Problems

1. Sketch the curve \(C_1\) with equation \[ \left(y^2 + (x-1)^2 - 1\right)\left(y^2 + (x+1)^2 - 1\right) = 0. \]
2. Consider the curve \(C_2\) with equation \[ \left(y^2 + (x-1)^2 - 1\right)\left(y^2 + (x+1)^2 - 1\right) = \tfrac{1}{16}. \]
   1. Show that the line \(y = k\) meets the curve \(C_2\) at points for which \[ x^4 + 2(k^2 - 2)x^2 + \left(k^4 - \tfrac{1}{16}\right) = 0. \] Hence determine the number of intersections between curve \(C_2\) and the line \(y = k\) for each positive value of \(k\).
   2. Determine whether the points on curve \(C_2\) with the greatest possible \(y\)-coordinate are further from, or closer to, the \(y\)-axis than those on curve \(C_1\).
   3. Show that it is not possible for both \(y^2 + (x-1)^2 - 1\) and \(y^2 + (x+1)^2 - 1\) to be negative, and deduce that curve \(C_2\) lies entirely outside curve \(C_1\).
   4. Sketch the curves \(C_1\) and \(C_2\) on the same axes.

In this question, the following theorem may be used without proof. Let \(u_1, u_2, \ldots\) be a sequence of real numbers. If the sequence is

• bounded above, so \(u_n \leqslant b\) for all \(n\), where \(b\) is some fixed number
• and increasing, so \(u_n \leqslant u_{n+1}\) for all \(n\)

then there is a number \(L \leqslant b\) such that \(u_n \to L\) as \(n \to \infty\). For positive real numbers \(x\) and \(y\), define \(\mathrm{a}(x,y) = \frac{1}{2}(x+y)\) and \(\mathrm{g}(x,y) = \sqrt{xy}\). Let \(x_0\) and \(y_0\) be two positive real numbers with \(y_0 < x_0\) and define, for \(n \geqslant 0\) \[ x_{n+1} = \mathrm{a}(x_n, y_n)\,, \] \[ y_{n+1} = \mathrm{g}(x_n, y_n)\,. \]

1. By considering \((\sqrt{x_n} - \sqrt{y_n})^2\), show that \(y_{n+1} < x_{n+1}\), for \(n \geqslant 0\). Show further that, for \(n \geqslant 0\)
   • \(x_{n+1} < x_n\)
   • \(y_n < y_{n+1}\).
   Deduce that there is a value \(M\) such that \(y_n \to M\) as \(n \to \infty\). Show that \(0 < x_{n+1} - y_{n+1} < \frac{1}{2}(x_n - y_n)\) and hence that \(x_n - y_n \to 0\) as \(n \to \infty\). Explain why \(x_n\) also tends to \(M\) as \(n \to \infty\).
2. Let \[ \mathrm{I}(p,q) = \int_0^{\infty} \frac{1}{\sqrt{(p^2 + x^2)(q^2 + x^2)}}\,\mathrm{d}x, \] where \(p\) and \(q\) are positive real numbers with \(q < p\). Show, using the substitution \(t = \frac{1}{2}\!\left(x - \dfrac{pq}{x}\right)\) in the integral \[ \int_{-\infty}^{\infty} \frac{1}{\sqrt{\left(\frac{1}{4}(p+q)^2 + t^2\right)(pq + t^2)}}\,\mathrm{d}t, \] that \[ \mathrm{I}(p,q) = \mathrm{I}\!\left(\mathrm{a}(p,q),\, \mathrm{g}(p,q)\right). \] Hence evaluate \(\mathrm{I}(x_0, y_0)\) in terms of \(M\).

A long straight trench, with rectangular cross section, has been dug in otherwise horizontal ground. The width of the trench is \(d\) and its depth \(2d\). A particle is projected at speed \(v\), where \(v^2 = \lambda dg\), at an angle \(\alpha\) to the horizontal, from a point on the ground a distance \(d\) from the nearer edge of the trench. The vertical plane in which it moves is perpendicular to the trench.

1. The particle lands on the base of the trench without first touching either of its sides.
   1. By considering the vertical displacement of the particle when its horizontal displacement is \(d\), show that \((\tan\alpha - \lambda)^2 < \lambda^2 - 1\) and deduce that \(\lambda > 1\).
   2. Show also that \((2\tan\alpha - \lambda)^2 > \lambda^2 + 4(\lambda - 1)\) and deduce that \(\alpha > 45^\circ\).
2. Show that, provided \(\lambda > 1\), \(\alpha\) can always be chosen so that the particle lands on the base of the trench without first touching either of its sides.

A triangular prism lies on a horizontal plane. One of the rectangular faces of the prism is vertical; the second is horizontal and in contact with the plane; the third, oblique rectangular face makes an angle \(\alpha\) with the horizontal. The two triangular faces of the prism are right angled triangles and are vertical. The prism has mass \(M\) and it can move without friction across the plane. A particle of mass \(m\) lies on the oblique surface of the prism. The contact between the particle and the plane is rough, with coefficient of friction \(\mu\).

1. Show that if \(\mu < \tan\alpha\), then the system cannot be in equilibrium.

Let \(\mu = \tan\lambda\), with \(0 < \lambda < \alpha < \frac{1}{4}\pi\). A force \(P\) is exerted on the vertical rectangular face of the prism, perpendicular to that face and directed towards the interior of the prism. The particle and prism accelerate, but the particle remains in the same position relative to the prism.

2. Show that the magnitude, \(F\), of the frictional force between the particle and the prism is \[ F = \frac{m}{M+m}\left|(M+m)g\sin\alpha - P\cos\alpha\right|. \] Find a similar expression for the magnitude, \(N\), of the normal reaction between the particle and the prism.
3. Hence show that the force \(P\) must satisfy \[ (M+m)g\tan(\alpha - \lambda) \leqslant P \leqslant (M+m)g\tan(\alpha + \lambda). \]

1. Sketch a graph of \(y = x^{\frac{1}{x}}\) for \(x > 0\), showing the location of any turning points. Find the maximum value of \(n^{\frac{1}{n}}\), where \(n\) is a positive integer.

\(N\) people are to have their blood tested for the presence or absence of an enzyme. Each person, independently of the others, has a probability \(p\) of having the enzyme present in a sample of their blood, where \(0 < p < 1\). The blood test always correctly determines whether the enzyme is present or absent in a sample. The following method is used.

• The people to be tested are split into \(r\) groups of size \(k\), with \(k > 1\) and \(rk = N\).
• In every group, a sample from each person in that group is mixed into one large sample, which is then tested.
• If the enzyme is not present in the combined sample from a group, no further testing of the people in that group is needed.
• If the enzyme is present in the combined sample from a group, a second sample from each person in that group is tested separately.

2. Find, in terms of \(N\), \(k\) and \(p\), the expected number of tests.
3. Given that \(N\) is a multiple of \(3\), find the largest value of \(p\) for which it is possible to find an integer value of \(k\) such that \(k > 1\) and the expected number of tests is at most \(N\). Show that this value of \(p\) is greater than \(\frac{1}{4}\).
4. Show that, if \(pk\) is sufficiently small, the expected number of tests is approximately \[ N\!\left(\frac{1}{k} + pk\right). \] In the case where \(p = 0.01\), show that choosing \(k = 10\) gives an expected number of tests which is only about \(20\%\) of \(N\).

In this question, you may use without proof the results \[ \sum_{i=1}^{n} i^2 = \tfrac{1}{6}n(n+1)(2n+1) \quad \text{and} \quad \sum_{i=1}^{n} i^3 = \tfrac{1}{4}n^2(n+1)^2. \] Throughout the question, \(n\) and \(k\) are integers with \(n \geqslant 3\) and \(k \geqslant 2\).

1. In a game, \(k\) players, including Ada, are each given a random whole number from \(1\) to \(n\) (that is, for each player, each of these numbers is equally likely and assigned independently of all the others). A player wins the game if they are given a smaller number than all the other players, so there may be no winner in this game. Find an expression, in terms of \(n\), \(k\) and \(a\), for the probability that Ada is given number \(a\), where \(1 \leqslant a \leqslant n-1\), and all the other players are given larger numbers. Hence show that, if \(k = 4\), the probability that there is a winner in this game is \[ \frac{(n-1)^2}{n^2}\,. \]
2. In a second game, \(k\) players, including Ada and Bob, are each given a random whole number from \(1\) to \(n\). A player wins the game if they are given a smaller number than all the other players or if they are given a larger number than all the other players, so it is possible for there to be zero, one or two winners in this game. Find an expression, in terms of \(n\), \(k\) and \(d\), for the probability that Ada is given number \(a\) and Bob is given number \(a + d + 1\), where \(1 \leqslant d \leqslant n-2\) and \(1 \leqslant a \leqslant n - d - 1\), and all the other players are given numbers greater than \(a\) and less than \(a + d + 1\). Hence show that, if \(k = 4\), the probability that there are two winners in this game is \[ \frac{(n-2)(n-1)^2}{n^3}\,. \] If \(k = 4\), what is the minimum value of \(n\) for which there are more likely to be exactly two winners than exactly one winner in this game?

Throughout this question, \(N\) is an integer with \(N \geqslant 1\) and \(S_N = \displaystyle\sum_{r=1}^{N} \frac{1}{r^2}\). You may assume that \(\displaystyle\lim_{N\to\infty} S_N\) exists and is equal to \(\frac{1}{6}\pi^2\).

1. Show that \[\frac{1}{r+1} - \frac{1}{r} + \frac{1}{r^2} = \frac{1}{r^2(r+1)}.\] Hence show that \[\sum_{r=1}^{N} \frac{1}{r^2(r+1)} = \sum_{r=1}^{N} \frac{1}{r^2} - 1 + \frac{1}{N+1}.\] Show further that \(\displaystyle\sum_{r=1}^{\infty} \frac{1}{r^2(r+1)} = \frac{1}{6}\pi^2 - 1\).
2. Find \(\displaystyle\sum_{r=1}^{N} \frac{1}{r^2(r+1)(r+2)}\) in terms of \(S_N\), and hence evaluate \(\displaystyle\sum_{r=1}^{\infty} \frac{1}{r^2(r+1)(r+2)}\).
3. Show that \[\sum_{r=1}^{\infty} \frac{1}{r^2(r+1)^2} = \sum_{r=1}^{\infty} \frac{2}{r^2(r+1)} - 1.\]

1. Solve the inequalities
   1. \(\sqrt{4x^2 - 8x + 64} \leqslant |x+8|\,\),
   2. \(\sqrt{4x^2 - 8x + 64} \leqslant |3x-8|\,\).
2. 1. Let \(\mathrm{f}(x) = \sqrt{4x^2 - 8x + 64} - 2(x-1)\). Show, by considering \(\bigl(\sqrt{4x^2 - 8x + 64} + 2(x-1)\bigr)\mathrm{f}(x)\) or otherwise, that \(\mathrm{f}(x) \to 0\) as \(x \to \infty\).
   2. Sketch \(y = \sqrt{4x^2 - 8x + 64}\) and \(y = 2(x-1)\) on the same axes.
3. Find a value of \(m\) and the corresponding value of \(c\) such that the solution set of the inequality \[\sqrt{4x^2 - 5x + 4} \leqslant |mx + c|\] is \(\{x : x \geqslant 3\}\).
4. Find values of \(p\), \(q\), \(m\) and \(c\) such that the solution set of the inequality \[|x^2 + px + q| \leqslant mx + c\] is \(\{x : -5 \leqslant x \leqslant 1\} \cup \{x : 5 \leqslant x \leqslant 7\}\).

Throughout this question, consider only \(x > 0\).

1. Let \[\mathrm{g}(x) = \ln\left(1 + \frac{1}{x}\right) - \frac{x+c}{x(x+1)}\] where \(c \geqslant 0\).
   1. Show that \(y = \mathrm{g}(x)\) has positive gradient for all \(x > 0\) when \(c \geqslant \frac{1}{2}\).
   2. Find the values of \(x\) for which \(y = \mathrm{g}(x)\) has negative gradient when \(0 \leqslant c < \frac{1}{2}\).
2. It is given that, for all \(c > 0\), \(\mathrm{g}(x) \to -\infty\) as \(x \to 0\). Sketch, for \(x > 0\), the graphs of \[y = \mathrm{g}(x)\] in the cases
   1. \(c = \frac{3}{4}\),
   2. \(c = \frac{1}{4}\).
3. The function \(\mathrm{f}\) is defined as \[\mathrm{f}(x) = \left(1 + \frac{1}{x}\right)^{x+c}.\] Show that, for \(x > 0\),
   1. \(\mathrm{f}\) is a decreasing function when \(c \geqslant \frac{1}{2}\);
   2. \(\mathrm{f}\) has a turning point when \(0 < c < \frac{1}{2}\);
   3. \(\mathrm{f}\) is an increasing function when \(c = 0\).

1. Show that if the acute angle between straight lines with gradients \(m_1\) and \(m_2\) is \(45^\circ\), then \[\frac{m_1 - m_2}{1 + m_1 m_2} = \pm 1.\]

The curve \(C\) has equation \(4ay = x^2\) (where \(a \neq 0\)).

2. If \(p \neq q\), show that the tangents to the curve \(C\) at the points with \(x\)-coordinates \(p\) and \(q\) meet at a point with \(x\)-coordinate \(\frac{1}{2}(p+q)\). Find the \(y\)-coordinate of this point in terms of \(p\) and \(q\). Show further that any two tangents to the curve \(C\) which are at \(45^\circ\) to each other meet on the curve \((y+3a)^2 = 8a^2 + x^2\).
3. Show that the acute angle between any two tangents to the curve \(C\) which meet on the curve \((y+7a)^2 = 48a^2 + 3x^2\) is constant. Find this acute angle.

In this question, \(\mathbf{M}\) and \(\mathbf{N}\) are non-singular \(2 \times 2\) matrices. The \emph{trace} of the matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is defined as \(\operatorname{tr}(\mathbf{M}) = a + d\).

1. Prove that, for any two matrices \(\mathbf{M}\) and \(\mathbf{N}\), \(\operatorname{tr}(\mathbf{MN}) = \operatorname{tr}(\mathbf{NM})\) and derive an expression for \(\operatorname{tr}(\mathbf{M}+\mathbf{N})\) in terms of \(\operatorname{tr}(\mathbf{M})\) and \(\operatorname{tr}(\mathbf{N})\).

The entries in matrix \(\mathbf{M}\) are functions of \(t\) and \(\dfrac{\mathrm{d}\mathbf{M}}{\mathrm{d}t}\) denotes the matrix whose entries are the derivatives of the corresponding entries in \(\mathbf{M}\).

2. Show that \[\frac{1}{\det \mathbf{M}} \frac{\mathrm{d}}{\mathrm{d}t}(\det \mathbf{M}) = \operatorname{tr}\!\left(\mathbf{M}^{-1} \frac{\mathrm{d}\mathbf{M}}{\mathrm{d}t}\right).\]
3. In this part, matrix \(\mathbf{M}\) satisfies the differential equation \[\frac{\mathrm{d}\mathbf{M}}{\mathrm{d}t} = \mathbf{MN} - \mathbf{NM},\] where the entries in matrix \(\mathbf{N}\) are also functions of \(t\). Show that \(\det \mathbf{M}\), \(\operatorname{tr}(\mathbf{M})\) and \(\operatorname{tr}(\mathbf{M}^2)\) are independent of \(t\). In the case \(\mathbf{N} = \begin{pmatrix} t & t \\ 0 & t \end{pmatrix}\), and given that \(\mathbf{M} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\) when \(t = 0\), find \(\mathbf{M}\) as a function of \(t\).
4. In this part, matrix \(\mathbf{M}\) satisfies the differential equation \[\frac{\mathrm{d}\mathbf{M}}{\mathrm{d}t} = \mathbf{MN},\] where the entries in matrix \(\mathbf{N}\) are again functions of \(t\). The trace of \(\mathbf{M}\) is non-zero and independent of \(t\). Is it necessarily true that \(\operatorname{tr}(\mathbf{N}) = 0\)?

1. A particle moves in two-dimensional space. Its position is given by coordinates \((x, y)\) which satisfy \[\frac{\mathrm{d}x}{\mathrm{d}t} = -x + 3y + u\] \[\frac{\mathrm{d}y}{\mathrm{d}t} = x + y + u\] where \(t\) is the time and \(u\) is a function of time. At time \(t = 0\), the particle has position \((x_0,\, y_0)\).
   1. By considering \(\dfrac{\mathrm{d}x}{\mathrm{d}t} - \dfrac{\mathrm{d}y}{\mathrm{d}t}\), show that if the particle is at the origin \((0,0)\) at some time \(t > 0\), then it is necessary that \(x_0 = y_0\).
   2. Given that \(x_0 = y_0\), find a constant value of \(u\) that ensures that the particle is at the origin at a time \(t = T\), where \(T > 0\).
2. A particle whose position in three-dimensional space is given by co-ordinates \((x, y, z)\) moves with time \(t\) such that \[\frac{\mathrm{d}x}{\mathrm{d}t} = 4y - 5z + u\] \[\frac{\mathrm{d}y}{\mathrm{d}t} = x - 2z + u\] \[\frac{\mathrm{d}z}{\mathrm{d}t} = x - 2y + u\] where \(u\) is a function of time. At time \(t = 0\), the particle has position \((x_0,\, y_0,\, z_0)\).
   1. Show that, if the particle is at the origin \((0,0,0)\) at some time \(t > 0\), it is necessary that \(y_0\) is the mean of \(x_0\) and \(z_0\).
   2. Show further that, if the particle is at the origin \((0,0,0)\) at some time \(t > 0\), it is necessary that \(x_0 = y_0 = z_0\).
   3. Given that \(x_0 = y_0 = z_0\), find a constant value of \(u\) that ensures that the particle is at the origin at a time \(t = T\), where \(T > 0\).

In this question, you need not consider issues of convergence. For positive integer \(n\) let \[\mathrm{f}(n) = \frac{1}{n+1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \ldots\] and \[\mathrm{g}(n) = \frac{1}{n+1} - \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} - \ldots\,.\]

1. Show, by considering a geometric series, that \(0 < \mathrm{f}(n) < \dfrac{1}{n}\).
2. Show, by comparing consecutive terms, that \(0 < \mathrm{g}(n) < \dfrac{1}{n+1}\).
3. Show, for positive integer \(n\), that \((2n)!\,\mathrm{e} - \mathrm{f}(2n)\) and \(\dfrac{(2n)!}{\mathrm{e}} + \mathrm{g}(2n)\) are both integers.
4. Show that if \(q\,\mathrm{e} = \dfrac{p}{\mathrm{e}}\) for some positive integers \(p\) and \(q\), then \(q\,\mathrm{f}(2n) + p\,\mathrm{g}(2n)\) is an integer for all positive integers \(n\).
5. Hence show that the number \(\mathrm{e}^2\) is irrational.

1. Explain why the equation \((y - x + 3)(y + x - 5) = 0\) represents a pair of straight lines with gradients \(1\) and \(-1\). Show further that the equation \[y^2 - x^2 + py + qx + r = 0\] represents a pair of straight lines with gradients \(1\) and \(-1\) if and only if \(p^2 - q^2 = 4r\).

In the remainder of this question, \(C_1\) is the curve with equation \(x = y^2 + 2sy + s(s+1)\) and \(C_2\) is the curve with equation \(y = x^2\).

2. Explain why the coordinates of any point which lies on both of the curves \(C_1\) and \(C_2\) also satisfy the equation \[y^2 + 2sy + s(s+1) - x + k(y - x^2) = 0\] for any real number \(k\). Given that \(s\) is such that \(C_1\) and \(C_2\) intersect at four distinct points, show that choosing \(k = 1\) gives an equation representing a pair of straight lines, with gradients \(1\) and \(-1\), on which all four points of intersection lie.
3. Show that if \(C_1\) and \(C_2\) intersect at four distinct points, then \(s < -\frac{3}{4}\).
4. Show that if \(s < -\frac{3}{4}\), then \(C_1\) and \(C_2\) intersect at four distinct points.

The origin \(O\) of coordinates lies on a smooth horizontal table and the \(x\)- and \(y\)-axes lie in the plane of the table. A smooth sphere \(A\) of mass \(m\) and radius \(r\) is at rest on the table with its lowest point at the origin. A second smooth sphere \(B\) has the same mass and radius and also lies on the table. Its lowest point has \(y\)-coordinate \(2r\sin\alpha\), where \(\alpha\) is an acute angle, and large positive \(x\)-coordinate. Sphere \(B\) is now projected parallel to the \(x\)-axis, with speed \(u\), so that it strikes sphere \(A\). The coefficient of restitution in this collision is \(\frac{1}{3}\).

1. Show that, after the collision, sphere \(B\) moves with velocity \[\begin{pmatrix} -\frac{1}{3}u\bigl(1 + 2\sin^2\alpha\bigr) \\ \frac{2}{3}u\sin\alpha\cos\alpha \end{pmatrix}.\]
2. Show further that the lowest point of sphere \(B\) crosses the \(y\)-axis at the point \((0, Y)\), where \(Y = 2r(\cos\alpha\tan\beta + \sin\alpha)\) and \[\tan\beta = \frac{2\sin\alpha\cos\alpha}{1 + 2\sin^2\alpha}.\]

A third sphere \(C\) of radius \(r\) is at rest with its lowest point at \((0, h)\) on the table, where \(h > 0\).

3. Show that, if \(h > Y + 2r\sec\beta\), sphere \(B\) will not strike sphere \(C\) in its motion after the collision with sphere \(A\).
4. Show that \(Y < 2r\sec\beta\). Hence show that sphere \(B\) will not strike sphere \(C\) for any value of \(\alpha\), if \(h > \dfrac{8r}{\sqrt{3}}\).