Problems

If \({}^nC_r\) denotes the number of combinations of \(n\) things taken \(r\) at a time, establish the following results by considering \((1-x^2)^{2n}\), or otherwise: \[ \frac{1}{2}{}^{2n}C_r({}^{2n}C_r-1) = {}^{2n}C_{r-1}{}^{2n}C_{r+1} - {}^{2n}C_{r-2}{}^{2n}C_{r+2} + \dots + (-1)^{r-1}{}^{2n}C_{2r}, \] \[ {}^{2n}C_r = {}^nC_r + 2^{r-2}{}^nC_{r-2}{}^{n-r+2}C_1 + 2^{r-4}{}^nC_{r-4}{}^{n-r+4}C_2 + \dots, \] the last term being \({}^nC_{\frac{r}{2}}\) or \(2{}^{n-1}C_{\frac{r-1}{2}}\) according as to whether \(r\) is even or odd.

Establish Newton's method of approximating to the roots of an equation. Shew that between any two consecutive even integers there is one and only one real root of the equation \(\frac{1}{2x}=\tan\frac{\pi x}{2}\). Prove that for a large value of \(n\), the root between \(2n\) and \(2(n+1)\) is approximately \(2n+\frac{1}{2\pi n}\). Prove similar results for the equation \(4x=\tan\frac{\pi x}{2}\), with the result \(2n+1-\frac{1}{2\pi(2n+1)}\).

(a) If \(z\) is a function of two variables \(x,y\) in the form \(z=f(x,y)\), and if \[ f(Kx, Ky) = K^n f(x,y) \] where \(K\) is any constant, shew that \[ x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = nz. \] (b) If \(z\) is a function of \(\frac{x}{y}\), shew that \[ x^r\frac{\partial^r z}{\partial x^r} + {}^rC_1 x^{r-1}y\frac{\partial^r z}{\partial y \partial x^{r-1}} + \dots + {}^rC_s x^{r-s}y^s \frac{\partial^r z}{(\partial y)^s (\partial x)^{r-s}} + \dots + y^r\frac{\partial^r z}{\partial y^r} = 0. \]

\(A\) is the vertex and \(P\) any other point on a uniform catenary. The normals to the catenary at \(A\) and \(P\) intersect its directrix in \(O, N\), respectively. Draw the perpendicular bisector of \(ON\) to intersect \(PN\) internally at a point \(L\) say. If the tangents to the catenary at \(A\) and \(P\) intersect in \(K\), shew that the centroid of the arc \(AP\) is the point of intersection of straight lines drawn through \(K, L\), parallel respectively to the axis and directrix of the catenary.

Prove that the radius of curvature of a plane curve may be expressed in the form \(r\frac{dr}{dp}\). Shew that if for a curve the segment of the normal between any point on the curve and the corresponding centre of curvature subtends a constant angle \(\alpha\) (\(\ne \frac{\pi}{2}\)) at a fixed point, then for a suitable system of coordinates, \(r e^{-\phi\tan\alpha}\) is constant, where \(r\) is the length of the radius vector, and \(\phi\) one of the angles between it and the tangent to the curve. Discuss the case \(\alpha = \frac{\pi}{2}\).

Find a reduction formula for \(I_n = \int \frac{dx}{(ax^2+2bx+c)^n}\) in terms of \(I_n, I_{n-1}\). Hence or otherwise evaluate \[ \int_0^\infty \frac{dx}{(x^2+1)^n}, \quad \int_1^{\sqrt{3}} \frac{x^3-1}{x^2(x^2+1)^2}dx, \] where \(n\) is a positive integer.

Sketch the curve \(ay^2 = x(x-a)(x-b)\), where \(a\) and \(b\) are both positive. Prove that there are two and only two real points of inflexion. If \(a=b\), shew that the area of the loop is \(\frac{8}{15}a^2\).

Prove that the homogeneous coordinates of any point on a conic may be taken to be \((t^2, t, 1)\), where \(t\) is a parameter. The quadratic equations \(Q \equiv at^2+2bt+c=0\), \(Q' \equiv a't^2+2b't+c'=0\) determine two pairs of points on the conic; find

1. the geometrical relation between these pairs of points, when \[ ca'+c'a-2bb'=0, \]
2. the geometrical property of the pairs of points given by \(Q+\lambda Q'=0\), where \(\lambda\) is a parameter.

The equation of a conic referred to rectangular Cartesian coordinate axes is \[ ax^2+2hxy+by^2+2gx+2fy+c \equiv (a,b,c,f,g,h)(x,y,1)^2=0; \] prove that the line \(lx+my+1=0\) touches the conic, if \(\Sigma \equiv (A,B,C,F,G,H)(l,m,1)^2=0\), where \(A, B, \dots\) are the cofactors of \(a, b, \dots\) in the determinant \[ \delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} \] which is assumed not to be zero. Prove also that, if the line \(lx+my+1=0\) is a directrix of the conic, \[ \frac{l^2-m^2}{a-b} = \frac{lm}{h} = \frac{\Sigma}{\delta}, \] and interpret these equations geometrically when \(a=b, h=0\).

The set of numbers \(x_1, x_2, \dots, x_n\) are transformed into the set of numbers \(\xi_1, \xi_2, \dots, \xi_n\) by means of the equations \begin{align*} \xi_1 &= a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n, \\ &\vdots \\ \xi_n &= a_{n1}x_1 + a_{n2}x_2 + \dots + a_{nn}x_n; \end{align*} the set of numbers \(y_1, y_2, \dots, y_n\) are transformed into the set of numbers \(\eta_1, \eta_2, \dots, \eta_n\) in the same way, and the set of numbers \(\pounds_1, \pounds_2, \dots, \pounds_n\) are similarly transformed into the set of numbers \(X_1, X_2, \dots, X_n\). Shew that, if all the coefficients \(a_{ij}\) are real, and if \(a_{ij}=a_{ji}\), then

1. \(y_1\xi_1+y_2\xi_2+\dots+y_n\xi_n = x_1\eta_1+x_2\eta_2+\dots+x_n\eta_n\),
2. \(X_1X_1+x_2X_2+\dots+x_nX_n \geq 0\).

Shew, further, that if \[ (x_1\eta_1+x_2\eta_2+\dots+x_n\eta_n)^2 \leq M^2(x_1^2+x_2^2+\dots+x_n^2).(y_1^2+y_2^2+\dots+y_n^2), \] for all sets of numbers \(x_1, x_2, \dots, x_n\) and \(y_1, y_2, \dots, y_n\), then \[ X_1X_1+x_2X_2+\dots+x_nX_n \leq M^2(x_1^2+x_2^2+\dots+x_n^2). \]