Prove the convergency of the series whose \(n\)th term is \(\dfrac{1 \cdot 3 \cdot 5 \dots (2n-1)}{3^{n+1} \cdot n+2}\) and find the sum of an infinite number of terms of the series.
Prove that \(\Sigma[\cos 2A-\cos(B+C)](\cos B-\cos C) = \Sigma\sin(C-B)(\sin B+\sin C)\).
Two straight rulers with inches marked on them are laid across one another at a given angle so that the zero points do not coincide. Shew that perpendiculars drawn to the rulers at points having the same marks intersect on a line parallel to the bisector of the angle between the rulers.
Through a point \(O\) any two lines are drawn to cut, in \(P, Q\) and \(P', Q'\), any conic which touches two fixed lines through \(O\) at given points. Prove that \(PP'\) meets \(QQ'\) on a fixed line.
A pentagon \(ABCDE\) is formed of rods whose weight is \(w\) per unit length. The rods are freely jointed together and stand in a vertical plane with the lowest rod \(AB\) fixed horizontally, while the joints \(C, E\) are connected by a string. If \(BC\) and \(AE\) are of length \(a\), \(CD\) and \(DE\) of length \(b\) and the angles at \(A\) and \(B\) are each \(120^\circ\) and the angles at \(C\) and \(E\) are each \(90^\circ\), shew that the tension of the string is \(w\dfrac{a+5b}{2\sqrt{3}}\).
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 \[ \begin{vmatrix} a+x & h+x\cos\gamma & g+x\cos\beta \\ h+x\cos\gamma & b+x & f+x\cos\alpha \\ g+x\cos\beta & f+x\cos\alpha & c+x \end{vmatrix} = 0. \]