) \] explaining what conditions must be satisfied by the function \(f(x)\) in order that the theorem may be valid. If \(f(x) = \sin x\), find all the values of \(\xi\) between \(a\) and \(b\) which satisfy the equation (1) when \(a=0\) and \(b=3\pi/2\). Illustrate the result with reference to the graph of \(\sin x\).
If \[ \left(\frac{d}{dx}\right)^n e^{-x^2} = \phi_n(x)e^{-x^2}, \] shew that \[ \phi_n + 2x\phi_{n-1} + 2(n-1)\phi_{n-2} = 0, \] and that \[ \phi_n'' - 2x\phi_n' + 2n\phi_n = 0. \] Shew also that \[ \phi_{2n}(0) = 1.3.5 \dots (2n-1).(-2)^n, \] and \[ \phi_{2n}'(0)=0. \]
(i) Evaluate \[ \int_1^e \left(\log \frac{e}{x}\right)^2 dx, \quad \int_0^\pi \frac{dx}{a+b\cos x}, \quad (a>|b|). \] (ii) If \[ I_n = \int_0^\pi \frac{\sin^2 nx}{\sin^2 x} dx, \] where \(n\) is a positive integer, shew that \(I_n - I_{n-1} = I_{n-1} - I_{n-2}\), and hence evaluate \(I_n\).
Determine \(A, B, C\) and \(D\) such that \[ \frac{x^2}{(x^2+1)^4} = \frac{d}{dx} \frac{Ax^5+Bx^3+Cx}{(x^2+1)^3} + \frac{D}{x^2+1}, \] and shew that \[ \int_0^1 \frac{x^2}{(x^2+1)^4}dx = \frac{1}{48} + \frac{\pi}{64}. \]