Mock Exam Set 1 - Paper 2

[Maximum mark: 5]

ABCD is a quadrilateral where $\text{AB} = 9$ cm, $\text{BC} = 12$ cm, $\text{CD} = 11$ cm, $\text{\(\text{DA} = 8.5\)\hspace{0.25em}cm}$ and $\text{A\^{B}C} = \ang{90}$. Find $\text{A\^{D}C}$, giving your answer correct to the nearest degree.

[Maximum mark: 7]

A particle moves along a straight line so that its velocity, $v$ ms$^{-1}$, after $t$ seconds is given by $v(t) = 1.5^t - 4.9$, for $0 \leq t \leq 6$.

1. Find when the particle is at rest. [2]
   

2. Find the acceleration of the particle when $t = 3$. [2]
   

3. Find the total distance travelled by the particle. [3]
   

[Maximum mark: 6]

On Gary's $50$th birthday, he invests $\$P$ in an account that pays a nominal annual interest rate of $5$ %, compounded monthly.

The amount of money in Gary's account at the end of each year follows a geometric sequence with common ratio, $\alpha$.

1. Find the value of $\alpha$, giving your answer to four significant figures. [3]
   

Gary makes no further deposits or withdrawals from the account.

2. Find the age Gary will be when the amount of money in his account will be double the amount he invested. [3]
   

[Maximum mark: 6]

It is given that three in ten men take less than $220$ minutes to run a marathon. It is also known that one in five men take more than $274$ minutes to run the same marathon.

Assume that the time taken, in minutes, for men to run a marathon is modelled by a normal distribution.

Find the mean and standard deviation.

[Maximum mark: 6]

Ten students are to be arranged in a new chemistry lab. The chemistry lab is set out in two rows of five desks as shown in the following diagram.

1. Find the number of ways the ten students may be arranged in the lab. [1]
   

Two of the students, Hugo and Leo, were noticed to talk to each other during previous lab sessions.

2. Find the number of ways the students may be arranged if Hugo and Leo must sit so that one is directly behind the other. For example, Dest $1$ and Desk $6$. [2]
   

3. Find the number of ways the students may be arranged if Hugo and Leo must not sit next to each other in the same row. [3]
   

[Maximum mark: 6]

The following table below shows the points scored by eight basketball players in two games.

Let $L_1$ be the regression line of $x$ on $y$. The equation of the line $L_1$ can be written in the form $x = ay + b$.

1. Find the value of $a$ and the value of $b$. [2]
   

Let $L_2$ be the regression line of $y$ on $x$. The lines $L_1$ and $L_2$ both pass through the same point with coordinates $(p,q)$.

2. Find the value of $p$ and the value of $q$. [2]
   

3. Andrew's injury didn't allow him to play in the first game but he played and scored $10$ points in the second game. Use an appropriate regression equation to estimate the number of points that Andrew would have scored in the first game. Round your answer to the nearest point. [2]
   

[Maximum mark: 6]

The diagram below shows a fenced triangular enclosure in the middle of a grassy field where $\mathrm{AC} = 3$ m, $\mathrm{DC} = 2$ m, $\text{C\^{B}A} = \alpha$ radians and $\text{A\^{C}B} = \frac{\pi}{3}$ radians.

One end of a rope is attached at point D on the outside edge of the enclosure, and the other end is attached to a goat G.

Given that the rope is $6$ m long and the area of field outside the enclosure that the goat is able to graze is $74$ m$^2$, find the value of $\alpha$.

[Maximum mark: 9]

The function $f$ is defined by $f(x)=e^x\cos x$, $x\in \mathbb{R}$.

1. By finding a suitable number of derivatives of $f$, determine the Maclaurin series for $f(x)$ as far as the term $x^4$.[6]
   

2. Hence, or otherwise, determine the exact value of $\displaystyle \lim_{x\to 0} \dfrac{e^x\cos x-x-1}{x^3}$.[3]
   

[Maximum mark: 12]

The points A and B have coordinates A$(1,0,4)$ and B$(2,2,3)$ relative to an origin O.

1. 1. Find $\vv{\text{OA}}\times\vv{\text{OB}}$.

   2. Determine the area of the triangle OAB, giving your answer correct to two decimal places.

   3. Find the Cartesian equation of the plane OAB. [5]
      

2. 1. Find the vector equation of the line $\mathit{L}_1$ containing the points A and B.

   The line $\mathit{L}_2$ has vector equation $\mathbf{r} = \begin{pmatrix*}[c] 0 \\ 12 \\ -9 \end{pmatrix*}\hspace{-0.15em} + \hspace{0.1em}s\hspace{-0.1em}\begin{pmatrix*}[c] -3 \\ 1 \\ -4 \end{pmatrix*}$, $s\in\mathbb{R}$.

   2. Determine whether the lines $\mathit{L}_1$ and $\mathit{L}_2$ are parallel, skew or intersecting. If they intersect, find the coordinates of the point of intersection. [7]
      

[Maximum mark: 13]

The continuous random variable $X$ has probability density function $f$ given by

1. Show that $p = \dfrac{1}{6}$. [3]
   

2. Find $\mathrm{P}(X < 2)$. [3]
   

3. Given that $\mathrm{P}(a < X < 2) = 3\hspace{0.05em}\mathrm{P}(3\hspace{0.02em}a < X < 2)$, where $\dfrac{1}{3} < a < \dfrac{2}{3}$,

   find the value of $a$. [7]
   

[Maximum mark: 14]

The curves $y=f(x)$ and $y=g(x)$ both pass through the point $(1,0)$ and are defined by the differential equations $\dfrac{\mathrm{d}y}{\mathrm{d}x}=2x-y^2$ and $\dfrac{\mathrm{d}y}{\mathrm{d}x}=3y-\dfrac{x}{2}$ respectively.

1. Show that the tangent to the curve $y=f(x)$ at the point $(1,0)$ is normal to the curve $y=g(x)$ at the point $(1,0)$.[2]
   

2. Find $g(x)$.[7]
   

3. Use Euler's method with steps of $0.2$ to estimate $f(2)$ to 5 decimal places.[5]
   

[Maximum mark: 20]

An ice cream company advertises free gifts to customers who collect four coupons. The coupons are placed at random into $20\%$ of all bags of ice cream sticks. Oliver buys ice cream sticks and he opens them one at a time to see if they contain a coupon.

Let $\mathrm{P}(X = n)$ be the probability that Oliver obtains his $4$th coupon on the $n$th ice cream stick opened.

It is assumed that the probability that an ice cream stick contains a coupon stays at $20\%$ throughout the question.

1. Show that $\mathrm{P}(X = 4) = 0.0016$ and $\mathrm{P}(X = 5) = 0.00512$. [3]
   

It is given that $\mathrm{P}(X = n) = \dfrac{1}{3750}(n-1)(n^2+an+b)(0.8)^{n-4}$ for $n \geq 4$, $n \in \mathbb{N}$,

where $a,b \in \mathbb{Z}$.

2. Find the values of $a$ and $b$. [5]
   

3. Deduce that $\dfrac{\mathrm{P}(X = n)}{\mathrm{P}(X = n-1)} = \dfrac{4(n-1)}{5(n-4)}$ for $n > 4$. [4]
   

4. 1. Hence show that $X$ has two modes $m_1$ and $m_2$.

   2. Write down the values of $m_1$ and $m_2$. [5]
      

Oliver's mother goes to the market and buys $N$ ice cream sticks. She takes the ice cream sticks home for Oliver to open.

5. Determine the minimum value of $N$ such that the probability Oliver receives at least one free gift is greater than $0.5$. [3]