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Pre-U Pre-U 9795/2 Specimen Q5

8 marks Moderate -0.3

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. 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]
Calculate the times taken for each of the two cases in part (i). [3]

Pre-U Pre-U 9795/2 Specimen Q6

12 marks Challenging +1.8

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.

Prove that, while one part of the string is taut and the other part is slack, the particle is describing simple harmonic motion. [2]
Find the speed of the particle when the slack part of the string becomes taut. [2]
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]

Pre-U Pre-U 9795/2 Specimen Q7

6 marks Challenging +1.2

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]

Pre-U Pre-U 9795/2 Specimen Q8

9 marks Standard +0.3

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]
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]

Pre-U Pre-U 9795/2 Specimen Q9

10 marks Standard +0.3

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

State an assumption that must be made so that the above situation can be modelled by a Poisson distribution. [1]
Find the probability of at least 7 of these fossils occurring in an area of $10 \text{ m}^2$. [2]
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]
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]

Pre-U Pre-U 9795/2 Specimen Q10

10 marks Standard +0.3

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.

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

Pre-U Pre-U 9795/2 Specimen Q11

12 marks Standard +0.3

State briefly the conditions under which the binomial distribution $\text{B}(n, p)$ may be approximated by a normal distribution. [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]
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]

Pre-U Pre-U 9795/2 Specimen Q12

13 marks Standard +0.8

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 $$f(t) = \begin{cases} \frac{2}{5}t e^{-\frac{t^2}{5}} & t \geq 0, \\ 0 & \text{otherwise}. \end{cases}$$

Sketch the graph of $f(t)$. [2]
Find the cumulative distribution function $F(t)$ and hence find the median value of $T$. [3]
Find the probability that $T$ is greater than the modal value of $T$. [5]
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]

Pre-U Pre-U 9794/2 Specimen Q1

4 marks Moderate -0.3

Show that $\binom{n}{n-2} = \frac{n(n-1)}{2}$, where the positive integer $n$ satisfies $n \geqslant 2$. [1]
Solve the equation $\binom{2n+1}{2n-1} - 2 \times \binom{n}{n-2} = 24$. [3]

Determine the area of the smaller segment. [4]

The line is rotated clockwise about $O$ through $45^{\circ}$ and then reflected in the $x$-axis.

Find the equation of the line in its final position. [3]

Pre-U Pre-U 9794/2 Specimen Q5

9 marks Standard +0.3

Divide the quartic $2x^4 - 5x^3 + 4x^2 + 2x - 3$ by the quadratic $x^2 + x - 2$, identifying the quotient and the remainder. [4]
1. Show that $(x - 1)$ is a factor of $nx^{n+1} - (n + 1)x^n + 1$, where $n$ is a positive integer. [1]
2. Hence, or otherwise, find all the roots of $3x^4 - 4x^3 + 1 = 0$. [4]

Pre-U Pre-U 9794/2 Specimen Q6

10 marks Standard +0.8

Express $y^3 - 3y - 2$ in terms of $x$, where $x = y + 1$. [1]
Hence express $$\frac{2y + 5}{y^3 - 3y - 2}$$ in partial fractions. [5]
Find the exact value of $$\int_0^1 \frac{2y + 5}{y^3 - 3y - 2} dy.$$ [4]

Pre-U Pre-U 9794/2 Specimen Q7

12 marks Moderate -0.3

Given that $\cos \theta = \frac{7}{25}$, where $\frac{3}{2}\pi < \theta < 2\pi$, determine the exact values of
1. $\sin \theta$, [3]
2. $\sin(\frac{1}{2}\theta)$, [3]
3. $\sec(\frac{1}{2}\theta)$. [1]
1. Express $4 \cos x - 3 \sin x$ in the form $A \cos(x + \alpha)$, where $A > 0$ and $0 < \alpha < \frac{1}{2}\pi$. [2]
2. Hence find the greatest and least values of $4 \cos x - 3 \sin x$ for $0 \leqslant x \leqslant \pi$. [3]

Pre-U Pre-U 9794/2 Specimen Q8

14 marks Standard +0.8

1. Find the general solution of the differential equation $$x \frac{dy}{dx} = y(1 + x \cot x),$$ expressing $y$ in terms of $x$. [5]
2. Find the particular solution given that $y = 1$ when $x = \frac{1}{2}\pi$. [2]
The real variables $x$ and $y$ are related by $x^2 - y^2 = 2ax - b$, where $a$ and $b$ are real constants.
1. Show that $\frac{dy}{dx} = 0$ can only be solved for $x$ and $y$ if $b \geqslant a^2$. [5]
2. Show that $y \frac{d^2y}{dx^2} = 1 - \left(\frac{dy}{dx}\right)^2$. [2]

Pre-U Pre-U 9794/2 Specimen Q9

15 marks Challenging +1.3

Two curves are defined by $y = x^k$ and $y = x^{\frac{1}{k}}$, for $x \geqslant 0$, where $k > 0$.

Prove that, except for one value of $k$, the curves intersect in exactly two points. [4]

The two curves enclose a finite region $R$.

Find the area, $A$, of $R$, giving your answer in the form $A = f(k)$ and distinguishing clearly between the cases $k < 1$ and $k > 1$. [4]
Determine the set of values of $k$ for which $A \leqslant 0.5$. [3]
The function $f$ is given by $f : x \mapsto A$ with $k > 1$. Prove that $f$ is one-one and determine its inverse. [4]

Pre-U Pre-U 9794/2 Specimen Q10

7 marks Moderate -0.3

Determine the impulse of a force of magnitude $6$ N that acts on a particle of mass $3$ kg for $1.5$ seconds. [1]

Particles $A$ and $B$, of masses $0.1$ kg and $0.2$ kg respectively, can move on a smooth horizontal table. Initially $A$ is moving with speed $3$ m s$^{-1}$ towards $B$, which is moving with speed $1$ m s$^{-1}$ in the same direction as the motion of $A$. During a collision $B$ experiences an impulse from $A$ of magnitude $0.2$ kg m s$^{-1}$.

Find the speeds of the particles immediately after the collision. [4]
Determine the coefficient of restitution between the particles. [2]

Pre-U Pre-U 9794/2 Specimen Q11

11 marks Standard +0.3

A particle $P$ of mass $1.5$ kg is placed on a smooth horizontal table. The particle is initially at the origin of a $2$-dimensional vector system defined by perpendicular unit vectors $\mathbf{i}$ and $\mathbf{j}$ in the plane of the table. The particle is subject to three forces of magnitudes $10$ N, $12$ N and $F$ N, acting in the directions of the vectors $3\mathbf{i} + 4\mathbf{j}$, $-\mathbf{j}$ and $-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}$ respectively, and no others.

Given that the system is in equilibrium, determine $F$ and $\theta$. [6]

The force of magnitude $12$ N is replaced by one of magnitude $4$ N, but in the opposite direction. The particle is initially at rest.

Find the position vector of the particle $3$ seconds later. [5]

Pre-U Pre-U 9794/2 Specimen Q12

11 marks Standard +0.3

A particle $P$ of mass $2$ kg rests on a long rough horizontal table. The coefficient of friction between $P$ and the table is $0.2$. A light inextensible string has one end attached to $P$ and the other end attached to a particle $Q$ of mass $4$ kg. The particle $Q$ is placed on a smooth plane inclined at $30^{\circ}$ to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time $t = 0$, where $t$ is measured in seconds. In the subsequent motion $P$ does not reach the pulley.

Show that the magnitude of the acceleration of the particles is $\frac{8}{3}$ m s$^{-2}$. [4]

After the particles have moved a distance of $12$ m the string is cut.

Find the corresponding value of $t$ and the speed of the particles at this instant. [4]
Find the value of $t$ when $P$ comes to rest. [3]

Pre-U Pre-U 9794/2 Specimen Q13

11 marks Standard +0.3

A gunner fires one shell from each of two guns on a stationary ship towards a vertical cliff $AB$ of height $100$ m whose foot $A$ is at a horizontal distance $600$ m from the point of projection.

Given that the shell from the first gun hits the cliff, travelling horizontally, at a point $45$ m above $A$, determine the initial velocity of the shell. Express your answer in the form $a\mathbf{i} + b\mathbf{j}$, where $a$ and $b$ are integers. [6]
The shell from the second gun hits the cliff at its top point $B$. Given that the initial speed of the shell is $300$ m s$^{-1}$, determine the possible angles of projection. [5]