Two adjacent sides of a parallelogram are of lengths \(a,b\) and include an angle \(\alpha\), and a rhombus is described with one angular point on each of the 4 sides of the parallelogram or on these sides produced. Prove that the least area of the rhombus is \[ ab\sin^2\alpha/(1+\sin\alpha). \]
If the bisectors of the angle \(A\) of the triangle \(ABC\) meet \(BC\) in \(D,D'\), prove that the radius of the circle inscribed in the triangle \(ADD'\) is \[ bc/4R \cos\tfrac{1}{2}(B-C)\{\cos\tfrac{1}{4}(B-C)+\sin\tfrac{1}{4}(B-C)\}, \] where \(R\) is the radius of the circumcircle of the triangle \(ABC\), and \(B>C\).
A variable chord \(PQ\) of a curve passes through a fixed point \(O\) and \(M\) is the middle point of \(PQ\); the normals at \(P,Q\) to the curve meet the line through \(O\) perpendicular to \(OPQ\) in \(P',Q'\). Prove that the middle point of \(P'Q'\) lies on the normal at \(M\) to the locus of \(M\).
Find the locus of centres of curvature of the curve given by the equations \[ x=\cos\theta+\theta\sin\theta, \quad y=\sin\theta-\theta\cos\theta, \] where \(\theta\) is a parameter. Draw a graph of the two curves for \(0\le\theta\le\pi\).
Two coplanar forces of magnitudes \(P,Q\) and inclined at an angle \(\alpha\) act through the fixed points \(A,B\) respectively. Prove that, if each force is rotated through any the same angle in the same sense, their resultant passes through a fixed point at a distance \[ \tfrac{1}{2}AB(P^2+Q^2-2PQ\cos\alpha)^{\frac{1}{2}}/(P^2+Q^2+2PQ\cos\alpha)^{\frac{1}{2}} \] from the middle point of \(AB\).
A thin smooth elliptic tube of axes \(2a, 2b\) (\(a>b\)) is attached by light spokes to a horizontal axis which passes through the centre of the ellipse and is perpendicular to its plane. The weight of the tube is \(W\) and its centre of gravity is on the major axis at a distance \(d\) from the centre; and a particle of weight \(w\) is placed in the tube. Prove that there are 2 or 4 positions of equilibrium according as \(d >\) or \(< (a^2-b^2)w/aW\).
A peg is fixed in a horizontal table and a lamina with a straight slot cut in it is placed on the table with the peg through the slot; if a given point of the slot is made to describe a straight line along the table, find the locus of the centre of instantaneous rotation (i) on the table, (ii) on the lamina.
Two vertical posts of heights \(a,b\) stand on level ground at a distance \(c\) apart; a stone is projected from the ground level with the least possible velocity consistent with its just clearing the two posts. Prove that the latus rectum of this parabolic trajectory is \(c^2/d\), where \(d^2=(a-b)^2+c^2\), and that the range on the ground level is \[ c\{d^2+2(a+b)d+(a-b)^2\}^{\frac{1}{2}}/2d. \]
A jointed framework \(ABCD\) consisting of four equal uniform rods is caused to rotate in a horizontal plane about the corner \(A\) with a given angular velocity. It is held rigid either (a) by a tie between \(C\) and \(A\) in which the tension is \(T\), or (b) by a light strut between \(B\) and \(D\) in which the thrust is \(P\). Show that \(\frac{T}{AC}\) in case (a) is equal to \(\frac{P}{BD}\) in case (b).
Find the cartesian equations for the smooth cycloid on which a particle will describe simple harmonic oscillations of period \(T\) under the action of gravity. Show that if the particle is projected horizontally at the lowest point with the velocity acquired in falling through a height equal to half the length of the cycloid from cusp to cusp it will take a time \(\frac{1}{4}T\) to reach the cusp.