Find the asymptotes of the curve \[ x^3+2x^2+x = xy^2 - 2xy + y. \] Show that there are values which \(y-x\) never takes, and sketch the curve.
Evaluate the following integrals (in which \(\sqrt{\phantom{x}}\) means the positive square root):
The roots of the cubic equation \[ ax^3+bx^2+cx+d=0 \quad (a\neq 0, d\neq 0) \] are \(\alpha, \beta, \gamma\). Find the equations whose roots are (i) \(\displaystyle\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\); (ii) \(\alpha^2, \beta^2, \gamma^2\); (iii) \(\beta\gamma/\alpha, \gamma\alpha/\beta, \alpha\beta/\gamma\). Deduce a necessary and sufficient condition that the product of two of the roots of the original equation should be equal to the third root.
The polynomials \(f(x), g(x)\) are of degrees \(m,n\) respectively, where \(m\ge n\ge 1\), and have real coefficients. Show that polynomials \(q(x), r(x)\) exist such that \(f(x)=g(x)q(x)+r(x)\), and such that the degree of \(r(x)\) is less than \(n\). Show also that \(q(x), r(x)\) are unique, and have real coefficients. If \(g(x) = x^2 - (\alpha+\beta)x + \alpha\beta\), prove that, if \(\alpha\neq\beta\), \[ r(x) = \frac{f(\alpha)-f(\beta)}{\alpha-\beta}x + \frac{\alpha f(\beta) - \beta f(\alpha)}{\alpha-\beta}. \] Verify that the coefficients of \(r(x)\) are real if \(\alpha, \beta\) are conjugate complex numbers. Find also the form of \(r(x)\) if \(\alpha=\beta\).
Prove that the geometric mean of a finite set of positive numbers cannot exceed the arithmetic mean, and deduce that it cannot be less than the harmonic mean. In what circumstances does equality occur? By considering the set of numbers \(a_r=(n+r)(n+r+1)\), where \(n\) is fixed and \(r\) takes the values \(1, 2, \dots, n\), prove that \[ (n+1)^{n+1}(2n+1)^{n-1} < \left(\frac{(2n)!}{n!}\right)^2 < \left(\frac{7n}{6}\right)^{2n}(n+1)^{n+1}(2n+1)^{n-1}. \]
Show that, if \(n>2\) and \(\theta\) is not an integral multiple of \(\displaystyle\frac{\pi}{n-1}\), a unique set of \(n\) numbers \(a_1, a_2, \dots, a_n\) can be found to satisfy the equations \[ a_1=a, \quad a_n=b, \] \[ a_{r+1} - 2a_r\cos\theta+a_{r-1}=0 \quad (1< r < n), \] and express \(a_r\) as a real function of \(a,b\) and \(\theta\). Investigate the exceptional cases, where \(\theta = \displaystyle\frac{k\pi}{n-1}\) (\(k\) an integer).
Find all the real solutions of the simultaneous equations \begin{align*} \sin^{-1}\tfrac{1}{2}x + \sin^{-1}\tfrac{1}{2}y &= \tfrac{1}{2}\pi \\ \tan^{-1}x + \tan^{-1}y &= \tan^{-1}\sqrt{5} \end{align*} where \(\sin^{-1}, \tan^{-1}\) are to be taken to mean the particular values of the functions which lie between \(-\frac{1}{2}\pi\) and \(\frac{1}{2}\pi\).
Prove that \[ \int_0^a f(x) \,dx = \frac{1}{2} \int_0^a \{f(x)+f(a-x)\} \,dx \] and give a geometrical interpretation of the result. Evaluate \[ \int_0^1 \frac{dx}{(x^2-x+1)(e^{2x-1}+1)}. \]
Evaluate:
The area \(\Delta\) of a triangle is expressed as a function of its sides \(a,b,c\). Show that \[ \Delta d\Delta = \frac{1}{8}\{(b^2+c^2-a^2)ada + (c^2+a^2-b^2)bdb + (a^2+b^2-c^2)cdc\}. \] The sides of a triangle are measured, the limits of error in these measurements being \(\pm\mu\) per cent., \(\mu\) being small. The area of the triangle is calculated from these data. Show that, if the triangle is acute-angled, the limits of error in the calculated area are approximately \(\pm 2\mu\) per cent. Is the result still true if the triangle is obtuse-angled? Give reasons for your answer.