\(A_1A_2A_3A_4A_5A_6A_7\) is a regular heptagon, and the lines \(A_2A_5, A_2A_6, A_2A_7\) meet \(A_1A_4\) in \(B_5, B_6, B_7\) respectively. Prove that
Five points in a plane are given, no three of them lying on a straight line. Prove that at least one of the quadrangles determined by a set of four out of the five points is convex.
Express the polynomials \(x^8-34x^4+1\), \(x^8+34x^4+1\) as the product of irreducible polynomials with integer coefficients.
Prove that the radius of curvature of the envelope of the line \[ x\cos\theta+y\sin\theta+f(\theta)=0 \] at its point of contact with the line is \(\pm[f(\theta)+f''(\theta)]\). Deduce that a circle of radius \(a (\ne 0)\) is the only curve whose radius of curvature at every point is equal to \(a\).
If \(x\) is any complex root of the equation \(x^{11}-1=0\), and if \[ a=x+x^3+x^4+x^5+x^9, \quad b=x^2+x^6+x^7+x^8+x^{10}, \] prove that \((a-b)^2 = -11\). Show further that \[ (x^3+1)[a-b-2(x-x^{10})] = x^3-1, \] and deduce that \[ \tan\frac{3\pi}{11} + 4\sin\frac{2\pi}{11} = \sqrt{11}. \]
If \[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^n, \] prove that \[ \int_{-1}^1 P_m(x)P_n(x) \, dx = 0 \text{ if } m \ne n \] \[ = \frac{2}{2n+1} \text{ if } m=n. \]
The diagram represents a girder bridge in which the horizontal and vertical girders are of equal length, and the remaining girders are inclined at an angle of 45\(^\circ\) to the horizontal. The bridge rests on smooth horizontal supports at \(A\) and \(G\) and carries loads at \(B, C, D, E, F\) as indicated. The weights of the girders may be neglected, and all the joints are assumed to be smooth. [A diagram shows a Warren truss with vertical members. A is the left support, G is the right support. Top chord nodes are H, K, L, M, N. Bottom chord nodes are A, B, C, D, E, F, G. The bays are square. From left to right, loads are 4W at B, 3W at C, 4W at D, W at E, 2W at F.] Draw the force diagram, and deduce from it the stress in the girder \(LD\). Indicate in which of the girders the stress due to the loads is a tension, and in which it is a thrust.
Three equal smooth uniform spheres \(A, B, C\) lie in that order on a smooth horizontal table, with their centres in a straight line. The coefficient of restitution between any pair of the spheres is \(e\). The sphere \(A\) is projected along the line in the direction of \(B\). Prove that there will be three, and only three collisions between the spheres if \(1 > e \ge 3 - 2\sqrt{2}\). If \(e=3-2\sqrt{2}\), show that the final kinetic energy of the system is approximately one-third of the initial kinetic energy.
A small raindrop falling through a cloud acquires moisture by condensation from the cloud. When the mass of the raindrop is \(m\), the rate of increase of mass per unit time is \(km\), where \(k\) is small. The raindrop starts from rest. Prove that when it has fallen a distance \(h\) through the cloud, its velocity is given approximately by \[ v^2 = 2gh(1-\tfrac{2}{3}k\sqrt{2h/g}). \] (The resistance of the cloud to the motion is to be neglected, and the cloud is assumed to be stationary and of infinite mass.)
A uniform spherical ball of radius \(a\) is at rest on a rough horizontal table, and is set in motion by a horizontal blow in a vertical plane through the centre at a distance \(\frac{2}{3}a\) above the table. Show that when the ball ceases to slip its linear velocity is \(5/14\) of its initial linear velocity.