Find the number of ways of distributing twelve similar coins among seven persons so that at least two persons receive none.
Using \(\binom{x}{r}\) to denote \(\frac{x(x-1)(x-2)\dots(x-r+1)}{1.2.3\dots r}\) for positive integral values of \(r\), prove that \[ \binom{x+y}{r} = \binom{x}{r} + \binom{x}{r-1}\binom{y}{1} + \binom{x}{r-2}\binom{y}{2} + \dots + \binom{y}{r}, \] \[ \binom{x+r+1}{r} = \binom{x+r-1}{r} + 2\binom{x+r-2}{r-1} + 3\binom{x+r-3}{r-2} + \dots + r\binom{x}{1} + r+1. \]
In the series \(u_0+u_1+u_2+\dots+u_r+\dots+u_n\) successive terms are connected by the relation \(u_r+pu_{r-1}+qu_{r-2}=0\), where \(p\) and \(q\) are independent of \(r\). Explain how to find a general expression for \(u_r\). Prove that the series whose general term is \((u_r u_{r+2}-u_{r+1}^2)x^r\) is a geometrical progression.
State and prove a theorem on the effect on the value of a determinant of interchanging two rows or two columns. If \(a_1, a_2, \dots, a_n\), are in arithmetical progression, shew that the ratio of \[ \begin{vmatrix} a_1, & a_2, & a_3, & \dots & a_n \\ a_n, & a_1, & a_2, & \dots & a_{n-1} \\ a_{n-1}, & a_n, & a_1, & \dots & a_{n-2} \\ \vdots & & & & \vdots \\ a_2, & a_3, & a_4, & \dots & a_1 \end{vmatrix} \] to the sum of these \(n\) quantities is expressible solely in terms of \(n\) and the common difference.
(i) Prove the formula \(\frac{1}{r}\frac{dp}{dr}\) for the curvature at a point of a plane curve. (ii) Investigate the curvature at the point \((1,2)\) of the curve \[ (y-2)^2 = x(x-1)^2. \]
Shew that the number of real roots of the algebraic equation \(f(x)=0\) cannot exceed by more than unity that of its derived equation \(f'(x)=0\). Find the necessary and sufficient condition for the cubic equation \(x^3+3ax+b=0\) to have three real roots. \(a\) and \(b\) are real.
Determine the surface area and volume of the solid figure obtained by revolving the curve \(r=a(1+2\cos\theta)\) about its axis of symmetry.
State and prove Leibnitz' Theorem on the \(n\)th differential coefficient of the product of two functions. If \(y_n = \frac{d^n}{dx^n}(x^n f(x))\), where \(f(x)\) is a differentiable function of \(x\), shew that
Find the integral \[ \int (1-x^2)^{\frac{3}{2}} dx, \] and evaluate \[ \int_2^3 \frac{dx}{[(x-1)(3-x)]^{\frac{3}{2}}} \quad \text{and} \quad \int_{a+b}^{a+2b} \frac{dx}{2a+b - \sqrt{(x-a)(x-b)}}. \]
Describe some method of investigating the behaviour of the function \(\frac{f(x)}{\phi(x)}\) as \(x\) tends to \(a\), where \(f(a)=\phi(a)=0\). Determine the limiting values as \(x \to 0\) of: