Pre-U Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) Specimen

A smooth sphere \(A\) of mass \(m\) is projected with speed \(u\) along a smooth horizontal surface and strikes a stationary smooth sphere \(B\) of equal radius but of mass \(M\). The direction of motion of \(A\) before the impact makes an acute angle \(\theta\) with the line of centres at the moment of contact. After the impact, the direction of motion of \(A\) is perpendicular to the initial direction of motion of \(A\). The coefficient of restitution between the two spheres is \(e\). Given that \(Me \geq m\), prove that $$\tan^2 \theta = \frac{Me - m}{m + M}.$$ [8]

One end of a light inextensible string of length \(l\) is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs at rest vertically below \(O\). The particle is then given a horizontal speed \(u\).

1. 1. Show that when \(OP\) has turned through an angle \(\theta\) the tension in the string is given by $$T = mg(3\cos \theta - 2) + \frac{mu^2}{l}$$ as long as the string remains taut. [5]
   2. Deduce that \(u^2 \geq 5gl\) in order for the particle to perform complete circles. [1]
2. 1. In the case \(u^2 = 3gl\), find the angle that \(OP\) makes with the downward vertical at \(O\) at the instant when the string becomes slack. [2]
   2. Describe the nature of the motion while the string is slack. [1]

A stone of mass \(m\) is projected vertically upwards with initial velocity \(u\). At time \(t\), the height risen above the point of projection is \(x\) and the resistance to motion is \(kv\) when the velocity of the stone is \(v\).

1. Write down a first-order differential equation relating \(v\) and \(t\) and hence find \(t\) in terms of \(v\). [5]
2. Write down a first-order differential equation relating \(v\) and \(x\) and hence find \(x\) in terms of \(v\). [6]

A particle is projected with velocity \(V\) at an angle \(\alpha\) to the horizontal up a plane inclined at \(\beta\) to the horizontal, where \(\alpha > \beta\).

1. Show that the time of flight is \(\frac{2V \sin(\alpha - \beta)}{g \cos \beta}\). [3]
2. Show that the range on the inclined plane is \(\frac{2V^2 \sin(\alpha - \beta) \cos \alpha}{g \cos^2 \beta}\). [4]
3. If the particle strikes the plane at right angles, prove that \(\tan \alpha = \cot \beta + 2 \tan \beta\). [5]

A girl can paddle her canoe at \(5 \text{ m s}^{-1}\) in still water. She wishes to cross a river which is \(100 \text{ m}\) wide and flowing at \(8 \text{ m s}^{-1}\).

1. 1. Write down the angle to the river bank at which the boat must head, in order to cross the river in the least possible time. [1]
   2. Find the acute angle to the river bank at which the boat must head, in order to cross the river by the shortest route. [4]
2. Calculate the times taken for each of the two cases in part (i). [3]

A light elastic string of natural length \(2a\) and modulus of elasticity \(\lambda\) is stretched between two points \(A\) and \(B\), which are \(3a\) apart on a smooth horizontal table. A particle of mass \(m\) is attached to the mid-point of the string, pulled aside to \(A\) and released.

1. Prove that, while one part of the string is taut and the other part is slack, the particle is describing simple harmonic motion. [2]
2. Find the speed of the particle when the slack part of the string becomes taut. [2]
3. Prove that the total time for the particle to reach the mid-point of the string for the first time is $$\sqrt{\frac{ma}{\lambda}} \left( \frac{\pi}{3} + \frac{1}{\sqrt{2}} \sin^{-1} \frac{1}{\sqrt{7}} \right).$$ [8]

The length \(M\) of male snakes of a certain species may be regarded as a normal random variable with mean \(0.45\) metres and standard deviation \(0.06\) metres. The length \(F\) of female snakes of the same species may be regarded as a normal random variable with mean \(0.55\) metres and standard deviation \(0.08\) metres. Assuming that \(M\) and \(F\) are independent, find the probability that a randomly chosen male snake of this species is more than three-quarters of the length of a randomly chosen female snake of this species. [6]

1. The random variable \(X\) is such that \(\text{E}(X) = a\theta + b\), where \(a\) and \(b\) are constants and \(\theta\) is a parameter. Show that \(\frac{X - b}{a}\) is an unbiased estimator of \(\theta\). [2]
2. The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{8}(\theta + 4 - x) & \theta \leq x \leq \theta + 4, \\ 0 & \text{otherwise}. \end{cases}$$ Find \(\text{E}(X)\) and hence find an unbiased estimator of \(\theta\). [7]

A certain type of fossil occurs at a mean rate of \(0.5\) per square metre at a particular location.

1. State an assumption that must be made so that the above situation can be modelled by a Poisson distribution. [1]
2. Find the probability of at least 7 of these fossils occurring in an area of \(10 \text{ m}^2\). [2]
3. Given that at least 4 such fossils have occurred in an area of \(5 \text{ m}^2\), find the probability that there will be more than 6 found in this area of \(5 \text{ m}^2\). [3]
4. Find the least area that must be searched in order that the probability of finding at least one fossil of this type is greater than \(0.999\). Give your answer to the nearest square metre. [4]

A biased tetrahedral die has faces numbered \(1\) to \(4\). The random variable \(X\) is the number on the face of the die which is in contact with the table after the die has been thrown. It is known, for this die, that \(\text{P}(X = x) = kx\) where \(k\) is a constant.

1. Determine the value of \(k\) and state the moment generating function of \(X\). [3]
2. Hence find \(\text{E}(X)\) and \(\text{Var}(X)\). [7]

1. State briefly the conditions under which the binomial distribution \(\text{B}(n, p)\) may be approximated by a normal distribution. [2]
2. A multiple-choice test has \(50\) questions. Each question has four possible answers. A student passes the test if answering \(36\%\) or more of the questions correctly. Using a suitable distributional approximation, estimate the probability that a student who selects answers to all the questions randomly will pass the test. [5]
3. A test similar to that in part (ii) has \(N\) questions instead of \(50\) questions. Estimate the least value of \(N\) so that the probability that a student gets \(36\%\) or more of the questions correct, by selecting answers to all questions randomly, will be less than \(0.01\). (A continuity correction is not required in this part of the question.) [5]

The length of time, in years, that a salesman keeps his company car may be modelled by the continuous random variable \(T\). The probability density function of \(T\) is given by $$f(t) = \begin{cases} \frac{2}{5}t e^{-\frac{t^2}{5}} & t \geq 0, \\ 0 & \text{otherwise}. \end{cases}$$

1. Sketch the graph of \(f(t)\). [2]
2. Find the cumulative distribution function \(F(t)\) and hence find the median value of \(T\). [3]
3. Find the probability that \(T\) is greater than the modal value of \(T\). [5]
4. The probability that a randomly chosen salesman keeps his car longer than \(N\) years is \(0.05\). Find the value of \(N\) correct to \(3\) significant figures. [3]