Prove that for odd values of \(n\), \[ \int_0^\pi \frac{\cos n\theta}{\cos\theta} d\theta = (-1)^{\frac{n-1}{2}}\pi. \] If, for odd values of \(n\), \(I_n = \int_0^\pi \frac{\cos^2 n\theta}{\cos^2\theta}d\theta\), shew that \begin{align*} I_n &= I_{n-2}+2\pi \\ &= n\pi. \end{align*}
Shew that the area of the surface of the prolate spheroid obtained by the rotation of an ellipse of eccentricity \(e\) about its major axis (\(2a\)) is \[ A = 2\pi a^2\left[ \sqrt{1-e^2} + \frac{\sin^{-1} e}{e} \right] \] and that the centroid of the half surface bounded by the central circular section is at a distance \(d\) from the plane of that section, where \[ Ad = \frac{4\pi a^3}{3} \frac{1}{e^2}\left[ \sqrt{1-e^2} - (1-e^2)^{\frac{3}{2}} \right]. \]
Find the maximum and minimum values for real values of \(x,y,z\), of the quantity \(x^2+y^2+z^2\), subject to the conditions that \begin{align*} lx+my+nz &= 0, \\ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} &= 1, \end{align*} where \(a,b,c\) are positive and \(l,m,n\) are real. Verify that the values determined are real and positive.
(a) Prove that the three lines joining the mid-points of opposite edges of a tetrahedron meet in a point. (b) \(ABCD\) is a tetrahedron in which the edge \(AD\) is at right angles to the edge \(BC\), and the edge \(BD\) to the edge \(CA\). Prove that the edge \(CD\) is at right angles to the edge \(AB\). Prove also that in this case the perpendiculars from the vertices on to the opposite faces meet in a point.
Shew that the locus of a point \(P\) whose rectangular Cartesian coordinates are given by \[ x:y:1 = at^2+2bt+c : a't^2+2b't+c' : a''t^2+2b''t+c'', \] where \(t\) is a variable parameter, is in general a conic. Examine the particular cases in which (i) the determinant \[ \Delta = \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} \] vanishes, (ii) all first minors of \(\Delta\) vanish. Find the equation of the tangent at the point \(t\), and prove that the coordinates \((\xi, \eta)\) of the pole of the line joining the points \(t_1\) and \(t_2\) are given by \[ \xi:\eta:1 = at_1t_2+b(t_1+t_2)+c : a't_1t_2+b'(t_1+t_2)+c' : a''t_1t_2+b''(t_1+t_2)+c''. \]
\(g(x), h(x)\) are given polynomials, of degrees \(m, n\) respectively (\(m \ge n\)). Prove that the degree of a polynomial (not vanishing identically) which can be written in the form \[ \text{(1)} \qquad G(x)g(x) + H(x)h(x), \] where \(G(x)\) and \(H(x)\) are polynomials, can be as small as, but not smaller than, a definite integer \(\nu (\ge 0)\). Prove also that the polynomial \(\chi(x)\), which is of the form (1) and of degree \(\nu\), and in which the coefficient of \(x^\nu\) is unity, is unique. Prove that \(\chi(x)\) is a common factor of \(g(x)\) and \(h(x)\). Prove also that in the expression of \(\chi(x)\) in the form (1), \(G(x)\) and \(H(x)\) can be found of degrees less than \(n\) and \(m\) respectively.
A function \(\psi_n(x)\) is defined by the equation \[ \psi_n(x) = \frac{d^n}{dx^n}\frac{\sqrt{x}}{1+x} = \frac{\sqrt{x}}{(x+1)^{n+1}}\Psi_n(x). \] Shew
The portion of the curve \(y=f(x)\) included between the ordinates \(x=a\) and \(x=b\) (\(a < b\)) is rotated about the axis of \(x\). Prove that the volume of the surface of revolution so obtained is \[ V = \pi \int_a^b \{f(x)\}^2 dx, \] and that the area of the curved surface is \[ S = 2\pi \int_a^b f(x) \{1+(f'(x))^2\}^{\frac{1}{2}} dx, \] where \(f'(x) = \dfrac{d}{dx}f(x)\). Find the volume and area of the surface of the solid obtained by rotating the portion of the cycloid \[ x=a(\theta+\sin\theta), \quad y=a(1+\cos\theta) \] between two consecutive cusps about the axis of \(x\).
A coplanar system of forces acts on a rigid body. Shew that the system is equivalent either to a single force, or to a single couple, stating clearly the assumptions which are needed in the course of the proof. Shew further that the work done by the system in an arbitrary infinitesimal displacement of the body is equal to the work done by the equivalent force or couple.
A uniform lamina in the shape of an equilateral triangle \(ABC\) of side \(a\) is free to move in a vertical plane, the edges \(AB, BC\) resting on two smooth pegs \(P, Q\) at the same level and at distance \(b\) apart. If \(G\) is the centroid of the lamina, shew that, whatever the inclination of the lamina, \(BG\) passes through a fixed point. Shew also that the height of \(G\) above this fixed point when \(BG\) makes an angle \(\theta\) with the vertical is \[ (a\cos\theta-b-b\cos 2\theta)/\sqrt{3}. \] Hence shew that the position of equilibrium with \(BG\) vertical is stable if \(a < 4b\), and unstable if \(a > 4b\). Is the equilibrium stable or unstable if \(a=4b\)?