\(ABCD\) is a cyclic quadrilateral whose diagonals \(AC, BD\) meet in \(X\). \(E\) and \(F\) are the feet of the perpendiculars from \(X\) to \(AD\) and \(BC\) respectively, and \(G\) is the mid-point of \(AB\). Points \(P\) and \(Q\) are taken on \(AD, BC\) respectively such that \(EP=AE, FQ=BF\). Show that the triangles \(AXQ\) and \(PXB\) are congruent, and hence prove that \(GE=GF\).
A man can walk at the rate of 100 yd. a minute, which is \(n\) times faster than he can swim. He stands at one corner of a rectangular pond 80 yd. long and 60 yd. wide. To get to the opposite corner he may walk round the edge, swim straight across or walk part of the way along the longer side and then swim the rest. If he is to make the trip in the least time, how should he proceed and how long does he take if (i) \(n=\frac{5}{3}\), (ii) \(n=\frac{4}{3}\), (iii) \(n=\frac{5}{4}\)?
A uniform heavy rod of weight \(6w\) and length \(3a\) is freely hinged at one end and kept horizontal by a support distant \(2a\) from the hinge. A weight \(w\) is suspended from the rod midway between the hinge and the support, and a weight \(w\) is suspended from the free end of the rod. Calculate the bending moment at all points of the rod and illustrate your result by a sketch. Show that the bending moment has a stationary value at a point distant less than \(a\) from the hinge, but attains its greatest numerical value at the support.
The ends \(A, B\) of a heavy uniform rod of weight \(w\) and length \(2a\) are attached by two light inextensible strings each of length \(b\) to two points \(C, D\) at the same level a distance \(2c\) apart where \(a+c>b>\sqrt{(a^2+c^2)}\). The rod is now moved in such a way that it always remains horizontal and so that the mid-point of \(AB\) remains vertically below the mid-point of \(CD\). Find the potential energy of the rod as a function of the angle between \(AB\) and \(CD\), if both strings remain taut and do not cross. Also find the couple necessary to keep \(AB\) perpendicular to \(CD\).
The point of suspension of a simple pendulum with a bob of mass \(m\) is made to move in a horizontal straight line with constant acceleration \(f\); if \(3f^2=g^2\), and if the string is initially at rest and vertical with the bob vertically below the point of suspension, find the greatest angle the string makes with the vertical in the motion. Also show that the maximum tension in the string is \(2(\sqrt{3}-1)mg\).
A particle of mass \(m\) is suspended from a fixed point \(O\) by a light elastic string of natural length \(l\). The particle hangs in equilibrium under gravity, and the length of the string is \(l+a\); an upward vertical impulse is then applied to the particle and it first comes to rest at \(O\). Show that the magnitude of the impulse is \(m[g(a+2l)]^{\frac{1}{2}}\), and find the time the particle takes to reach \(O\).
A reel of thread of radius \(a\) is unwound by moving the end of the thread in a plane \(p\) perpendicular to the axis of the reel in such a way that the free part of the thread is straight and moves with constant angular velocity \(\omega\); the reel is kept fixed, and it may be assumed that all the thread on the reel is in the plane \(p\). Find the magnitude and direction of the acceleration of the end of the thread when the free part of the thread is of length \(l\).
Prove that the geometric mean of a number of positive quantities can never exceed their arithmetic mean. Prove that if \(a_1, a_2, \dots, a_n\) are essentially positive but not all equal then \(\sum_{r \ne s} a_r/a_s > n(n-1)\).
Prove that \[ a^3+b^3+c^3-3abc = \tfrac{1}{2}(a+b+c)[(b-c)^2+(c-a)^2+(a-b)^2]. \] Hence, or otherwise, establish the identity \[ (a^3+b^3+c^3-3abc)^2 = (a^2-bc)^3 + (b^2-ca)^3 + (c^2-ab)^3 - 3(a^2-bc)(b^2-ca)(c^2-ab). \]
In the permutation (denoted by \(p\)) obtained by rearranging the integers 1 to \(n\) in any manner, the "number of inversions with respect to one of the given integers," say \(r\), is defined as the number of integers greater than \(r\) which precede it in the permutation \(p\), and is denoted by \(k_r\). The total number of inversions in the permutation is given by \(\sum_1^n k_r\) and is denoted by \(k\). Prove that if the permutation \(p\) is modified by the simple interchange of two integers, say \(r\) and \(s\), the change in the total number of inversions is \(2q+1\), where \(q\) is the number of integers that lie between \(r\) and \(s\) both in the original order of integers and in the permutation \(p\). Hence, or otherwise, show that if the permutation \(p\) had been effected by a number of simple interchanges, the total number of such interchanges is always odd or even with the total number of inversions.