Problems

Shew that the locus of the foot of the perpendicular from the centre of the ellipse \(x^2/a^2 + y^2/b^2 = 1\) on to a tangent is the curve \[ (x^2+y^2)^2 = a^2x^2+b^2y^2, \] and that the corresponding locus for the rectangular hyperbola \(xy = c^2\) is \[ (x^2+y^2)^2 = 4c^2xy. \]

Two particles of masses \(m\) and \(m'\) are connected by a fine thread passing over a small smooth pulley at the top of a smooth fixed solid whose vertical section is a quadrant of a circle, as in the figure. If motion begins when the radius to the particle \(m\) is horizontal, shew that when the radius has turned through an angle \(\theta\) the pressure between the mass \(m\) and the surface is \[ mg\{(3m+m')\sin\theta - 2m'\theta\}/(m+m'), \] so long as this expression remains positive.

\(A\) and \(B\) are two given points, and \(P\) a variable point on a given straight line parallel to \(AB\). Prove that (i) if the line cuts the circle on \(AB\) as diameter, \(AP \cdot BP\) is a maximum when \(P\) lies on the perpendicular bisector of \(AB\), and (ii) if the line does not meet the circle, \(AP \cdot BP\) is a minimum when \(P\) lies on this perpendicular bisector. Examine the case when the line touches the circle.

If \[ y = \frac{x-1}{(x+1)^2}, \] shew that \(y\) can never be greater than \(\frac{1}{8}\). Sketch the graph; and find the point in which the line which joins the points in which the curve meets the axes meets the curve again.

Three particles of equal mass are connected by light rods forming an equilateral triangle \(ABC\) with the particles at the corners and rest on a smooth horizontal plane. Shew that, if an impulse be applied to the particle at \(A\) in a direction parallel to \(BC\), the motion of the particles can be obtained by rolling the circumcircle of the triangle \(ABC\) along a certain straight line.

Find the asymptotes of the curve \[ x^4 + 3x^2y + 2x^2y^2 + 2xy + 3x+y = 0; \] determine on which side or sides the curve approaches each asymptote, and where it cuts the asymptotes.

Find the \(n\)th differential coefficients of \[ \cos x, \quad \cos^2 x, \quad \log(1+x), \quad \frac{x}{1+3x+2x^2}. \]

Two equal light strings of length \(l\) are hung at their upper ends from two fixed points distant \(a\) apart in the same horizontal line, \(a\) being small compared with \(l\). Their lower ends are joined to one another and to a third equal string, from the lower end of which a small mass is suspended. The mass is drawn aside in the vertical plane containing the two fixed points through a distance \(x\) from the position of equilibrium. Shew that the time of a complete oscillation is \[ 4\sqrt{\frac{l}{g}} \left\{\sqrt{2}\cos^{-1}\frac{2}{3} + \sin^{-1}\frac{2}{\sqrt{7}}\right\}. \]

Prove that, if \[ u_n = \int_{-a}^a (a^2-x^2)^n \cos bx \,dx, \] \[ b^2 u_{n+2} - 2(n+2)(2n+3)u_{n+1} + 4(n+1)(n+2)a^2 u_n = 0, \] where \(n\) may be assumed to be positive.

Find a formula of reduction for the integral \[ \int_0^{\pi/2} \sin^m x \cos^n x \,dx \] reducing one of the indices by two; and evaluate \(\int_0^{\pi/2} \sin^2 x \cos^6 x \,dx\). Find the integrals \[ \int \frac{dx}{\sqrt{1-x^2}}, \quad \int \frac{\sqrt{1-x^2}}{x} \,dx. \]