Questions Paper 2 - A-Level Maths

Edexcel Paper 2 Specimen Q5

The line $l$ has equation

$$3 x - 2 y = k$$ where $k$ is a real constant.
Given that the line $l$ intersects the curve with equation $$y = 2 x ^ { 2 } - 5$$ at two distinct points, find the range of possible values for $k$.

Edexcel Paper 2 Specimen Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-12_624_1057_258_504} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the $x$-axis at the points $A$ and $B$ and has a maximum turning point at $Q$, as shown in Figure 2.

Find the $x$ coordinate of $A$ and the $x$ coordinate of $B$.
Show that the $x$ coordinate of $Q$ satisfies $$x = \frac { 8 } { 1 + \ln x }$$
Show that the $x$ coordinate of $Q$ lies between 3.5 and 3.6
Use the iterative formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$ with $x _ { 1 } = 3.5$ to
1. find the value of $x _ { 5 }$ to 4 decimal places,
2. find the $x$ coordinate of $Q$ accurate to 2 decimal places.

Edexcel Paper 2 Specimen Q7

A bacterial culture has area $p \mathrm {~mm} ^ { 2 }$ at time $t$ hours after the culture was placed onto a circular dish.

A scientist states that at time $t$ hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture.

Show that the scientist's model for $p$ leads to the equation $$p = a \mathrm { e } ^ { k t }$$ where $a$ and $k$ are constants. The scientist measures the values for $p$ at regular intervals during the first 24 hours after the culture was placed onto the dish. She plots a graph of $\ln p$ against $t$ and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95
Estimate, to 2 significant figures, the value of $a$ and the value of $k$.
Hence show that the model for $p$ can be rewritten as $$p = a b ^ { t }$$ stating, to 3 significant figures, the value of the constant $b$. With reference to this model,
1. interpret the value of the constant $a$,
2. interpret the value of the constant $b$.
State a long term limitation of the model for $p$.

Edexcel Paper 2 Specimen Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-18_367_709_280_676} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl. When the depth of the water is $h \mathrm {~cm}$, the volume of water, $V \mathrm {~cm} ^ { 3 }$, according to the model is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$ The flow of water into the bowl is at a constant rate of $160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$ for $0 \leqslant h \leqslant 12$

Find the rate of change of the depth of the water, in $\mathrm { cm } \mathrm { s } ^ { - 1 }$, when $h = 10$ Given that the flow of water into the bowl is increased to a constant rate of $300 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$ for $12 < h \leqslant 24$
find the rate of change of the depth of the water, in $\mathrm { cms } ^ { - 1 }$, when $h = 20$

Edexcel Paper 2 Specimen Q9

A circle with centre $A ( 3 , - 1 )$ passes through the point $P ( - 9,8 )$ and the point $Q ( 15 , - 10 )$
1. Show that $P Q$ is a diameter of the circle.
2. Find an equation for the circle.
A point $R$ also lies on the circle. Given that the length of the chord $P R$ is 20 units,
find the length of the shortest distance from $A$ to the chord $P R$. Give your answer as a surd in its simplest form.
Find the size of angle $A R Q$, giving your answer to the nearest 0.1 of a degree.

Edexcel Paper 2 Specimen Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-22_554_862_260_603} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the curve $C$ with parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { t + 1 } , \quad t > - \frac { 2 } { 3 }$$

State the domain of values of $x$ for the curve $C$. The finite region $R$, shown shaded in Figure 4, is bounded by the curve $C$, the line with equation $x = \ln 2$, the $x$-axis and the line with equation $x = \ln 4$
Use calculus to show that the area of $R$ is $\ln \left( \frac { 3 } { 2 } \right)$.

Edexcel Paper 2 Specimen Q11

The second, third and fourth terms of an arithmetic sequence are $2 k , 5 k - 10$ and $7 k - 14$ respectively, where $k$ is a constant.

Show that the sum of the first $n$ terms of the sequence is a square number.

Edexcel Paper 2 Specimen Q12

A curve $C$ is given by the equation

$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point $P$ lies on $C$.
The tangent to $C$ at the point $P$ is parallel to the $x$-axis.
Find the exact coordinates of all possible points $P$, justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel Paper 2 Specimen Q13

13. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for $0 \leqslant \theta < 180 ^ { \circ }$, $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel Paper 2 Specimen Q14

(i) Kayden claims that

$$3 ^ { x } \geqslant 2 ^ { x }$$ Determine whether Kayden's claim is always true, sometimes true or never true, justifying your answer.
(ii) Prove that $\sqrt { 3 }$ is an irrational number.

OCR MEI Paper 2 2018 June Q1

1 Show that $\sqrt { 27 } + \sqrt { 192 } = a \sqrt { b }$, where $a$ and $b$ are prime numbers to be determined.

OCR MEI Paper 2 2018 June Q2

2 Solve the inequality $| 2 x + 1 | < 5$.

OCR MEI Paper 2 2018 June Q3

3 The probability that Chipping FC win a league football match is $\mathrm { P } ( W ) = 0.4$.

Calculate the probability that Chipping FC fail to win each of their next two league football matches. The probability that Chipping FC lose a league football match is $\mathrm { P } ( L ) = 0.3$.
Explain why $\mathrm { P } ( W ) + \mathrm { P } ( L ) \neq 1$.

OCR MEI Paper 2 2018 June Q4

4 A survey of the number of cars per household in a certain village generated the data in Fig. 4. \begin{table}[h]

Number of cars	0	1	2	3	4
Number of households	8	22	31	27	7

\captionsetup{labelformat=empty} \caption{Fig. 4}

\end{table}

Calculate the mean number of cars per household.
Calculate the standard deviation of the number of cars per household.

OCR MEI Paper 2 2018 June Q5

5

(A) Sketch the graph of $y = 3 ^ { x }$.
(B) Give the coordinates of any intercepts. The curve $y = \mathrm { f } ( x )$ is the reflection of the curve $y = 3 ^ { x }$ in the line $y = x$.
Find $\mathrm { f } ( x )$.

OCR MEI Paper 2 2018 June Q6

6

Express $7 \cos x - 24 \sin x$ in the form $R \cos ( x + \alpha )$, where $0 < \alpha < \frac { \pi } { 2 }$.
Write down the range of the function $$f ( x ) = 12 + 7 \cos x - 24 \sin x , \quad 0 \leqslant x \leqslant 2 \pi .$$

OCR MEI Paper 2 2018 June Q7

7 Find $\int \left( 4 \sqrt { x } - \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x$. Answer all the questions
Section B (79 marks)

OCR MEI Paper 2 2018 June Q8

8 Every morning before breakfast Laura and Mike play a game of chess. The probability that Laura wins is 0.7 . The outcome of any particular game is independent of the outcome of other games. Calculate the probability that, in the next 20 games,

Laura wins exactly 14 games,
Laura wins at least 14 games.

OCR MEI Paper 2 2018 June Q9

9 At the end of each school term at North End College all the science classes in year 10 are given a test. The marks out of 100 achieved by members of set 1 are shown in Fig. 9. \begin{table}[h]

3	5
4	0	9
5	2	3	6
6	0	1	3	5	6
7	0	1	2	5	6	8	9	9
8	3	4	6	6	8	8	9
9	5	5	5	6	7

\captionsetup{labelformat=empty} \caption{Fig. 9}

\end{table} Key $5 \quad$ 2 represents a mark of 52

Describe the shape of the distribution.
The teacher for set 1 claimed that a typical student in his class achieved a mark of 95. How did he justify this statement?
Another teacher said that the average mark in set 1 is 76 . How did she justify this statement? Benson's mark in the test is 35 . If the mark achieved by any student is an outlier in the lower tail of the distribution, the student is moved down to set 2 .
Determine whether Benson is moved down to set 2 .

OCR MEI Paper 2 2018 June Q10

10 The screenshot in Fig. 10 shows the probability distribution for the continuous random variable $X$, where $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-06_515_1009_338_529} \captionsetup{labelformat=empty} \caption{Fig. 10}

\end{figure} The area of each of the unshaded regions under the curve is 0.025 . The lower boundary of the shaded region is at 16.452 and the upper boundary of the shaded region is at 21.548 .

Calculate the value of $\mu$.
Calculate the value of $\sigma ^ { 2 }$.
$Y$ is the random variable given by $Y = 4 X + 5$.
(A) Write down the distribution of $Y$.
(B) Find $\mathrm { P } ( \mathrm { Y } > 90 )$.

OCR MEI Paper 2 2018 June Q11

11 The discrete random variable $X$ takes the values $0,1,2,3,4$ and 5 with probabilities given by the formula $$\mathrm { P } ( X = x ) = k ( x + 1 ) ( 6 - x ) .$$

Find the value of $k$. In one half-term Ben attends school on 40 days. The probability distribution above is used to model $X$, the number of lessons per day in which Ben receives a gold star for excellent work.
Find the probability that Ben receives no gold stars on each of the first 3 days of the half-term and two gold stars on each of the next 2 days.
Find the expected number of days in the half-term on which Ben receives no gold stars.

OCR MEI Paper 2 2018 June Q13

13 Each weekday Keira drives to work with her son Kaito. She always sets off at 8.00 a.m. She models her journey time, $x$ minutes, by the distribution $X \sim \mathrm {~N} ( 15,4 )$. Over a long period of time she notes that her journey takes less than 14 minutes on $7 \%$ of the journeys, and takes more than 18 minutes on $31 \%$ of the journeys.

Investigate whether Keira's model is a good fit for the data. Kaito believes that Keira’s value for the variance is correct, but realises that the mean is not correct.
Find, correct to two significant figures, the value of the mean that Keira should use in a refined model which does fit the data. Keira buys a new car. After driving to work in it each day for several weeks, she randomly selects the journey times for $n$ of these days. Her mean journey time for these $n$ days is 16 minutes. Using the refined model she conducts a hypothesis test to see if her mean journey time has changed, and finds that the result is significant at the $5 \%$ level.
Determine the smallest possible value of $n$.

OCR MEI Paper 2 2018 June Q14

14 The pre-release material includes data on unemployment rates in different countries. A sample from this material has been taken. All the countries in the sample are in Europe. The data have been grouped and are shown in Fig 14.1. \begin{table}[h]

Unemployment rate	$0 -$	$5 -$	$10 -$	$15 -$	$20 -$	$35 - 50$
Frequency	15	21	5	5	2	2

\captionsetup{labelformat=empty} \caption{Fig. 14.1}

\end{table} A cumulative frequency curve has been generated for the sample data using a spreadsheet. This is shown in Fig. 14.2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-08_639_1081_808_466} \captionsetup{labelformat=empty} \caption{Fig. 14.2}

\end{figure} Hodge used Fig. 14.2 to estimate the median unemployment rate in Europe. He obtained the answer 5.0. The correct value for this sample is 6.9.

(A) There is a systematic error in the diagram.
- Identify this error.
- State how this error affects Hodge’s estimate.
  (B) There is another factor which has affected Hodge’s estimate.
- Identify this factor.
- State how this factor affects Hodge’s estimate.
- Use your knowledge of the pre-release material to give another reason why any estimation of the median unemployment rate in Europe may be unreliable.
- Use your knowledge of the pre-release material to explain why it is very unlikely that the sample has been randomly selected from the pre-release material.
The scatter diagram shown in Fig. 14.3 shows the unemployment rate and life expectancy at birth for the 47 countries in the sample for which this information is available. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Scatter diagram to show life expectancy at birth against unemployment rate} \includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-09_627_1281_456_367}
\end{figure} Fig. 14.3 The product moment correlation coefficient for the 47 items in the sample is - 0.2607 .
The $p$-value associated with $r = - 0.2607$ and $n = 47$ is 0.0383 .
Does this information suggest that there is an association between unemployment rate and life expectancy at birth in countries in Europe? Hodge uses the spreadsheet tools to obtain the equation of a line of best fit for this data.
The unemployment rate in Kosovo is 35.3 , but there is no data available on life expectancy. Is it reasonable to use Hodge’s line of best fit to estimate life expectancy at birth in Kosovo?

OCR MEI Paper 2 2018 June Q15

15 You must show detailed reasoning in this question. The equation of a curve is $$y ^ { 3 } - x y + 4 \sqrt { x } = 4 .$$ Find the gradient of the curve at each of the points where $y = 1$.

OCR MEI Paper 2 2018 June Q16

16 In the first year of a course, an A-level student, Aaishah, has a mathematics test each week. The night before each test she revises for $t$ hours. Over the course of the year she realises that her percentage mark for a test, $p$, may be modelled by the following formula, where $A , B$ and $C$ are constants. $$p = A - B ( t - C ) ^ { 2 }$$

Aaishah finds that, however much she revises, her maximum mark is achieved when she does 2 hours revision. This maximum mark is 62 .
Aaishah had a mark of 22 when she didn't spend any time revising.
1. Find the values of $A , B$ and $C$.
2. According to the model, if Aaishah revises for 45 minutes on the night before the test, what mark will she achieve?
3. What is the maximum amount of time that Aaishah could have spent revising for the model to work?

In an attempt to improve her marks Aaishah now works through problems for a total of $t$ hours over the three nights before the test. After taking a number of tests, she proposes the following new formula for $p$. $$p = 22 + 68 \left( 1 - \mathrm { e } ^ { - 0.8 t } \right)$$ For the next three tests she recorded the data in Fig. 16. \begin{table}[h]

$t$	1	3	5
$p$	59	84	89

\captionsetup{labelformat=empty} \caption{Fig. 16}