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2005 Paper 3 Q12
Five independent timers time a runner as she runs four laps of a track. Four of the timers measure the individual lap times, the results of the measurements being the random variables \(T_1\) to \(T_4\), each of which has variance \(\sigma^2\) and expectation equal to the true time for the lap. The fifth timer measures the total time for the race, the result of the measurement being the random variable \(T\) which has variance \(\sigma^2\) and expectation equal to the true race time (which is equal to the sum of the four true lap times). Find a random variable \(X\) of the form \(aT+b(T_1+T_2+T_3+T_4)\), where \(a\) and \(b\) are constants independent of the true lap times, with the two properties:
- whatever the true lap times, the expectation of \(X\) is equal to the true race time;
- the variance of \(X\) is as small as possible.
Solution: Let the expected total time for the race be \(\mu\). Let \(X = aT + b(T_1 + T_2+T_3+T_4)\) then \(\E[X] = a\E[T] + b\E[T_1+\cdots+T_4] = a \mu + b \mu = (a+b)\mu\). So \(a+b=1\). \begin{align*} && \var[X] &= a^2\var[T] + b^2(\var[T_1] + \var[T_2] + \var[T_3] + \var[T_4]) \\ &&&= a^2\sigma^2 + 4b^2 \sigma^2 \\ &&& = \sigma^2 (a^2 + 4(1-a)^2 ) \\ &&&= \sigma^2 (5a^2 - 8a + 4) \\ &&&= \sigma^2 \left ( 5 \left ( a - \frac45 \right)^2 - \frac{16}{5}+4 \right)\\ &&&= \sigma^2 \left ( 5 \left ( a - \frac45 \right)^2 + \frac{4}{5}\right) \end{align*} Therefore variance is minimised when \(a = \frac45, b = \frac15\). Let \(Y = cT + d(T_1 + T_2+T_3+T_4)\) then \begin{align*} && \E[Y^2] &= \E \left [c^2T^2 + 2cd T(T_1+T_2+T_3+T_4) + d^2(T_1+T_2+T_3+T_4)^2 \right] \\ &&&= c^2 (\mu^2 + \sigma^2) + 2cd \mu^2 + d^2 (\var[T_1 + \cdots + T_4] + \mu^2) \\ &&&= c^2(\mu^2+\sigma^2) + 2cd \mu^2 + d^2(4\sigma^2 + \mu^2) \\ &&&= (c^2 + 2cd + d^2) \mu^2 + (c^2+4d^2) \sigma^2 \\ &&&= (c+d)^2 \mu^2 + (c^2+4d^2) \sigma^2 \\ \\ \Rightarrow && d &= -c \\ && 1 &= c^2 + 4d^2 \\ \Rightarrow && c &= \pm \frac{1}{\sqrt5} \\ && d &= \mp \frac{1}{\sqrt5} \end{align*} Given our results, our best estimate for \(\mu\) is \(\frac45 \cdot 220 + \frac15 220.5 = 220.1\). Our estimate for \(\sigma^2 = \left( \frac{1}{\sqrt{5}}(220.5-220) \right)^2 = \frac{1}{20}\). Note that \(\var[X] = \frac45\sigma^2 \approx \frac{1}{25}\) so we are looking at an interval \((220.1 - 0.4, 220.1 + 0.4) = (219.7, 220.5)\) using an interval of two standard errors.
2004 Paper 2 Q6
The vectors \({\bf a}\) and \({\bf b}\) lie in the plane \(\Pi\,\). Given that \(\vert {\bf a} \vert= 1\) and \({\bf a}.{\bf b} = 3,\) find, in terms of \({\bf a}\) and \({\bf b}\,\), a vector \({\bf p}\) parallel to \({\bf a}\) and a vector \({\bf q}\) perpendicular to \({\bf a}\,\), both lying in the plane \(\Pi\,\), such that $${\bf p}+{\bf q}={\bf a}+{\bf b}\;.$$ The vector \({\bf c}\) is not parallel to the plane \(\Pi\) and is such that \({\bf a}.{\bf c} = -2\) and \({\bf b}.{\bf c} = 2\,\). Given that \(\vert {\bf b} \vert = 5\,\), find, in terms of \({\bf a}, {\bf b}\) and \({\bf c},\) vectors \({\bf P}\), \({\bf Q}\) and \({\bf R}\) such that \({\bf P}\) and \({\bf Q}\) are parallel to \({\bf p}\) and \({\bf q},\) respectively, \({\bf R}\) is perpendicular to the plane \(\Pi\) and $${\bf P} + {\bf Q} + {\bf R} = {\bf a}+{\bf b}+{\bf c}\;.$$
Solution: Suppose \({\bf p} = \lambda {\bf a}\) and \({\bf p} + {\bf q} = {\bf a} + {\bf b}\) then \begin{align*} {\bf a} \cdot : && {\bf a} \cdot {\bf p} + {\bf a} \cdot {\bf p} &= {\bf a} \cdot {\bf a} + {\bf a} \cdot {\bf b} \\ && \lambda + 0 &= 1 + 3 = 4 \\ \Rightarrow && \mathbf{p} &= 4 \mathbf{a} \\ && \mathbf{q} &= \mathbf{b} - 3\mathbf{a} \\ \\ && \mathbf{P} &= 4p\mathbf{a} \\ && \mathbf{Q} &= q\mathbf{b} - 3q\mathbf{a} \\ \\ \mathbf{a} \cdot : && \mathbf{a} \cdot \mathbf{P} + \mathbf{a} \cdot \mathbf{Q} + \mathbf{a} \cdot \mathbf{R} &= \mathbf{a} \cdot \mathbf{a} + \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \\ && 4p &= 1+3-2 \\ \Rightarrow && p &= \tfrac12 \\ \\ && {\bf P} + {\bf Q} + {\bf R} &= {\bf a}+{\bf b}+{\bf c} \\ \mathbf{b} \cdot : && \mathbf{b} \cdot \mathbf{P} + \mathbf{b} \cdot \mathbf{Q} + \mathbf{b} \cdot \mathbf{R} &= \mathbf{b} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} \\ && 12p + 25q - 9q &= 3+25+2 \\ && 6+16q &= 30 \\ \Rightarrow && q &= \tfrac{3}{2}\\ && \\ && \mathbf{P} &= 2\mathbf{a} \\ && \mathbf{Q} &= \tfrac32 \mathbf{b} - \tfrac92 \mathbf{a} \\ && \mathbf{R} &= \tfrac72\mathbf{a} -\tfrac12 \mathbf{b} + \mathbf{c} \end{align*}
2002 Paper 1 Q1
Show that the equation of any circle passing through the points of intersection of the ellipse \((x+2)^2 +2y^2 =18\) and the ellipse \(9(x-1)^2 +16y^2 = 25\) can be written in the form \[ x^2-2ax +y^2 =5-4a\;. \]
Solution: \begin{align*} && (x+2)^2 +2y^2 &=18 \\ && 9(x-1)^2 +16y^2 &= 25 \\ \Rightarrow && 2y^2 &= 18 - (x+2)^2 \\ && 16y^2 &= 25 - 9(x-1)^2 \\ \Rightarrow && 25-9(x-1)^2 &= 8 \cdot 18 - 8(x+2)^2 \\ \Rightarrow && 25 -9+18x-9x^2 &= 144 -32- 32x +8x^2 \\ \Rightarrow && 0 &= 96 - 50x+x^2 \\ &&&= (x-48)(x-2) \\ \Rightarrow && x &= 2,48 \\ \Rightarrow && 2y^2 &= 2, 18-50^2 \\ \Rightarrow && (x,y) &= (2,\pm1) \end{align*} Therefore any circle must have it's centre on there perpendicular bisector of \((2, \pm 1)\), ie on the \(x\)-axis. Therefore it will have equation \((x-a)^2+y^2 = r^2\) and also contain the point \((2,1)\), therefore: \begin{align*} r^2 &= (2-a)^2 + 1^2 \\ &= 4 -2a+a^2 + 1 \\ &= 5-2a+a^2 \end{align*} and the equation is: \begin{align*} && (x-a)^2 + y^2 &= 5-4a+a^2 \\ \Rightarrow && x^2-2ax+a^2 +y^2 &= 5-4a+a^2 \\ \Rightarrow && x^2-2ax+y^2 &= 5-4a \end{align*} as required.
2001 Paper 2 Q14
Two coins \(A\) and \(B\) are tossed together. \(A\) has probability \(p\) of showing a head, and \(B\) has probability \(2p\), independent of \(A\), of showing a head, where \(0 < p < \frac12\). The random variable \(X\) takes the value 1 if \(A\) shows a head and it takes the value \(0\) if \(A\) shows a tail. The random variable \(Y\) takes the value 1 if \(B\) shows a head and it takes the value \(0\) if \(B\) shows a tail. The random variable \(T\) is defined by \[ T= \lambda X + {\textstyle\frac12} (1-\lambda)Y. \] Show that \(\E(T)=p\) and find an expression for \(\var(T)\) in terms of \(p\) and \(\lambda\). Show that as \(\lambda\) varies, the minimum of \(\var(T)\) occurs when \[ \lambda =\frac{1-2p}{3-4p}\;. \] The two coins are tossed \(n\) times, where \(n>30\), and \(\overline{T}\) is the mean value of \(T\). Let \(b\) be a fixed positive number. Show that the maximum value of \(\P\big(\vert \overline{T}-p\vert < b\big)\) as \(\lambda\) varies is approximately \(2\Phi(b/s)-1\), where \(\Phi\) is the cumulative distribution function of a standard normal variate and \[ s^2= \frac{p(1-p)(1-2p)}{(3-4p)n}\;. \]
Solution: \begin{align*} && \E[T] &= \E[\lambda X + \tfrac12(1-\lambda)Y] \\ &&&= \lambda \E[X] + \tfrac12(1-\lambda) \E[Y] \\ &&&= \lambda p + \tfrac12 (1-\lambda) 2p \\ &&&= p \\ \\ && \var[T] &= \var[\lambda X + \tfrac12(1-\lambda)Y] \\ &&&= \lambda^2 \var[X] + \tfrac14(1-\lambda)^2 \var[Y] \\ &&&= \lambda^2 p(1-p) + \tfrac14(1-\lambda)^22p(1-2p) \\ &&&= p(\lambda^2 + \tfrac12(1-\lambda)^2) -p^2(\lambda^2+(1-\lambda)^2)\\ &&&= p(\tfrac32\lambda^2 - \lambda + \tfrac12) -p^2(2\lambda^2 -2\lambda + 2) \end{align*} Differentiating \(\var[T]\) with respect to \(\lambda\), and noting it is a quadratic with positive leading coefficient, we get \begin{align*} && \frac{\d \var[T]}{\d \lambda} &= p(2\lambda -(1-\lambda)) - p^2(2 \lambda -2(1-\lambda)) \\ &&&= p(3\lambda - 1)-p^2(4\lambda - 2) \\ \Rightarrow && \lambda(4p-3) &= 2p-1 \\ \Rightarrow && \lambda &= \frac{1-2p}{3-4p} \end{align*} By the central limit theorem \(\overline{T} \sim N(p, \frac{\sigma^2}{n})\) in particular, \(\mathbb{P}(|\overline{T} - p| < b) = \mathbb{P}(\left \lvert |\frac{\overline{T}-p}{\frac{\sigma}{\sqrt{n}}} \right \lvert < \frac{b}{\frac{\sigma}{\sqrt{n}}}) = \mathbb{P}(|Z| < \frac{b\sqrt{n}}{\sigma}) = 2\Phi(b/s) - 1\) where \(s = \frac{\sigma}{\sqrt{n}}\) so \begin{align*} && s^2 &= \frac1n \sigma^2 \\ &&&= \frac1n \left ( \left (\left ( \frac{1-2p}{3-4p} \right)^2 + \tfrac12 \left (1-\frac{1-2p}{3-4p} \right)^2 \right)p - \left ( \left ( \frac{1-2p}{3-4p} \right)^2 + \left (1-\frac{1-2p}{3-4p} \right)^2\right)p^2 \right) \\ &&&= \frac1n \left ( \left (\left ( \frac{1-2p}{3-4p} \right)^2 + \tfrac12 \left (\frac{2-2p}{3-4p} \right)^2 \right)p - \left ( \left ( \frac{1-2p}{3-4p} \right)^2 + \left (\frac{2-2p}{3-4p} \right)^2\right)p^2 \right) \\ &&&= \frac{p}{n(3-4p)^2} \left ( (1 -4p + 4p^2 + 2-4p+2p^2) - (1-4p+4p^2+4-8p+4p^2)p \right) \\ &&&= \frac{p}{n(3-4p)^2} \left (3-13p+18p^2-8p^3 \right) \\ &&&= \frac{p}{n(3-4p)^2} (3-4p)(1-2p)(1-p) \\ &&&= \frac{p(1-p)(1-2p)}{(3-4p)n} \end{align*}
1995 Paper 1 Q8
Find functions \(\mathrm{f,g}\) and \(\mathrm{h}\) such that \[ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+\mathrm{f}(x)\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{g}(x)y=\mathrm{h}(x)\tag{\ensuremath{*}} \] is satisfied by all three of the solutions \(y=x,y=1\) and \(y=x^{-1}\) for \(0 < x < 1.\) If \(\mathrm{f,g}\) and \(\mathrm{h}\) are the functions you have found in the first paragraph, what condition must the real numbers \(a,b\) and \(c\) satisfy in order that \[ y=ax+b+\frac{c}{x} \] should be a solution of \((*)\)?
1991 Paper 2 Q1
Let \(\mathrm{h}(x)=ax^{2}+bx+c,\) where \(a,b\) and \(c\) are constants, and \(a\neq0\). Give a condition which \(a,b\) and \(c\) must satisfy in order that \(\mathrm{h}(x)\) can be written in the form \[ a(x+k)^{2},\tag{*} \] where \(k\) is a constant. If \(\mathrm{f}(x)=3x^{2}+4x\) and \(\mathrm{g}(x)=x^{2}-2\), find the two constant values of \(\lambda\) such that \(\mathrm{f}(x)+\lambda\mathrm{g}(x)\) can be written in the form \((*)\). Hence, or otherwise, find constants \(A,B,C,D,m\) and \(n\) such that \begin{alignat*}{1} \mathrm{f}(x) & =A(x+m)^{2}+B(x+n)^{2}\\ \mathrm{g}(x) & =C(x+m)^{2}+D(x+n)^{2}. \end{alignat*} If \(\mathrm{f}(x)=3x^{2}+4x\) and \(\mathrm{g}(x)=x^{2}+\alpha\) and it is given by that there is only one value of \(\lambda\) for which \(\mathrm{f}(x)+\lambda\mathrm{g}(x)\) can be written in the form \((*)\), find \(\alpha\).
Solution: For \(h(x)\) to be written in this form \(b^2=4ac\). Suppose \(f(x) = 3x^2+4x\), \(g(x) = x^2-2\). then, \begin{align*} && f(x) + \lambda g(x) &= (3+\lambda)x^2+4x - 2 \lambda \\ \Rightarrow && 0 &= 16 + 8(3+\lambda) \lambda \\ \Rightarrow && 0 &= 2+ 3 \lambda + \lambda^2 \\ &&&= (\lambda +1)(\lambda + 2) \\ \Rightarrow && \lambda &= -1 , -2 \\ \end{align*} \begin{align*} && f(x) - g(x) &= 2(x+1)^2 \\ && f(x) -2g(x) &= (x+2)^2 \\ \Rightarrow && g(x) &= 2(x+1)^2 - (x+2)^2 \\ && f(x) &= 4(x+1)^2 - (x+2)^2 \end{align*} Suppose \(f(x) = 3x^2+4x, g(x) = x^2 + \alpha\), then \begin{align*} && f(x) + \lambda g(x) &= (3+\lambda)x^2+4x+\lambda \alpha \\ \Rightarrow && 0 &= 16 -2\lambda \alpha(\lambda + 3) \\ && 0 &= \alpha \lambda^2 +3\lambda-8 \\ \Rightarrow && 0 &= 9 +32 \alpha \\ \Rightarrow && \alpha &= -\frac{9}{32} \end{align*}
1988 Paper 2 Q8
In a crude model of population dynamics of a community of aardvarks and buffaloes, it is assumed that, if the numbers of aardvarks and buffaloes in any year are \(A\) and \(B\) respectively, then the numbers in the following year at \(\frac{1}{4}A+\frac{3}{4}B\) and \(\frac{3}{2}B-\frac{1}{2}A\) respectively. It does not matter if the model predicts fractions of animals, but a non-positive number of buffaloes means that the species has become extinct, and the model ceases to apply. Using matrices or otherwise, show that the ratio of the number of aardvarks to the number of buffaloes can remain the same each year, provided it takes one of two possible values. Let these two possible values be \(x\) and \(y\), and let the numbers of aardvarks and buffaloes in a given year be \(a\) and \(b\) respectively. By writing the vector \((a,b)\) as a linear combination of the vectors \((x,1)\) and \((y,1),\) or otherwise, show how the numbers of aardvarks and buffaloes in subsequent years may be found. On a sketch of the \(a\)-\(b\) plane, mark the regions which correspond to the following situations
- an equilibrium population is reached as time \(t\rightarrow\infty\);
- buffaloes become extinct after a finite time;
- buffaloes approach extinction as \(t\rightarrow\infty.\)
Solution: If the population in a given year is \(\mathbf{v} = \begin{pmatrix}A \\ B \end{pmatrix}\) then the population the next year is \(\mathbf{Mv}\) where \(\mathbf{M} = \begin{pmatrix} \frac14 & \frac34 \\ -\frac12 &\frac32 \end{pmatrix}\) The ratio is the same if \(\mathbf{Mv} = \lambda \mathbf{v}\) ie if \(\mathbf{v}\) is an eigenvector of \(\mathbf{M}\). The eigenvalues will be \(1\) and \(\frac38\) (by inspection) so we should be able to solve for the eigenvectors: \(\lambda = 1\) we have \(\frac14A + \frac34B = A \Rightarrow A = B\) a ratio of \(1\). \(\lambda = \frac38\) we have \(\frac14A + \frac34B = \frac38A \Rightarrow \frac34B = \frac18A \Rightarrow A = 6B\) a ratio of \(6\). If we write \(\begin{pmatrix} a \\ b \end{pmatrix}\) as \(x_1 \begin{pmatrix} 1 \\ 1 \end{pmatrix} + x_6 \begin{pmatrix} 6 \\ 1 \end{pmatrix}\) we find that after \(n\) years, we have: \(x_1 \begin{pmatrix} 1 \\ 1 \end{pmatrix} + \l \frac38 \r^n x_6 \begin{pmatrix} 6 \\ 1 \end{pmatrix}\) for the populations. Therefore if \(x_1\) is \(< 0\) then in finite time we will end up with one population being 0. If \(x_1 > 0\) are positive we tend to a finite population and if \(x_1 = 0\) then over time the population will tend to \(0\) at infinity. In our diagram these areas correspond to (red) - die out in finite time, (green) population stable and the thick black line where the population goes extinct as \(t \to \infty\)
