Problems

Show that, if \(p>q>0\) and if \(x>0\), then \[ \frac{x^p-1}{p} > \frac{x^q-1}{q}. \] Prove further that, for \(n\) a positive integer and \(s>0\), \[ \frac{1}{p(p+s)^n} \left|x^p - \frac{1}{s^n}\right| > \frac{1}{q(q+s)^n} \left|x^q - \frac{1}{s^n}\right|. \]

If \(\frac{\sin \theta}{x} = \frac{\sinh \phi}{y} = \cos \theta + \cosh \phi\), prove that \[ \frac{\partial x}{\partial \theta} = \frac{\partial y}{\partial \phi}, \quad \frac{\partial x}{\partial \phi} = -\frac{\partial y}{\partial \theta}. \] The function \(U(x,y)\) transforms by means of the above relations into \(V(\theta, \phi)\). Prove that \[ \left(\frac{\partial U}{\partial x}\right)^2 + \left(\frac{\partial U}{\partial y}\right)^2 = \left\{\left(\frac{\partial V}{\partial \theta}\right)^2 + \left(\frac{\partial V}{\partial \phi}\right)^2\right\} / (\cos \theta + \cosh \phi)^2. \]

(i) Find the limit of \((\cos x)^{\cot^2 x}\) as \(x \to 0\). \newline (ii) Determine constants \(a\) and \(b\) in order that \((1+a \cos 2x + b \cos 4x)/x^4\) may have a finite limit as \(x \to 0\), and find the value of the limit.

A tank in the form of a rectangular parallelepiped open at the top is to be made of uniform thin sheet metal and is to contain a given volume of water. What proportions must the depth bear to the length and breadth in order that the amount of metal required shall be least? \newline If, instead, the amount of metal is given and it is required to construct the tank of greatest cubic capacity, what must be the appropriate proportions? Would it be possible to deduce the result from the former result without detailed calculations?

Prove that \[ \int_0^{\pi/3} \sqrt{\cos 2x - \cos 4x} \, dx = \frac{1}{4}\sqrt{6} - \frac{1}{2}\sqrt{2} \log(2+\sqrt{3}). \]

Trace the curve \((x^2+y^2)^2 = 16axy^2\), and find the areas of its loops. \newline Prove that the smallest circle that will completely circumscribe the curve has radius \(3\sqrt{3}a\).

The centre of a circular disc of radius \(r\) is \(O\), and \(P\) is a point on the line through \(O\) perpendicular to the plane of the disc. If \(OP=p\), prove that the mean distance (with respect to area) of points of the disc from \(P\) is \(2\{(p^2+r^2)^{3/2}-p^3\}/3r^2\). \newline Find the mean distance (with respect to volume) of the interior points of a sphere of radius \(a\) from a fixed point of its surface.

Solve the equations:

1. [(i)] \(\frac{dy}{dx} = \frac{1-x-y}{1+x+y}\),
2. [(ii)] \(\left(\frac{dy}{dx}\right)^2 + (x+y)\frac{dy}{dx} + xy = 0\).

If \(P\) and \(Q\) are polynomials and if the degree of \(Q\) is less than the degree of \(P\), show that polynomials \(P_0, P_1, P_2, \dots\) all of degree less than \(Q\) can be found such that \[ P = \Sigma P_i Q^i. \] Prove that the polynomials \(P_i\) are unique. \newline If the roots \(\alpha_1, \alpha_2, \dots, \alpha_n\) of the equation \(Q=0\) are all different, find the polynomial of least degree which takes the value \(a\) whenever \(Q=0\) and whose derived polynomial takes the value \(b\) whenever \(Q=0\). \newline [The derived polynomial of \(P(x)\) is the coefficient of \(h\) in the expansion of \(P(x+h)\) in powers of \(h\).]

State a necessary and sufficient condition that \[ z^2+4axz+6byz+4cxy+dy^2+2\lambda(x^2-yz) \] shall be the product of two linear factors. \newline By taking \(z=x^2, y=1\), state briefly the steps to be taken in order to find the roots of the quartic equation \[ x^4+4ax^3+6bx^2+4cx+d=0. \] Hence find the roots of the equation \[ x^4-x^3-4x^2+x+1=0. \]