\(A\) makes a statement which is overheard by \(B\), who reports on its truth to \(C\). \(A\) and \(C\) each independently tell the truth once in three times and lie twice. \(B\) says that \(A\) was lying. By considering the eight combinations of truth and falsehood, or otherwise, find the probability that \(A\) was in fact telling the truth.
Six equal rods are joined together to form a regular tetrahedron. Two scorpions are placed at the midpoints of two opposite edges of this framework, and a beetle is placed at some point of this framework. The scorpions can move along the rods with maximum speed \(s\), and the beetle with maximum speed \(b\). Show that if \(b < 2s\) the scorpions can always catch the beetle; and explain in detail how they should manoeuvre to do so.
Let \(N = p_1^{a_1}p_2^{a_2}\ldots p_n^{a_n}\) be the representation of \(N\) as a product of powers of distinct primes. How many proper factors has \(N\)? (A proper factor of \(N\) is an integer \(M\) which exactly divides \(N\) and which satisfies \(1 < M < N\).) Hence or otherwise find
For any continuous function \(g(x)\) write \[Y(x) = \int_0^x g(t)\,dt.\] Prove the identity \[\int_0^{1\pi} (g^3 - Y^3)\,dx = \int_0^{1\pi} (g - Y\cot x)^2\,dx\] and deduce the inequality \[\int_0^{1\pi} Y^2\,dx \leq \int_0^{1\pi} g^2\,dx.\] For what functions \(g(x)\) are the two sides equal? [Problems of convergence may be ignored.]
One of the ways of sorting a list of distinct numbers, initially in a random order, involves arranging them in a tree-like structure which satisfies the following rules. The tree consists of nodes, at each of which one of the numbers is placed, and branches each of which join two nodes. There is one special node, called the base, at the foot of the tree; any other node is at the top of just one branch. From any node there is at most one branch which grows upwards to the left, and at most one branch which grows upwards to the right. If a branch grows upwards to the left, all the numbers accessible from the top of the branch by proceeding upwards (including the number at the top of the branch itself) are less than the number at the bottom of the branch; and for a branch that grows upwards to the right they are all greater. (A typical tree is illustrated below.) [Tree diagram showing nodes with numbers: 13, 24 at top level; 22 below them; 34 below 22; 39 at bottom center; 44, 72, 57 on right side; 43, 75 connected to right structure; 45 connecting parts] Numbers are supplied one by one from a list. By means of a flow diagram, or otherwise, describe how to add a new number to an existing tree. (The operations available are to locate the base node, to move up or down an existing branch, to grow a new branch from the node which you are at, to compare the new number with the number placed at the node which you are at, and such other similarly simple operations as you may require.) Draw the tree which should be formed from the list 28, 79, 18, 45, 60, 63, 54, 33, 11, 55, 98, 27, 47, 20.
Denote by \(g_1, g_2, \ldots, g_n\) the elements of a given finite multiplicative group \(G\), not necessarily commutative, and let \(\mathscr{S}\) be the set of all formal expressions \[a_1g_1 + a_2g_2 + \ldots + a_ng_n,\] where the \(g_i\) are the elements of \(G\) and the \(a_i\) are any real numbers. Addition and multiplication are defined on the set \(\mathscr{S}\) by the rules \[\{\sum a_ig_i\} + \{\sum b_ig_i\} = \sum (a_i + b_i)g_i\] and \[\{\sum a_ig_i\} \times \{ \sum b_jg_j\} = \sum \sum (a_ib_j)(g_ig_j)\] where in the second equation the dot denotes multiplication in \(G\). Prove that \[0 = 0g_1 + 0g_2 + \ldots + 0g_n\] has in \(\mathscr{S}\) the properties normally associated with the symbol zero. Writing \[s = 1g_1 + 1g_2 + \ldots + 1g_n\] prove that for any \(i, j\) \[s \times \{1g_i - 1g_j\} = 0.\] In the special case where \(G\) contains just two elements \(g_1\) and \(g_2\), of which \(g_1\) is the identity, find all expressions \(x\) in \(\mathscr{S}\) which satisfy \[x \times x = 1g_1 + 0g_2.\]
Sketch the curve whose equation is \[y^2(1+x^2) = x^2(1-x^2),\] and find the area of a loop of the curve.
According to the Special Theory of Relativity, the dynamics of a particle, moving on a straight line, may be treated in a given frame of reference by solving the equation \[\frac{dp}{dt} = F\] where \(p\) is the momentum and \(F\) the force, the only difference between relativistic and ordinary mechanics being that the formula for the momentum is \[p = \frac{mv}{(1-v^2/c^2)^{1/2}}\] instead of \(mv\). Here \(m\) is the mass (a given constant), \(v\) is the velocity observed in that frame, and \(c\) the speed of light. Show that \[Fv = \frac{d}{dt}\left[\frac{mc^2}{(1-v^2/c^2)^{1/2}}\right].\] In the case where \(F\) is a constant force, and the particle starts from rest at the origin at time \(t = 0\), show that the distance covered after time \(t\) is \[x = \frac{c^2}{a}\left[\left(1 + \frac{a^2t^2}{c^4}\right)^{1/2} - 1\right],\] where \(a = F/m\). Give approximations to this result for \(at \ll c\) and \(at \gg c\) respectively, and comment on them.
Two astronomical bodies may be regarded as particles of masses \(M_1\) and \(M_2\), and attract each other according to the inverse square law. Prove that a possible solution of their equations of motion is one in which they move steadily on circles centred on their mass-centre, and give the relation between the radii and the period of rotation. Explain qualitatively why there are two tides per day rather than one.
An electric hand drill consists of a rigid casing held by the user, and in it are two parallel spindles \(S_1\) and \(S_2\) mounted on frictionless bearings. \(S_1\) is driven by a motor mounted rigidly in the casing. \(S_2\) incorporates the drill bit. The two spindles are coupled by gear wheels with \(n_1\) and \(n_2\) teeth respectively, which mesh externally without friction. The rotating parts are rigid, with moments of inertia \(I_1\) and \(I_2\) respectively about their axes, and their mass centres lying on the axes. The motor is designed to provide a constant torque with moment \(G\) (independent of its angular velocity). When drilling into a certain material, the bit experiences a resistive couple of moment \(k\omega_2\) where \(k\) is constant and \(\omega_2\) is the angular velocity of \(S_2\). At time \(t = 0\) the spindles are at rest and the motor is switched on to start drilling into that material. Show that at any later time \(t\) \[\omega _2 = \frac{n_2G}{n_1k}\left[1 - e^{-kt/A}\right],\] where \(A = I_2 + (n_2/n_1)^2I_1\). Find also, as a function of time, the couple which the user must exert to hold the casing stationary.