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 \[ \int_1^\infty \frac{x^2+2}{x^4(x^2+1)}dx = \frac{\pi}{4}-\frac{1}{3}. \]
Prove that the area of the curved surface and the volume of a segment of height \(h\) of a sphere of radius \(a\) are \(2\pi ah\) and \(\frac{1}{3}\pi h^2(3a-h)\). The whole area (curved and plane) of a segment of a sphere is given to be equal to \(\pi c^2\). Prove that when the volume is greatest the height of the segment is \(c\).
Trace the curve \(x^3+y^3-2ax^2=0\).
Find an expression for all the values of \(\theta\) satisfying the equation \(\sin\theta=\sin\alpha\). If \(\theta_1, \theta_2\) are the two values of \(\theta\) not differing by a multiple of \(\pi\) which satisfy the equation \[ \frac{\sin\theta\sin\phi}{\sin\alpha} + \frac{\cos\theta\cos\phi}{\cos\alpha} + 1 = 0, \] prove that \[ \cos(\theta_1+\theta_2) = \frac{\sin^2\alpha-\sin^2\phi}{\sin^2\alpha\cos^2\phi+\cos^2\alpha\sin^2\phi}. \]