Find the necessary and sufficient conditions that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] represent (i) two straight lines, (ii) two coincident straight lines. Prove that in general three of the conics represented by the equation \[ ax^2+2hxy+by^2+2gx+2fy+c+\lambda(a'x^2+2h'xy+b'y^2+2g'x+2f'y+c') = 0, \] for varying values of \(\lambda\), are line-pairs. State geometrically the conditions that more than three of the conics be line-pairs.
(\(\alpha\)) Prove that the arithmetic mean of any number of positive quantities is not less than their geometric mean. When are the two means equal? (\(\beta\)) Prove the inequalities:
Shew that if the elements of the determinant \(\Delta\) are functions of \(x\), \(d\Delta/dx\) is the sum of the determinants formed by differentiating the separate rows of \(\Delta\). If \(\Delta(x)\) is formed from the \(n\) functions \(f_1(x), f_2(x), \dots, f_n(x)\) as follows: \[ \begin{vmatrix} f_1(x), & f_2(x), & \dots, & f_n(x) \\ f_1'(x), & f_2'(x), & \dots, & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x), & f_2^{(n-1)}(x), & \dots, & f_n^{(n-1)}(x) \end{vmatrix} \] find \(d\Delta/dx\). If \(D(x)\) is formed in the same way from the \(n\) functions \[ \phi(x)f_1(x), \phi(x)f_2(x), \dots, \phi(x)f_n(x), \] shew that \(D(x) = \{\phi(x)\}^n \Delta(x)\). By considering the case \(\phi(x) = 1/f_1(x)\) prove by induction that if \(\Delta(x) = 0\) for all values of \(x\), there are constants \(c_1, c_2, \dots, c_n\), not all zero, such that \(c_1f_1(x)+c_2f_2(x)+\dots+c_n f_n(x) = 0\).
If \(P_0, P_1, \dots, P_{n-1}\) are \(n\) equidistant points round a circle of unit radius, and \(a_r\) is the distance \(P_0P_r\), prove that \(a_1a_2\dots a_{n-1} = n\). Find also \(a_1+a_2+\dots+a_{n-1}\) and deduce that when \(n\) is large the average distance of the points from \(P_0\) is approximately \(4/\pi\).
A curve \(C\) touches the \(x\)-axis at the origin. Obtain the expansions \[ x=s-\frac{1}{6}\kappa^2 s^3 + \dots, \quad y=\frac{1}{2}\kappa s^2 + a s^3 + \dots, \] where \(\kappa\) is the curvature (\(=1/\rho\)). Find the coefficient \(a\). Hence shew that for a short arc of length \(s\) \[ \text{arc} - \text{chord} = \frac{1}{24}\kappa^2 s^3, \] if higher powers of \(s\) are neglected. Let the tangent at \(P\) on the curve cut the \(x\)-axis in \(T\), and suppose that \(C\) has a point of inflexion at \(O\), but \(d\kappa/ds \ne 0\). Shew that \[ \lim_{P\to O} TO/PT = 2. \]
A coplanar system of forces acts on a rigid body. Shew that in general the system can be reduced to parallel forces through two given points \(A, B\) of the plane. In what cases is the reduction (i) impossible, (ii) possible in more than one way? \(P, Q, R, S\) are the mid-points of the sides \(AB, BC, CD, DA\) respectively of a square \(ABCD\), and forces \(\alpha, \beta, \gamma, \delta\) act in the lines \(PQ, QR, RS, SP\) respectively. It is required to reduce the system to a pair of parallel forces through \(A\) and \(B\). Shew that in general the direction of these forces makes an angle \((\frac{\pi}{4}+\theta)\) with \(AB\) (measured in the direction from \(AB\) to \(AD\)), where \(\tan\theta=\frac{\beta-\delta}{\alpha-\gamma}\), and that their magnitudes are \[ \frac{\alpha-\beta-3\gamma-\delta}{2(\cos\theta+\sin\theta)}, \quad \frac{\alpha+3\beta+\gamma-\delta}{2(\cos\theta+\sin\theta)} \] respectively. Discuss the exceptional cases.
Two similar uniform rods \(AB, AC\), each of length \(a\) and weight \(w\), are freely hinged together at \(A\). The ends \(B, C\) can move, without friction, on a horizontal line, and the system is free to move in the vertical plane through this line. \(B\) and \(C\) are joined by a light spring of natural length \(b\), the tension needed to double the length of the spring being \(\frac{1}{2}w\). If the angle \(BAC\) is denoted by \(2\theta\) (\(\theta\) being acute when \(A\) is above \(BC\)), shew that in a position of equilibrium \[ 2a\cos\theta\sin\theta = b(\cos\theta+\sin\theta). \] By expressing the equation as a quadratic equation for \((\cos\theta+\sin\theta)\), or otherwise, shew that there are one, two or three positions of equilibrium according as \(a/b <, = \text{ or } > \sqrt{2}\).
A bead slides under gravity along a smooth straight wire. Shew that if the bead starts from rest at a given origin \(P\) the locus of positions after a given time for different directions of the wire in a vertical plane through \(P\) is a circle. Find the straight line of quickest descent to a circle in a vertical plane from a point \(P\) of the plane. Shew also that if the time of quickest descent from \(P\) to the circle is prescribed the locus of \(P\) is a circle. Give a geometrical construction for the straight line of quickest descent from one circle to another, external to it, in the same vertical plane.
Two billiard balls, each of diameter \(b\), rest on a smooth table with their centres at a distance \(c\) apart. The balls are equidistant from a smooth vertical cushion, the shortest distance between the cushion and a ball being \(h\). One ball is projected along the table so as to strike the other after reflexion at the cushion. If \(e\) denotes the coefficient of restitution between the cushion and a ball, shew that the direction of projection must lie within an angle \(\alpha\), where \[ \tan\alpha = \frac{2be\sqrt{h^2(1+e)^2+c^2-b^2}}{h^2(1+e)^2+c^2e^2-b^2(1+e^2)}. \]
Explain briefly the principle of the conservation of energy in dynamics. A bead of mass \(m\) slides on a smooth parabolic wire whose axis is vertical and vertex upwards. It is connected to a particle of mass \(3m\) by a light inelastic string passing through a smooth ring at the focus. The system is let go from rest when the bead is at a depth \(\frac{1}{2}a\) below the vertex, where \(4a\) is the latus rectum of the parabola. Shew that in the subsequent motion the greatest velocity of the particle is \(\frac{1}{2}\sqrt{ga}\), and that the velocity of the bead when it passes through the vertex is \(\frac{1}{2}\sqrt{ga}\).