State and prove a rule for expressing \[ \frac{P(x)}{Q(x)} \] as the sum of a polynomial and partial fractions, where \(P\) and \(Q\) are polynomials, and \(Q\) has no repeated factors. \par Express in this form \[ \frac{(x-a)(x-b)(x-c)(x-d)}{(x+a)(x+b)(x+c)(x+d)}, \] (i) when \(a,b,c,d\) are all unequal, (ii) when they are all equal.
Shew that if \(x\) is added to all the elements of any determinant, the resulting determinant has the value \(A+Bx\), where \(A\) and \(B\) are independent of \(x\). \par Shew that for the determinant \[ \begin{vmatrix} a & b & b & b \\ p & a & b & b \\ q & r & a & b \\ s & t & u & a \end{vmatrix} \] \(A = \Delta\) and \(B=\Delta-(a-b)^4\), where \(\Delta\) is the value of the determinant; and hence evaluate \[ \begin{vmatrix} a & b & b & b \\ c & a & b & b \\ c & c & a & b \\ c & c & c & a \end{vmatrix}. \]
Find the condition that the \(n\)th term in the expansion of \((1-x)^{-k}\) exceed the next, assuming that \(k > 0\) and \(0 < x < 1\). Hence find the position and value of the greatest terms of the expansions of
Solution: The ratio of the \((n+1)\)th term in the expansion of \((1-x)^{-k}\) to the \(n\)th term is \(\displaystyle \frac{(-k-n)}{n+1}(-1)x = \frac{n+k}{n+1}x\), which is increasing if \((n+k)x > n+1 \Leftrightarrow kx-1 > (1-x)n \Leftrightarrow \frac{kx-1}{1-x} > n\). Therefore the largest element is when \(\displaystyle \left \lfloor \frac{kx-1}{1-x} \right \rfloor\).
The function \(\cot\theta + k\sec\theta\), (\(k>0\)), has a turning value when \(\theta=\alpha\). Find a cubic satisfied by \(\sin\alpha\), and shew, by a graph or otherwise, that just one root of this cubic gives real values of \(\alpha\). \par Shew that of the two turning values of the function between \(0\) and \(\pi\) one is a minimum and the other a maximum, and sketch the graph of \(\cot\theta+2\sec\theta\) for the range \(-\pi < \theta < \pi\).
The points \(P_1, P_2, \dots, P_n\) are the vertices of a regular \(n\)-agon inscribed in a circle \(C_0\), and \(A\) is a point on a concentric circle \(C_1\). Shew that \[ AP_1^2 + AP_2^2 + \dots + AP_n^2 \] remains fixed when \(A\) moves round the circle \(C_1\).
Shew that if \(u=(1-x^2)^n\), \[ u'(1-x^2) + 2nxu = 0; \] and by differentiating this equation \(n+1\) times, shew that \[ (1-x^2)\frac{d^2P}{dx^2} - 2x\frac{dP}{dx} + n(n+1)P=0, \] where \[ P(x) = \left(\frac{d}{dx}\right)^n \{(1-x^2)^n\}. \] By proceeding similarly with \[ v=e^{-x^2}, \] shew that if \[ H(x)e^{-x^2} = \left(\frac{d}{dx}\right)^n e^{-x^2}, \] then \[ \frac{d^2H}{dx^2} - 2x\frac{dH}{dx} + 2nH = 0. \]
Find the equations of the tangent and normal to the curve \(\phi(x,y)=0\) at the point \((x_0, y_0)\) on it. \par Shew that the normal distance of a point \((x_0, y_0)\) on \(\phi(x,y)=a\) from the curve \(\phi(x,y)=a+\delta\) is \[ |\delta|\left[\left(\frac{\partial\phi}{\partial x_0}\right)^2 + \left(\frac{\partial\phi}{\partial y_0}\right)^2\right]^{-\frac{1}{2}}, \] if \(\delta^2\) is neglected, and provided that \(\frac{\partial\phi}{\partial x_0}\) and \(\frac{\partial\phi}{\partial y_0}\) do not vanish.
Give an account of the method of finding the asymptotes of the curve \(P(x,y)=0\), where \(P\) is a polynomial in \(x\) and \(y\). \par Shew that \(x-y=3\) is an asymptote of \[ (x-y+1)(x-y-2)(x+y) = 8x-1, \] find the other asymptotes, and sketch the curve.
Shew that in the range \(a < x < b\),
\[ \frac{d}{dx}\left( -2\tan^{-1}\sqrt{\frac{b-x}{x-a}} \right) = \frac{1}{\sqrt{(b-x)(x-a)}}, \]
and integrate with respect to \(t\)
\[ \frac{1}{(1-kt)\sqrt{1-t^2}} \quad (0
Shew that if \[ I_m = \int_0^\infty e^{-x}\sin^m x dx \] and \(m\ge 2\), then \[ (1+m^2)I_m = m(m-1)I_{m-2}; \] and hence evaluate \(I_4\).