Express \[ \frac{3x^2+1}{(x-1)^3(x^2+2)(x-3)} \] in terms of partial fractions, and expand as a series of increasing powers of \(x\), stating the coefficients of \(x^{2n}\) and \(x^{2n+1}\).
Find the limitations on the value of \(a\), in order that \(\dfrac{x^2+4x-5}{x^2+2x+a}\) may take every real value for real values of \(x\). \par Determine the restriction on the values of the function for all real values of \(x\), when \(a\) does not satisfy the limitations.
Shew how to find the equation whose roots are the squares of those of a given algebraic equation. \par Hence or otherwise find the equation whose roots are \[ (b+c+d-a)^2, (c+d+a-b)^2, (d+a+b-c)^2, (a+b+c-d)^2, \] where \(a,b,c,d\) are the roots of the equation \(x^4+px^3+qx+r=0\). \par Verify for the case \(p=r=0\), that the equation obtained is \(x^4-64q^2x=0\).
By the method of differences, or otherwise, shew that the series \[ 1+5+15+35+70+126+\dots, \] can be represented as a series whose \(n\)th term is a polynomial in \(n\) with rational coefficients. Find the \(n\)th term and the appropriate scale of relation.
If \(x>0\), prove that \((x-1)^2\) is not less than \(x(\log x)^2\). \par Discuss the general behaviour of the function \((\log x)^{-1}-(x-1)^{-1}\) for positive values of \(x\) and with special reference to \(x=1\). \par Sketch the graph of the function.
Prove that, if \(u\) and \(v\) are functions of \(x\) and \(y\) such that \(\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial y}-\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial x}=0\), then \(v\) may be expressed in terms of \(u\) alone. \par Verify that this condition is satisfied in the case \(u=\dfrac{x-y}{1+xy}, v=\dfrac{(x-y)(1+xy)}{(1+x^2)(1+y^2)}\), and express \(v\) in terms of \(u\).
A family of conics having a fixed point S as one focus and major axes of given length \(2a\) along a given line through S is inverted with respect to a circle, centre S and radius \(k\). Find the envelope of the inverse curves.
Find the equation of the normal and the centre and radius of curvature of the curve \(ay^2=x^3\) at the point \((at^2, at^3)\). \par Shew that the length of the arc of the evolute between the points corresponding to \(t=0\) and \(t=1\) is \(\dfrac{13\sqrt{13}-8}{6}a\).
Determine, in terms of \(\theta\) and the length of the latus rectum, the area of the region bounded by a parabola and a focal chord inclined at an angle \(\theta\) to the axis of the parabola. \par Find the volume swept out by this region when revolved about the directrix through an angle \(2\pi\). \par Check both results by considering the special case \(\theta=\dfrac{\pi}{2}\).
O is the centre of a rectangle ABCD. E is the mid-point of CD and F is the mid-point of AD. AB, BC are of length \(2a, 2b\) respectively. \par A uniform lamina in the shape ABCEFA is of mass M. Find the moment of inertia of the lamina about the line through its mass centre and parallel to AB.