Problems

Two particles \(A, B\) of masses \(m_1, m_2\) rest on a smooth horizontal plane connected by an elastic string of modulus \(\lambda\), natural length \(l\) and negligible mass. The particle \(A\) is given a velocity \(u\) in direction \(BA\). Find the velocities of the particles \(t\) seconds after the string begins to stretch, and shew that there is a continuous flux of momentum between the particles amounting to \(2m_1m_2u/(m_1+m_2)\) in \(\pi/n\) seconds, where \[ n^2 = \frac{\lambda}{l} \left( \frac{1}{m_1} + \frac{1}{m_2} \right). \]

The cross-section of a wedge of mass \(M\) is an isosceles triangle of base angles \(\alpha\). It is placed on a rough horizontal plane. Particles of masses \(m,m'\) (\(m>m'\)) are placed one on each of the smooth slant faces and connected by a fine inextensible thread which passes over and at right angles to the top edge of the wedge. Motion is possible in vertical planes through lines of greatest slope. Prove that if the coefficient of friction exceeds \[ (m^2-m'^2)\sin\alpha\cos\alpha / \{(m+m')(M+m+m') - (m-m')^2\sin^2\alpha\} \] the wedge will not move; and write down sufficient equations of motion to determine the acceleration of the wedge and of the particles relative to the wedge when the coefficient of friction has a smaller value.

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.