Prove that if two coplanar triangles are such that the lines joining corresponding vertices are concurrent, then the points of intersection of corresponding sides are collinear. \(A, B, C\), and \(D\) are the vertices of a quadrangle. \(AB\) and \(CD\) meet at \(F\); \(AC, BD\) at \(G\); \(AD, BC\) at \(H\). \(CD\) meets \(GH\) at \(P\), \(DB\) meets \(HF\) at \(Q\), \(BC\) meets \(FG\) at \(R\). Prove that \(P, Q\) and \(R\) are collinear. Shew that there are four such lines, forming a quadrilateral, whose diagonal triangle is \(FGH\).
Prove that the lines \(\alpha=0, \alpha-\lambda\beta=0, \beta=0, \alpha+\lambda\beta=0\), where \(\lambda\) is a constant, form an harmonic pencil. If two conics, \(S\) and \(S'\), each have double contact with a third conic, prove that the chords of contact meet at the point of intersection of a pair of common chords of \(S\) and \(S'\), the four lines forming an harmonic pencil. Conversely, if \(\alpha=0\) and \(\beta=0\) are a pair of common chords of \(S\) and \(S'\), shew that there is a conic having double contact with \(S\) along \(\alpha-\lambda\beta=0\), and with \(S'\) along \(\alpha+\lambda\beta=0\).
The function \(y=\sin x\) satisfies the differential equation \(\frac{d^2y}{dx^2}+y=0\). Assuming that \(\sin x\) can be expanded as a series in ascending powers of \(x\), deduce the series. Prove that, if \(y=\sin(n\sin^{-1}x)\), then \[ (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + n^2y = 0. \] Deduce the expansion of \(y\) as a series in ascending powers of \(x\), when such an expansion is possible. Discuss what happens when \(n\) is an integer.
Prove by induction or otherwise that \begin{align*} \cos(\alpha_1+\alpha_2+\alpha_3+\dots+\alpha_n) &= \{1-s_2+s_4-s_6+\dots\}\cos\alpha_1\cos\alpha_2\dots\cos\alpha_n, \\ \sin(\alpha_1+\alpha_2+\alpha_3+\dots+\alpha_n) &= \{s_1-s_3+s_5-s_7+\dots\}\cos\alpha_1\cos\alpha_2\dots\cos\alpha_n, \end{align*} where \(s_r\) denotes the sum of the products of \(\tan\alpha_1, \tan\alpha_2, \dots \tan\alpha_n\) taken \(r\) at a time. If \(\theta_1, \theta_2, \theta_3\) and \(\theta_4\) are the four roots of \(a\cos2\theta+b\sin2\theta+c\cos\theta+d\sin\theta+e=0\) between 0 and \(2\pi\), prove that \[ \frac{\cos\frac{1}{2}(\theta_1+\theta_2+\theta_3+\theta_4)}{a} = \frac{\sin\frac{1}{2}(\theta_1+\theta_2+\theta_3+\theta_4)}{b} = \frac{\sum \cos\frac{1}{2}(-\theta_1+\theta_2+\theta_3+\theta_4)}{-c} = \frac{\sum\sin\frac{1}{2}(-\theta_1+\theta_2+\theta_3+\theta_4)}{-d}. \]
Prove that the equations \begin{align*} a(x) \equiv a_0x^3+a_1x^2+a_2x+a_3=0 \\ \text{and} \quad b(x) \equiv b_0x^3+b_1x^2+b_2x+b_3=0 \end{align*} will have a common root if \[ \Delta \equiv \begin{vmatrix} a_0 & a_1 & a_2 & a_3 & 0 & 0 \\ 0 & a_0 & a_1 & a_2 & a_3 & 0 \\ 0 & 0 & a_0 & a_1 & a_2 & a_3 \\ 0 & 0 & b_0 & b_1 & b_2 & b_3 \\ 0 & b_0 & b_1 & b_2 & b_3 & 0 \\ b_0 & b_1 & b_2 & b_3 & 0 & 0 \end{vmatrix} = 0. \] If \begin{align*} a(x) &= (x-\lambda)(a_0'x^2+a_1'x+a_2') \\ b(x) &= (x-\lambda)(b_0'x^2+b_1'x+b_2') \end{align*} and \[ \Delta_1 \equiv \begin{vmatrix} a_0 & a_1 & a_2 & a_3 \\ 0 & a_0 & a_1 & a_2 \\ 0 & b_0 & b_1 & b_2 \\ b_0 & b_1 & b_2 & b_3 \end{vmatrix} \] prove that \[ \Delta_1 \times \begin{vmatrix} 1 & \lambda & \lambda^2 & \lambda^3 \\ 0 & 1 & \lambda & \lambda^2 \\ 0 & 0 & 1 & \lambda \\ 0 & 0 & 0 & 1 \end{vmatrix} = \begin{vmatrix} a_0' & a_1' & a_2' & 0 \\ 0 & a_0' & a_1' & a_2' \\ 0 & b_0' & b_1' & b_2' \\ b_0' & b_1' & b_2' & 0 \end{vmatrix} \] and hence that the equations \(a(x)=0, b(x)=0\) will have two common roots if \(\Delta=0\) and \(\Delta_1=0\).
Shew that a system of coplanar forces is equivalent to a couple if the geometric sum of the forces is zero, i.e. if the same forces acting on a particle would be in equilibrium. The coplanar forces \(P_1, P_2, \dots P_n\), whose geometric sum is not zero, act at points \(A_1, A_2, \dots A_n\) respectively. Shew that if the direction of each force is turned through an angle \(\theta\) the resultant force passes through a point \(C\) for all values of \(\theta\). Shew further that if \(A_2, A_3, \dots A_n\) are fixed, and \(A_1\) moves on a given curve, then \(C\) traces out a similar (but not in general similarly situated) curve. Explain how the exceptional cases (in which the curves are similarly situated) arise.
Two circular cylinders, \(A\) and \(B\), have their axes parallel in the same horizontal plane, \(A\) being fixed and \(B\) free to turn about its axis. A uniform heavy circular cylinder \(C\), having its axis parallel to those of \(A\) and \(B\), is placed in contact with them. The angle that the plane containing the axes of \(A\) and \(C\) makes with the horizontal is denoted by \(\alpha\), and the corresponding angle for \(B\) and \(C\) by \(\beta\). The angle of friction between \(A\) and \(C\) is denoted by \(\theta\), and that between \(B\) and \(C\) by \(\phi\). A couple is applied to \(B\) (in the sense which tends to move the highest point of \(B\) away from \(A\)) and gradually increased. Shew that equilibrium is broken by slipping between \(B\) and \(C\) if \[ \sin\alpha - \sin\beta > \cos\alpha\cot\theta - \cos\beta\cot\phi, \] and that this occurs when the couple attains the value \[ \frac{bW\sin\phi\cos\alpha}{\sin(\alpha+\beta+\phi)+\sin\phi}, \] where \(b\) denotes the radius of \(B\), and \(W\) the weight of \(C\).
Three equal particles \(A, B, C\) rest on a smooth table, \(A\) being joined to \(B\), and \(B\) to \(C\), by tight inelastic strings. The angle between the strings is \(\beta(<\pi/2)\). \(A\) is given a velocity \(u\) in the direction parallel to \(CB\). Shew that when the string \(AB\) again tightens, \(C\) starts off with velocity \(u/(3+4\tan^2\beta)\).
A man of mass \(m\) stands on an escalator of inclination \(\alpha\) which ascends with uniform velocity. He walks up the escalator, and finally comes to rest again relative to the escalator. Shew that if \(a\) is the distance travelled by the escalator, and \(b\) the distance travelled by the man relative to the escalator, then of the total work done \(mga\sin\alpha\) is supplied by the engine and \(mgb\sin\alpha\) by the man. (The man is to be considered as a particle, and the escalator as a continuous gradient without steps. The acceleration of the man relative to the escalator is assumed to be a continuous function of the time.) Consider next the more general problem in which the velocity of the escalator at any instant is \(u\), and of the man relative to the escalator \(v\), \(u\) and \(v\) being functions of the time having continuous derivatives. Shew that of the total work done on the man in any interval of time \(t_1\) to \(t_2\) the amount supplied by the engine is \[ mga\sin\alpha + \frac{1}{2}m(u_2^2-u_1^2) + m\int_{t_1}^{t_2} u\frac{dv}{dt}dt, \] and by the man \[ mgb\sin\alpha + \frac{1}{2}m(v_2^2-v_1^2) + m\int_{t_1}^{t_2} v\frac{du}{dt}dt, \] where \(a\) is the distance travelled by the escalator, and \(b\) by the man relative to the escalator, in the given interval.
A bead of mass \(m\) is free to slide on a smooth horizontal wire. A light rod of length \(a\) is freely attached to the bead, and carries a particle of mass \(m\) at the other end. The system rests in equilibrium, and the particle is struck a blow \(mu\) parallel to the wire. Shew that in the subsequent motion the rod will just become horizontal if \(u=2\sqrt{ga}\). Shew further that if \(u\) has this value the angular velocity of the rod when it makes an angle \(\theta\) with the horizontal is \[ \sqrt{\frac{4g\sin\theta}{a(2-\sin^2\theta)}}. \]