Problems

A bag contains a large number of red, white and blue dice in equal numbers. If \(n\) are drawn at random, show that the probability \(P(n,r)\) of drawing exactly \(r\) red dice is equal to the term containing \((\frac{1}{3})^r(\frac{2}{3})^{n-r}\) in the binomial expansion of \((\frac{1}{3} + \frac{2}{3})^n\). If \(r\) dice are thrown, find the probability \(Q(r,s)\) of throwing exactly \(s\) sixes. If \(n\) dice are drawn from the bag and the red dice drawn are thrown, show that the probability of throwing exactly \(s\) sixes is $$\sum_{t=0}^{n-s} P(n, s+t)Q(s+t, s)$$ and prove that this is equal to a term in a binomial expansion. Explain why a binomial distribution is obtained.

Objects \(\ldots, \langle -2 \rangle, \langle -1 \rangle, \langle 0 \rangle, \langle 1 \rangle, \langle 2 \rangle, \ldots\) are given. Two objects \(\langle p \rangle\) and \(\langle q \rangle\) are equal if \(p-q\) is a multiple of 4. Prove that

1. [(i)] \(\langle p \rangle = \langle q \rangle\) and \(\langle q \rangle = \langle r \rangle\) imply that \(\langle q \rangle = \langle p \rangle\) and \(\langle p \rangle = \langle r \rangle\);
2. [(ii)] each \(\langle p \rangle\) is equal to a member of the set \(S\) consisting of \(\langle 0 \rangle, \langle 1 \rangle, \langle 2 \rangle\) and \(\langle 3 \rangle\).

The product \(\langle p \rangle \bullet \langle q \rangle\) is defined as \(\langle p + q + 1 \rangle\). Prove that

1. [(iii)] \(\langle p \rangle \bullet \langle q \rangle \bullet \langle r \rangle = \langle p \rangle \bullet (\langle q \rangle \bullet \langle r \rangle)\);
2. [(iv)] there is a member \(E\) of \(S\) such that, for every \(p\), $$\langle p \rangle \bullet E = E \bullet \langle p \rangle = \langle p \rangle;$$
3. [(v)] every member \(\langle p \rangle\) of \(S\) has an inverse \(\langle p \rangle^{-1}\), also a member of \(S\), such that $$\langle p \rangle \bullet \langle p \rangle^{-1} = \langle p \rangle^{-1} \bullet \langle p \rangle = E.$$

Complete the multiplication table below: Evaluate $$(\langle 1 \rangle \bullet \langle 2 \rangle^{-1}) \bullet (\langle 2 \rangle \bullet \langle 3 \rangle) \quad \text{and} \quad \langle 1 \rangle \bullet (\langle 2 \rangle^{-1} \bullet (\langle 2 \rangle \bullet \langle 1 \rangle^{-1})).$$

The equations of motion of a particle in a plane, referred to rectangular axes \(Ox, Oy\) in the plane, are $$\ddot{x} = ky, \quad \ddot{y} = -kx.$$ Show that the equations of motion become $$\ddot{x'} = ky', \quad \ddot{y'} = -kx',$$ and are thus unaltered in form, when referred to axes \(Ox', Oy'\) derived from the first by rotation about \(O\) through an arbitrary angle \(\alpha\). Are the equations of motion unaltered in form when referred to axes \(Ox, Oy\) derived from \(Ox, Oy\) by reflection in an arbitrary line \(x/\cos\beta = y/\sin\beta\) through the origin? Make a similar investigation for the equations $$\ddot{x} = k(x\ddot{y} - y\ddot{x})y, \quad \ddot{y} = -k(x\ddot{y} - y\ddot{x})x.$$

The equation of a conic referred to rectangular cartesian axes is $$ax^2 + 2hxy + by^2 = 1.$$ Show that the locus of the mid-points of chords parallel to the diameter \(x/l = y/m\) is the line \((al + hm)x + (hl + bm)y = 0\) (the conjugate diameter). Prove the equivalence of the following definitions of the principal axes:

1. [(i)] an axis is a diameter that is orthogonal to its conjugate diameter;
2. [(ii)] the extremities of an axis are points on the conic whose distances from the centre are maximal or minimal.

Show that the function $$f(x) = e^{-x} \int_{-\infty}^{x} e^{s} F(s) ds$$ satisfies the differential equation $$f'(x) + f(x) = F(x).$$ The function \(\phi(x)\) is defined as follows: $$\phi(x) = 0 \quad \text{for } x < 0$$ $$x \quad \text{for } x > 0$$ Given that $$f'(x) + f(x) = \phi(x) \quad \text{and} \quad f(-\infty) = 0,$$ find \(f(x)\). Show graphically the forms of the functions \(\phi(x)\) and \(f(x)\). Given that $$g'(x) + g(x) = \phi(x) - \phi(x-1) \quad \text{and} \quad g(-\infty) = 0,$$ find the function \(g(x)\) and show graphically the forms of the functions \(\phi(x) - \phi(x-1)\) and \(g(x)\).

A function \(y\) of \(x\) and \(\lambda\) is defined by the equation $$y = x^2 + \lambda x^2 y^{-\frac{1}{2}}$$ where \(\lambda\) is small. Assuming that \(y\) may be expressed in the form $$p(x) + \lambda q(x) + \lambda^2 r(x) + \lambda^3 s(x) + \ldots$$ find the functions \(p(x), q(x), r(x)\) and \(s(x)\). Convergence need not be discussed.

By considering $$\int_a^b \{f(x) + \lambda g(x)\}^2 dx$$ show that $$\left\{\int_a^b fg dx\right\}^2 \leq \int_a^b f^2 dx \cdot \int_a^b g^2 dx.$$ By applying this inequality to the integrals $$\int_0^1 (x^{\frac{1}{2}})(x^2-1) dx \quad \text{and} \quad \int_0^1 (x^{\frac{1}{2}})(x^{\frac{1}{2}} e^{-x}) dx,$$ show that $$\int_0^1 x^{\frac{1}{2}} e^x dx$$ lies between 1.11 and 1.13. \([e = 2.71828; e^2 = 7.38906; \sqrt{e} = 1.64872.]\)

A particle of mass \(m\) falls in a vertical plane from rest under the influence of constant gravitational force \(mg\) and a force \(mkv\) perpendicular to its velocity, where \(k\) is a constant. Write down expressions for \(v\) and the curvature of its path after it has fallen through a vertical distance \(y\). Show that \(y\) never exceeds \(2g/k^2\).

The mass per unit surface area of a thin spherical shell of radius \(a\) is proportional to the square of the distance from a point on its surface. It is suspended from by light rods from a fixed point at its centre. If \(k_1\) is the radius of gyration about a horizontal axis through the centre and \(k_2\) is the radius of gyration about the vertical axis through the centre, prove that $$2k_1^2 + k_2^2 = 2a^2.$$ Show that the period of small planar oscillations under gravity is the same as that of a simple pendulum of length \(2a\).

A light rod \(OA\) of length \(l\) rotates freely about a fixed point \(O\). A point particle of mass \(m\) attached to the rod at \(A\) is initially at rest vertically below \(O\). A projectile of mass \(m\) moving horizontally with speed \(v\) (\(v^2 < 16gl\)) embeds itself instantaneously in the target. Obtain the height \(h\) through which the target would rise before first coming to rest if undisturbed in the subsequent motion. However, after rising through a height \(3h/4\) another similar projectile embeds itself in the target. How much further will the target rise? If the total height through which the target rises is \(3h/4 + h'\), show that \(h'\) is greatest (for variable \(v\)) if \(v^2 = 16gl/3\).