Mock Exam Set 1 - Paper 2

[Maximum mark: 4]

The following table shows the number of overtime hours worked by employees in a company.

It is known that the mean number of overtime hours is $1$.

1. Find the value of $x$. [2]
   

2. Find the standard deviation of the data. [2]
   

[Maximum mark: 6]

Greg has saved $2000$ British pounds (GBP) over the last six months. He decided to deposit his savings in a bank which offers a nominal annual interest rate of $\text{\(8\)\hspace{0.05em}\%}$, compounded monthly, for two years.

1. Calculate the total amount of interest Greg would earn over the two years. Give your answer correct to two decimal places. [3]
   

Greg would earn the same amount of interest, compounded semi-annually, for two years if he deposits his savings in a second bank.

2. Calculate the nominal annual interest rate the second bank offers. [3]
   

[Maximum mark: 6]

A farmer is going to fence two equal adjacent parcels of land. These parcels share one side (which also requires fencing) as shown in the following diagram. The farmer has only $80$ metres of fence.

1. Write down the equation for the total length of the fence, $80$ m, in terms of $x$ and $y$. [1]
   

2. Write down the total area of both parcels of land in terms of $x$. [2]
   

3. Find the maximum area, in m$^2$, of one parcel of land. [3]
   

[Maximum mark: 6]

$A$ and $B$ are independent events such that $\mathrm{P}(A \cap {B\,} ^\prime) = 0.09$ and $\mathrm{P}({A\,}^\prime \cap B) = 0.49$.

Let $x = \mathrm{P}(A\cap B)$.

1. 1. Express $\mathrm{P}(A)$ in terms of $x$.

   2. Express $\mathrm{P}(B)$ in terms of $x$. [2]
      

2. Find the value of $x$. [2]
   

3. Find $\mathrm{P}(B\,|\,A)$. [2]
   

[Maximum mark: 6]

A $3$D printer builds a set of $49$ Ei$\text{f}$fel Tower Replicas in different sizes. The height of the largest tower in this set is $64$ cm. The heights of successive smaller towers are $95$ % of the preceding larger tower, as shown in the diagram below.

1. Find the height of the smallest tower in this set. [3]
   

2. Find the total height if all $49$ towers were placed one on top of another. [3]
   

[Maximum mark: 6]

Consider the function $f(x)=\dfrac{e^x-e^{-x}}{2}$, $x\in \mathbb{R}$.

1. Show that $f$ is an odd function. [2]
   

Now, consider the function $g$ given by $g(x) = \dfrac{x^4+2}{2x}$ , $x\in \mathbb{R}, x \neq 0$.

2. By considering the graph of $y=f(x)-g(x)$, solve $f(x)>g(x)$ for $x\in \mathbb{R}$. [4]
   

[Maximum mark: 6]

The following diagram shows the curve $\dfrac{x^2}{400}+\dfrac{(y+5)^2}{225}=1$, where $0 \leq y \leq h$.

The curve from point C to point P is rotated $360^\circ$ about the $y$-axis to form a lamp shade. The rectangle ABCD, of height $(10-h)$ cm, is rotated $360^\circ$ about the $y$-axis to form a solid ceiling fixture.

The lamp shade is assumed to have a negligible thickness.

Given that the interior volume of the lamp shade is to be $6\hspace{0.15em}000\hspace{0.15em}\text{cm}^3$, determine the height of the ceiling fixture, length AD in the diagram.

[Maximum mark: 7]

A professor and five of his students attend a talk given in a lecture series. They have a row of 8 seats to themselves.

Find the number of ways the professor and his students can sit if

1. the professor and his students sit together. [3]
   

2. the students decide to sit at least one seat apart from their professor. [4]
   

[Maximum mark: 9]

Consider two vectors $\mathbf{u}$ and $\mathbf{v}$ such that $\mathbf{u} = \begin{pmatrix*}[c] -6 \\ 8 \end{pmatrix*}$ and $|\mathbf{v}|=20$.

1. Find the possible range of values for $|\mathbf{u}+\mathbf{v}|$. [2]
   

2. Given that $\mathbf{v}=k \mathbf{u}$ for some $k\in\mathbb{R}$, find $\mathbf{v}$ when $|\mathbf{u}+\mathbf{v}|$ is a minimum. [2]
   

3. Find the vector $\mathbf{w}=\begin{pmatrix*}[c] a \\ b \end{pmatrix*}$ such that $a,b \in \mathbb{R}^+$, $|\mathbf{w}|=|\mathbf{v}|$ and $\mathbf{w}$ is perpendicular to $\mathbf{u}$. [5]
   

[Maximum mark: 19]

Consider the differential equation

for $x>0$ and $y>3x$. It is given that $y=4$ when $x=1$.

1. Use Euler's method, with a step length of $0.08$, to find an approximate value for $y$ when $x=1.4$. [4]
   

2. Use the substitution $y=vx$ to show that $x\hspace{0.15em}\dfrac{\mathrm{d}v}{\mathrm{d}x}=v^2-v-6$. [3]
   

3. By solving the differential equation, show that $y=\dfrac{18x+2x^6}{6-x^5}\,$. [10]
   

4. 1. Find the actual value of $y$ when $x=1.4$.

   2. Using the graph of $y=\dfrac{18x+2x^6}{6-x^5}\,$, suggest a reason why the approximation given by Euler's method in part (a) is not a good estimate to the actual value of $y$ at $x=1.4$. [2]
      

[Maximum mark: 15]

A physicist is studying the motion of two separate particles moving in a straight line. She measures the displacement of each particle from a fixed origin over the course of 10 seconds.

The physicist found that the displacement of particle $A$, $s_A$ cm, at time $t$ seconds can be modelled by the function $s_A(t)=7t+9$, where $0 \leq t \leq 10$.

The physicist found that the displacement of particle $B$, $s_B$ cm, at time $t$ seconds can be modelled by the function $s_B(t)=\mathrm{cos}(3t+5)+8t+4$.

1. Use the physicist's models to find the initial displacement of

   1. Particle $A$;

   2. Particle $B$ correct to three significant figures. [3]
      

2. Find the values of $t$ when $s_A(t)=s_B(t)$. [3]
   

3. For $t>6$, prove that particle $B$ was always further away from the fixed origin than particle $A$. [3]
   

4. For $0 \leq t \leq 10$, find the total amount of time that the velocity of particle $A$ was greater than the velocity of particle $B$. [6]
   

[Maximum mark: 20]

Consider the function $f(x)=\sqrt{\dfrac{10}{x^2}-1}$, where $1 \leq x \leq \sqrt{10}$.

1. Sketch the curve $y=f(x)$, indicating the coordinates of the endpoints. [2]
   

2. 1. Show that $f^{-1}(x)=\sqrt{\dfrac{10}{x^2+1}}$.

   2. State the domain and range of $f^{-1}$. [5]
      

The curve $y=f(x)$ is rotated through $2\pi$ about the $y$-axis to form a solid of revolution that is used to model a vase.

3. 1. Show that the volume $V$ cm$^3$, of liquid in the vase when it is filled to a height of $h$ centimetres is given by $V=10\pi\hspace{0.15em}\text{arctan}(h)$.

   2. Hence, determine the volume of the vase. [5]
      

At $t=0$, the vase is filled to its maximum volume with water. Water is then removed from the vase at a constant rate of $4\hspace{0.15em}\text{cm}^3\hspace{0.15em}\text{s}^{-1}$.

4. Find the time it takes to completely empty the vase. [2]
   

5. Find the rate of change of the height of the water when half of the water has been emptied from the vase. [6]