Questions - A-Level Maths

A company sells seeds and claims that $55 \%$ of its pea seeds germinate.
1. Write down a reason why the company should not justify their claim by testing all the pea seeds they produce.
A random selection of the pea seeds is planted in 10 trays with 24 seeds in each tray.
Assuming that the company's claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.
Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution. A random sample of 240 pea seeds was planted and 150 of these seeds germinated.
Assuming that the company's claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.
Using your answer to part (d), comment on whether or not the proportion of the company's pea seeds that germinate is different from the company's claim of $55 \%$

Edexcel Paper 3 Specimen Q6

6 marks Moderate -0.3

6. At time $t$ seconds, where $t \geqslant 0$, a particle $P$ moves so that its acceleration $\mathbf { a } \mathrm { m } \mathrm { s } ^ { - 2 }$ is given by $$\mathbf { a } = 5 t \mathbf { i } - 15 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$ When $t = 0$, the velocity of $P$ is $20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Find the speed of $P$ when $t = 4$

Edexcel Paper 3 Specimen Q7

8 marks Standard +0.3

A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$.

A particle of mass $m$ is placed on the plane and then projected up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is $\mu$.
The particle moves up the plane with a constant deceleration of $\frac { 4 } { 5 } \mathrm {~g}$.

Find the value of $\mu$. The particle comes to rest at the point $A$ on the plane.
Determine whether the particle will remain at $A$, carefully justifying your answer.

Edexcel Paper 3 Specimen Q8

10 marks Moderate -0.3

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively]

A radio controlled model boat is placed on the surface of a large pond.
The boat is modelled as a particle.
At time $t = 0$, the boat is at the fixed point $O$ and is moving due north with speed $0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Relative to $O$, the position vector of the boat at time $t$ seconds is $\mathbf { r }$ metres.
At time $t = 15$, the velocity of the boat is $( 10.5 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
The acceleration of the boat is constant.

Show that the acceleration of the boat is $( 0.7 \mathbf { i } - 0.1 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$.
Find $\mathbf { r }$ in terms of $t$.
Find the value of $t$ when the boat is north-east of $O$.
Find the value of $t$ when the boat is moving in a north-east direction.

Edexcel Paper 3 Specimen Q9

13 marks Challenging +1.2

9. Figure 1 A uniform ladder $A B$, of length $2 a$ and weight $W$, has its end $A$ on rough horizontal ground. The coefficient of friction between the ladder and the ground is $\frac { 1 } { 4 }$.
The end $B$ of the ladder is resting against a smooth vertical wall, as shown in Figure 1.
A builder of weight $7 W$ stands at the top of the ladder.
To stop the ladder from slipping, the builder's assistant applies a horizontal force of magnitude $P$ to the ladder at $A$, towards the wall.
The force acts in a direction which is perpendicular to the wall.
The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle $\alpha$ with the horizontal ground, where $\tan \alpha = \frac { 5 } { 2 }$.
The builder is modelled as a particle and the ladder is modelled as a uniform rod.

Show that the reaction of the wall on the ladder at $B$ has magnitude $3 W$.
Find, in terms of $W$, the range of possible values of $P$ for which the ladder remains in equilibrium. Often in practice, the builder's assistant will simply stand on the bottom of the ladder.
Explain briefly how this helps to stop the ladder from slipping.

Edexcel Paper 3 Specimen Q10

13 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e678bf51-6dca-4ad7-808b-dfa31b04dc63-22_719_1333_246_365} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A boy throws a stone with speed $U \mathrm {~ms} ^ { - 1 }$ from a point $O$ at the top of a vertical cliff. The point $O$ is 18 m above sea level.
The stone is thrown at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$.
The stone hits the sea at the point $S$ which is at a horizontal distance of 36 m from the foot of the cliff, as shown in Figure 2.
The stone is modelled as a particle moving freely under gravity with $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
Find

the value of $U$,
the speed of the stone when it is 10.8 m above sea level, giving your answer to 2 significant figures.
Suggest two improvements that could be made to the model.

Edexcel Paper 3 Specimen Q1

14 marks Standard +0.3

Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.

Mass $w ( \mathrm {~g} )$	Midpoint $y ( \mathrm {~g} )$	Frequency f
$240 \leq w < 245$	242.5	8
$245 \leq w < 248$	246.5	15
$248 \leq w < 252$	250.0	35
$252 \leq w < 255$	253.5	23
$255 \leq w < 260$	257.5	9

$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$ A histogram is drawn and the class $245 \leq w < 248$ is represented by a rectangle of width 1.2 cm and height 10 cm .

Calculate the width and the height of the rectangle representing the class $255 \leq w < 260$.
Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place. The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
(You may assume that the mass of the contents of a packet is normally distributed.)
Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
(Total 14 marks)

Edexcel Paper 3 Specimen Q2

7 marks Moderate -0.3

2. A researcher believes that there is a linear relationship between daily mean temperature and daily total rainfall. The 7 places in the northern hemisphere from the large data set are used. The mean of the daily mean temperatures, $t ^ { \circ } \mathrm { C }$, and the mean of the daily total rainfall, $s \mathrm {~mm}$, for the month of July in 2015 are shown on the scatter diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-03_844_1339_497_372}

With reference to the scatter diagram, explain why a linear regression model may not be suitable for the relationship between $t$ and s .
(1) The researcher calculated the product moment correlation coefficient for the 7 places and obtained $r = 0.658$.
Stating your hypotheses clearly, test at the $10 \%$ level of significance, whether or not the product moment correlation coefficient for the population is greater than zero.
(3)
Using your knowledge of the large data set, suggest the names of the 2 places labelled $G$ and $H$.
(1)
Using your knowledge from the large data set, and with reference to the locations of the two places labelled $G$ and $H$, give a reason why these places have the highest temperatures in July.
(2)
Suggest how you could make better use of the large data set to investigate the relationship between daily mean temperature and daily total rainfall.
(1)
(Total 7 marks)

Edexcel Paper 3 Specimen Q3

10 marks Standard +0.8

3. For a particular type of bulb, $36 \%$ grow into plants with blue flowers and the remainder grow into plants with white flowers. Bulbs are sold in mixed bags of 40 Russell selects a random sample of 5 bags of bulbs.

Find the probability that fewer than 2 of these bags will contain more bulbs that grow into plants with blue flowers than grow into plants with white flowers.
(4) Maggie takes a random sample of $n$ bulbs.
Using a normal approximation, the probability that more than 244 of these $n$ bulbs will grow into blue flowers is 0.0521 to 4 decimal places.
Find the value of $n$.
(6)
(Total 10 marks)

Edexcel Paper 3 Specimen Q4

11 marks Standard +0.3

4. The Venn diagram shows the probabilities of students' lunch boxes containing a drink, sandwiches and a chocolate bar.
\includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-05_655_899_392_484}
$D$ is the event that a lunch box contains a drink, $S$ is the event that a lunch box contains sandwiches,
$C$ is the event that a lunch box contains a chocolate bar, $u , v$ and $w$ are probabilities.

Write down $\mathrm { P } \left( S \cap D ^ { \prime } \right)$. One day, 80 students each bring in a lunch box.
Given that all 80 lunch boxes contain sandwiches and a drink,
estimate how many of these 80 lunch boxes will contain a chocolate bar. Given that the events $S$ and $C$ are independent and that $\mathrm { P } ( D \mid C ) = \frac { 14 } { 15 }$,
calculate the value of $u$, the value of $v$ and the value of $w$.
(7)
(Total 11 marks)

Edexcel Paper 3 Specimen Q5

8 marks Moderate -0.8

5. The lifetimes of batteries sold by company $X$ are normally distributed, with mean 150 hours and standard deviation 25 hours. A box contains 12 batteries from company $X$.

Find the expected number of these batteries that have a lifetime of more than 160 hours. The lifetimes of batteries sold by company $Y$ are normally distributed, with mean 160 hours and $80 \%$ of these batteries have a lifetime of less than 180 hours.
Find the standard deviation of the lifetimes of batteries from company $Y$. Both companies sell their batteries for the same price.
State which company you would recommend. Give reasons for your answer.

OCR MEI Paper 3 2019 June Q1

6 marks Moderate -0.8

1 The function $\mathrm { f } ( x )$ is defined for all real $x$ by
$f ( x ) = 3 x - 2$.

Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
Sketch the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ on the same diagram.
Find the set of values of $x$ for which $\mathrm { f } ( x ) > \mathrm { f } ^ { - 1 } ( x )$.

OCR MEI Paper 3 2019 June Q2

5 marks Easy -1.2

Find the transformation which maps the curve $y = x ^ { 2 }$ to the curve $y = x ^ { 2 } + 8 x - 7$.
Write down the coordinates of the turning point of $y = x ^ { 2 } + 8 x - 7$.

OCR MEI Paper 3 2019 June Q3

6 marks Moderate -0.8

Express $\frac { 1 } { ( x + 2 ) ( x + 3 ) }$ in partial fractions.
Find $\int \frac { 1 } { ( x + 2 ) ( x + 3 ) } \mathrm { d } x$ in the form $\ln ( \mathrm { f } ( x ) ) + c$, where $c$ is the constant of integration and $\mathrm { f } ( x )$ is a function to be determined.

OCR MEI Paper 3 2019 June Q4

3 marks Standard +0.3

4 In this question you must show detailed reasoning.
Show that $\frac { 1 } { \sqrt { 10 } + \sqrt { 11 } } + \frac { 1 } { \sqrt { 11 } + \sqrt { 12 } } + \frac { 1 } { \sqrt { 12 } + \sqrt { 13 } } = \frac { 3 } { \sqrt { 10 } + \sqrt { 13 } }$.

OCR MEI Paper 3 2019 June Q5

3 marks Challenging +1.2

5 A student's attempt to prove by contradiction that there is no largest prime number is shown below.
If there is a largest prime, list all the primes.
Multiply all the primes and add 1.
The new number is not divisible by any of the primes in the list and so it must be a new prime. The proof is incorrect and incomplete.
Write a correct version of the proof.

OCR MEI Paper 3 2019 June Q6

7 marks Moderate -0.8

6 A circle has centre $C ( 10,4 )$. The $x$-axis is a tangent to the circle, as shown in Fig. 6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{99485c27-9ff8-4bdb-a7e6-49dfcaedc579-5_605_828_979_255} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure}

Find the equation of the circle.
Show that the line $y = x$ is not a tangent to the circle.
Write down the position vector of the midpoint of OC.

OCR MEI Paper 3 2019 June Q7

4 marks Standard +0.3

7 In this question you must show detailed reasoning.

Express $\ln 3 \times \ln 9 \times \ln 27$ in terms of $\ln 3$.
Hence show that $\ln 3 \times \ln 9 \times \ln 27 > 6$.

OCR MEI Paper 3 2019 June Q8

10 marks Standard +0.3

8 In this question you must show detailed reasoning. A is the point $( 1,0 ) , B$ is the point $( 1,1 )$ and $D$ is the point where the tangent to the curve $y = x ^ { 3 }$ at B crosses the $x$-axis, as shown in Fig. 8. The tangent meets the $y$-axis at E. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{99485c27-9ff8-4bdb-a7e6-49dfcaedc579-6_1154_832_450_242} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Find the area of triangle ODE.
Find the area of the region bounded by the curve $y = x ^ { 3 }$, the tangent at B and the $y$-axis.

OCR MEI Paper 3 2019 June Q9

6 marks Standard +0.8

9 In this question you must show detailed reasoning.
The curve $x y + y ^ { 2 } = 8$ is shown in Fig. 9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{99485c27-9ff8-4bdb-a7e6-49dfcaedc579-7_734_750_397_244} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure} Find the coordinates of the points on the curve at which the normal has gradient 2.

OCR MEI Paper 3 2019 June Q10

4 marks Moderate -0.3

10 Show that $\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$ is an increasing function for all values of $x$.

OCR MEI Paper 3 2019 June Q11

6 marks Standard +0.3

11 By using the substitution $u = 1 + \sqrt { x }$, find $\int \frac { x } { 1 + \sqrt { x } } \mathrm {~d} x$. Answer all the questions.

OCR MEI Paper 3 2019 June Q12

2 marks Moderate -0.5

12 Show that the equation of the line in Fig. C2 is $r y + h x = h r$, as given in line 24.

OCR MEI Paper 3 2019 June Q13

6 marks Standard +0.3

1. Show that the cross-sectional area in Fig. C3.2 is $\pi x ( 2 r - x )$.
2. Hence show that the cross-sectional area is $\frac { \pi r ^ { 2 } } { h ^ { 2 } } \left( h ^ { 2 } - y ^ { 2 } \right)$, as given in line 37 .
Verify that the formula $\frac { \pi r ^ { 2 } } { h ^ { 2 } } \left( h ^ { 2 } - y ^ { 2 } \right)$ for the cross-sectional area is also valid for
1. Fig. C3.1,
2. Fig. C3.3.

OCR MEI Paper 3 2019 June Q14

4 marks Challenging +1.2

Express $\lim _ { \delta y \rightarrow 0 } \sum _ { 0 } ^ { h } \left( h ^ { 2 } - y ^ { 2 } \right) \delta y$ as an integral.
Hence show that $V = \frac { 2 } { 3 } \pi r ^ { 2 } h$, as given in line 41 .

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Mass \(w ( \mathrm {~g} )\)	Midpoint \(y ( \mathrm {~g} )\)	Frequency f
\(240 \leq w < 245\)	242.5	8
\(245 \leq w < 248\)	246.5	15
\(248 \leq w < 252\)	250.0	35
\(252 \leq w < 255\)	253.5	23
\(255 \leq w < 260\)	257.5	9

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