Three equal particles \(A, B, C\) of mass \(m\) are placed on a smooth horizontal plane. \(A\) is joined to \(B\) and \(C\) by light threads \(AB, AC\) and the angle \(BAC\) is \(60^\circ\). An impulse \(I\) is applied to \(A\) in the direction \(BA\). Find the initial velocities of the particles and show that \(A\) begins to move in a direction making an angle \(\tan^{-1}\sqrt{3}/7\) with \(BA\).
Prove that \[ \left(\frac{d}{dx}\right)^n \tan^{-1}x = P_{n-1}(x)/(x^2+1)^n, \] where \(P_{n-1}\) is a polynomial in \(x\) of degree \(n-1\), and \[ P_{n+1} + 2(n+1)xP_n + n(n+1)(x^2+1)P_{n-1} = 0. \]
Shew that all chords of an ellipse which subtend a right angle at a given point on the ellipse meet in a point \(P\). Shew also that the locus of \(P\) is a concentric, similar and similarly situated ellipse.
Prove that if a weight be hung upon the lower end of a vertical spiral spring, it will oscillate vertically with a periodic time equal to that of a simple pendulum of length equal to the static extension of the spring which the weight produces when at rest.
Prove that the curve \[ 2x^2y^2+x^3-y^3-2xy=0 \] has (1) a double-point at the origin, each branch having a point of inflexion there, (2) an inflexion at each of the four points \(x=\pm 1, y=\pm 1\), the tangents being parallel to the axes, (3) no rectilinear asymptotes. Give a sketch of the curve.
Prove that the radius of curvature at any point of a curve \(y = f(x)\) is \[ \frac{\left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}}}{\frac{d^2y}{dx^2}}. \] Prove that \(\frac{7}{10}a\) is the length of the least radius of curvature of the curve given by \[ x^2y = a\left(x^2 + \frac{1}{\sqrt{5}}a^2\right). \]
A ring of mass \(m\) can slide on a smooth circular wire of radius \(a\) in a horizontal plane. The ring is fastened by an elastic string to a point in the plane of the circle at a distance \(c (> a)\) from its centre. Show that if the ring makes small oscillations about its position of equilibrium the period is \(2\pi \left\{\frac{mla(c-a)}{\lambda c(c-a-l)}\right\}^{\frac{1}{2}}\), where \(\lambda\) is the modulus of elasticity of the string and \(l (
Prove by induction or otherwise that \[ \int_0^\pi \{\sin n\theta / \sin\theta\} \, d\theta = 0 \text{ or } \pi \] according as \(n\) is an even or odd positive integer.
Evaluate \[ \int \frac{x^2+x+1}{x^2(x+1)^2} dx, \quad \int a^x \sin bx \, dx, \quad \int_0^\pi x \sin^2 x \, dx. \]
Find the area of a loop of the curve \[ r = 3 \sin 2\theta + 4 \cos 2\theta. \]