State the principles by which we are enabled to calculate the changes in velocity produced by the impact of two smooth elastic spheres. A smooth elastic sphere falls vertically with velocity \(u\) on a smooth wedge which lies on a smooth table. Calculate the velocity of the wedge after the impact, the wedge and the table being supposed inelastic.
An elastic string of natural length \(a\) has one end fixed and a weight attached to the other. When it hangs vertically it is stretched a length \(c\), and when it revolves as a conical pendulum making \(n\) revolutions per second it is stretched a length \(z\). Prove that \(gz = 4\pi^2 n^2 c(a+z)\).
Find the conditions that \(ax^2+bx+c\) may be positive for all real values of \(x\). Shew that for real values of \(x\) the fraction \((2x^2-5x+2)/(x^2-4x+3)\) assumes all values from \(-\infty\) to \(+\infty\). Draw a graph of the function for all values of \(x\) from \(-\infty\) to \(+\infty\).
Prove that if \((x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)\) be a perfect square in \(x\), then \(a=b=c\). Determine \(\lambda\) so that \[ (3x+2y-1)(2x+3y-1) + \lambda(x+4y-1)(4x+y-1)=0 \] may be the product of linear factors.
Prove that an infinite series \(u_1+u_2+u_3+\dots\) is convergent or divergent according as when \(n\) tends to infinity \(u_{n+1}/u_n\) tends to a limit less than or greater than unity. State and prove a test for the case in which the limit of \(u_{n+1}/u_n\) is unity. Examine the convergency or divergency of the series whose \(n\)th terms are \(n^4/n!, (n!)^2 x^n/3n!\).
Prove that, if \(\omega\) is an imaginary cube root of unity, then \(1+\omega+\omega^2=0\). Shew how to use the cube roots of unity to find the sum of a series obtained by picking out every third term from a known series; and prove that \[ 1+\frac{x^3}{3!} + \frac{x^6}{6!} + \frac{x^9}{9!} + \dots = \frac{1}{3}\left\{e^x+2e^{-x/2}\cos\frac{\sqrt{3}}{2}x\right\}. \]
Shew, graphically or otherwise, that the cubic equation in \(\theta\), \[ \frac{x^2}{a^2-\theta} + \frac{y^2}{b^2-\theta} + \frac{z^2}{c^2-\theta} = 1, \quad a>b>c, \] has three real roots \(\lambda, \mu, \nu\) which are such that \(a^2>\lambda>b^2>\mu>c^2>\nu\). Also shew that \[ x^2 = \frac{(a^2-\lambda)(a^2-\mu)(a^2-\nu)}{(a^2-b^2)(a^2-c^2)}. \]
Find the fourth differential coefficient of \(\frac{\sin x}{x}\); and deduce that as \(x\to 0\), \[ \frac{x^4-12x^2+24}{x^5}\sin x + \frac{4x^2-24}{x^4}\cos x \to -\frac{1}{5}. \]
A triangle is circumscribed to a circle of given radius \(r\), and the sides of the triangle are to be determined in terms of \(r\) and the angles by the formula \[ r = a(\cot\frac{1}{2}B + \cot\frac{1}{2}C), \] and others like it. A first measurement makes the triangle equilateral. Shew that, if there is a possible error of \(10'\) in each of the angles \(B\) and \(C\), the percentage of error in the determination of the side \(a\) cannot exceed \(\cdot34\).
Evaluate \(\int\sec^3 x dx, \int\frac{3x+2}{\sqrt{\{x^2+4x+1\}}}dx\). Prove that \[ \int_1^\infty \frac{x^2+2}{x^4(x^2+1)}dx = \frac{\pi}{4}-\frac{1}{3}. \]