A board in the shape of a right-angled isosceles triangle rests in a vertical plane with its equal sides in contact with two rough pegs \(P, Q\). If \(b\) is the length of \(PQ\), which makes an angle \(\alpha\) with the vertical, \(a\) the length of the hypotenuse and \(\psi\) the angle which the side in contact with the lower peg makes with the vertical, shew that the board rests in limiting equilibrium if \[ a\cos\left(\frac{\pi}{4}\pm\psi\right)=3b\cos\alpha\sin(2\psi+\alpha\pm\lambda), \] where \(\lambda\) is the angle of friction.
A particle is projected under gravity from a given point and at the same instant a small particle, whose mass is \(\mu\) times that of the other particle, is dropped so as to meet it at the highest point of its path. If the two particles then coalesce, shew that, neglecting higher powers of \(\mu\) than the second, the range on the horizontal plane through the point of projection is diminished by \((\mu-\frac{1}{4}\mu^2)\) times the range which the particle would have had by itself.
A point \(P\) moves in a plane with a velocity compounded of two equal constant velocities, one in a fixed direction and the other in the direction of the line joining a given fixed point \(S\) to \(P\). Shew that the particle describes a parabola and that the acceleration at any instant is proportional to the angular velocity of the line \(SP\).
The normal at a point \(P\) of a parabola touches the evolute at \(Q\), and \(R\) is the centre of curvature of the evolute at \(Q\). Prove that the straight line \(PR\) makes an angle \(\cot^{-1}(\frac{1}{2}\cot\psi+\cot 2\psi)\) with the axis, where \(\psi\) is the inclination of the tangent at \(P\) to the axis.
Prove that the equation \[ \begin{vmatrix} a+x & h & g \\ h & b+x & f \\ g & f & c+x \end{vmatrix} = 0 \] has in general three real and distinct roots. Prove that the same holds of the equation \[ \begin{vmatrix} a+x & h+x\cos\gamma & g+x\cos\beta \\ h+x\cos\gamma & b+x & f+x\cos\alpha \\ g+x\cos\beta & f+x\cos\alpha & c+x \end{vmatrix} = 0. \]
Prove that, if \(n\) be an odd integer, \[ \sin n\theta = n\sin\theta - \frac{n(n^2-1^2)}{3!}\sin^3\theta + \frac{n(n^2-1^2)(n^2-3^2)}{5!}\sin^5\theta - \dots. \] Deduce, by considering the equation \(\sin n\theta=1\), that \[ \text{cosec}\frac{\pi}{2n} - \text{cosec}\frac{3\pi}{2n} + \text{cosec}\frac{5\pi}{2n} - \dots \pm \text{cosec}\frac{(n-2)\pi}{2n} = \frac{n\pm 1}{2}, \] according as \(n\) is of the form \(4m\pm 1\).
Shew that if \(f(x,y)\) is a function of \(x\) and \(y\) with continuous first derivatives, and if \(f=0\) and \(\dfrac{\partial f}{\partial y} \neq 0\) for \(x=a, y=b\), then there is a unique function \(y=\phi(x)\) which satisfies the equation \(f(x,y)=0\) identically in the neighbourhood of \(x=a, y=b\); and that it has a continuous derivative given by \[ \frac{dy}{dx} = -\frac{\partial f}{\partial x} \bigg/ \frac{\partial f}{\partial y}. \]
Prove that the necessary and sufficient condition for the integrability of \[ Pdx+Qdy+Rdz=0 \] is \[ P\left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) + Q\left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) + R\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) = 0. \] Solve the equation \[ (y-z)\frac{\partial z}{\partial x} + (x^2+y)\frac{\partial z}{\partial y} + x^3+z=0. \]
Shew that \(\log|f(x+iy)|\), where \(f\) is an analytic function, is a solution of Laplace's equation \[ \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = 0. \] Transform this equation to polar coordinates \((r,\theta)\); and shew that \[ \left(\frac{d}{d\log r}\right)^2 \log|f(x+iy)| \geq 0 \] at the point of the circle \(x+iy=r\) at which \(|f(x+iy)|\) assumes its greatest value.
Shew that the coordinates of any point on the developable surface, which is the envelope of the polar planes of a fixed point \((f,g,h)\) with respect to quadrics confocal with \(x^2/a+y^2/b+z^2/c=1\), may be written in the form \[ xf = \frac{(a+\lambda)^2(a+\mu)}{(a-b)(a-c)}, \quad yg = \frac{(b+\lambda)^2(b+\mu)}{(b-a)(b-c)}, \quad zh = \frac{(c+\lambda)^2(c+\mu)}{(c-a)(c-b)}, \] where \(\lambda, \mu\) are parameters. Shew that the equation \(\lambda=\) constant defines a generator, and that the equation \(\mu=\) constant defines a parabola, which together with a generator makes up the complete intersection of the surface by the polar plane with respect to the confocal of parameter \(\mu\).