Solve for \(x, y, z\) in terms of \(p, q, r\) the simultaneous equations \begin{align*} x+y+z &= 1, \\ x+py+3z &= 2, \\ x+y+qz &= r, \end{align*} and obtain the solutions in the special cases: (i) \(p=1, q \ne 1\), (ii) \(p \ne 1, q=1\), (iii) \(p=q=1\), giving any additional condition required for the existence of the solutions.
Prove that the geometric mean of a number of positive quantities is never greater than their arithmetic mean. If \(a_1, a_2, \dots, a_n\) are positive quantities, and \(n\) and \(r\) are positive integers (\(n>r\)), prove that if \[ a_1 + a_2 \cdot 2^{n-r} + a_3 \cdot 3^{n-r} + \dots + a_n \cdot n^{n-r} \le 1, \] then \[ \frac{1^{r}}{a_1} + \frac{2^r}{a_2} + \frac{3^r}{a_3} + \dots + \frac{n^r}{a_n} \ge n^2 n^r. \]
By consideration of \(\frac{1+x}{1+x^3}\), or otherwise, prove that \[ 1-3n + \frac{3n(3n-3)}{2!} - \frac{3n(3n-4)(3n-5)}{3!} + \dots \] \[ \quad + (-1)^r \frac{3n(3n-r-1)\dots(3n-2r+1)}{r!} + \dots \] \[ = \begin{cases} (-1)^{\frac{3n}{2}} \cdot 2 \text{ if } n \text{ is an even integer or} \\ (-1)^{\frac{3n-1}{2}} \cdot 3n \text{ if } n \text{ is an odd integer} \end{cases} \] \[ = (-1)^n \cdot 2. \qquad \text{(OCR error in problem statement)} \]
Show that the cubic equation \(x^3+3px+q=0\) can be expressed in the form \(a(x+b)^3 - b(x+a)^3=0\) by proper choice of \(a\) and \(b\). Hence solve the equation \(x^3-9x+28=0\).
If \(y=a^{x^x}\), where \(a\) is a positive constant, prove that \(y\) has a minimum value and that \(x\) has a maximum value. Find the limit of \(y\) as \(x \to 0\) through positive values, and sketch the graph of \(y\) for positive values of \(x\).
By taking \(u=x+y, v=x-y\) as new variables, or otherwise, show that, if \(f\) is a function of the variables \(x\) and \(y\) such that \(\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}\), then \(f\) is expressible completely in terms of \(x+y\). Without assuming the properties of the circular functions, show that, if \[ f'(x)=1+\{f(x)\}^2 \quad \text{and} \quad f(0)=0, \] then \[ \frac{f(x)+f(y)}{1-f(x)f(y)} = f(x+y). \]
As \(t\) varies, the line \(x-t^2y+2at^3=0\) envelops a curve \(C\). Show that for each value of \(t\) other than \(t=0\) the line cuts \(C\) at a point \(P\) distinct from the point at which the line touches \(C\). Find the equation of the normal to \(C\) at \(P\) and deduce that the centre of curvature at \(P\) is \[ \left( -20at^3 - \frac{3a}{4t}, 48at^5 - 3at \right). \] Discuss the case \(t=0\).
Assuming the earth's surface to be spherical, show that the mean distance from the north pole of all points on and inside the surface is 1.2 times the radius of the earth. For all points on the surface in the southern hemisphere show that the mean distance from the north pole as measured by the shortest path lying wholly on the surface is approximately 1.24 times the mean distance of the same points as measured in a straight line.
If \(0 < \theta_1 < \theta_2 < \pi\), prove that the volume swept out in one complete revolution about the line \(\theta=0\) by the plane region bounded by the curve \(r=f(\theta)\) and the lines \(\theta=\theta_1, \theta=\theta_2\) is \[ \frac{2\pi}{3} \int_{\theta_1}^{\theta_2} r^3 \sin\theta \, d\theta. \] Prove that the locus of a point which moves so that the product of its distances from two fixed points at a distance \(2c\) apart is \(c^2\) is a surface of revolution enclosing a volume \[ \pi c^3 \left\{ \log(1+\sqrt{2}) - \frac{\sqrt{2}}{3} \right\}. \]
Find the values of: