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2002 Paper 1 Q11
D: 1500.0 B: 1484.0

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A particle \(P_1\) of mass \(m\) collides with a particle \(P_2\) of mass \(km\) which is at rest. No energy is lost in the collision. The direction of motion of \(P_1\) and \(P_2\) after the collision make non-zero angles of \(\theta\) and \(\phi\), respectively, with the direction of motion of \(P_1\) before the collision, as shown. Show that \[ \sin^2\theta + k\sin^2\phi = k\sin^2(\theta+\phi) \;. \] Show that, if the angle between the particles after the collision is a right angle, then \(k=1\,\).

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2002 Paper 1 Q12
D: 1484.0 B: 1469.5

Harry the Calculating Horse will do any mathematical problem I set him, providing the answer is 1, 2, 3 or 4. When I set him a problem, he places a hoof on a large grid consisting of unit squares and his answer is the number of squares partly covered by his hoof. Harry has circular hoofs, of radius \(1/4\) unit. After many years of collaboration, I suspect that Harry no longer bothers to do the calculations, instead merely placing his hoof on the grid completely at random. I often ask him to divide 4 by 4, but only about \(1/4\) of his answers are right; I often ask him to add 2 and 2, but disappointingly only about \(\pi/16\) of his answers are right. Is this consistent with my suspicions? I decide to investigate further by setting Harry many problems, the answers to which are 1, 2, 3, or 4 with equal frequency. If Harry is placing his hoof at random, find the expected value of his answers. The average of Harry's answers turns out to be 2. Should I get a new horse?

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2002 Paper 1 Q13
D: 1484.0 B: 1443.0

The random variable \(U\) takes the values \(+1\), \(0\) and \(-1\,\), each with probability \(\frac13\,\). The random variable \(V\) takes the values \(+1\) and \(-1\) as follows:

\begin{tabular}{ll} if \(U=1\,\),&then \(\P(V=1)= \frac13\) and \(\P(V=-1)=\frac23\,\);\\[2mm] if \(U=0\,\),&then \(\P(V=1)= \frac12\) and \(\P(V=-1)=\frac12\,\);\\[2mm] if \(U=-1\,\),&then \(\P(V=1)= \frac23\) and \(\P(V=-1)=\frac13\,\). \end{tabular}
  1. Show that the probability that both roots of the equation \(x^2+Ux+V=0\) are real is \(\frac12\;\).
  2. Find the expected value of the larger root of the equation \(x^2+Ux+V=0\,\), given that both roots are real.
  3. Find the probability that the roots of the equation $$x^3+(U-2V)x^2+(1-2UV)x + U=0$$ are all positive.

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2002 Paper 1 Q14
D: 1500.0 B: 1516.0

In order to get money from a cash dispenser I have to punch in an identification number. I have forgotten my identification number, but I do know that it is equally likely to be any one of the integers \(1\), \(2\), \ldots , \(n\). I plan to punch in integers in order until I get the right one. I can do this at the rate of \(r\) integers per minute. As soon as I punch in the first wrong number, the police will be alerted. The probability that they will arrive within a time \(t\) minutes is \(1-\e^{-\lambda t}\), where \(\lambda\) is a positive constant. If I follow my plan, show that the probability of the police arriving before I get my money is \[ \sum_{k=1}^n \frac{1-\e^{-\lambda(k-1)/r}}n\;. \] Simplify the sum. On past experience, I know that I will be so flustered that I will just punch in possible integers at random, without noticing which I have already tried. Show that the probability of the police arriving before I get my money is \[ 1-\frac1{n-(n-1)\e^{-\lambda/r}} \;. \]

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