Questions - OCR - A-Level Maths

Find the vector product $\overrightarrow { A B } \times \overrightarrow { A C }$.
Hence find the equation of the plane $A B C$ in the form $a x + b y + c z = d$.

OCR Further Pure Core 1 2018 December Q2

2 The equation of the curve shown on the graph is, in polar coordinates, $r = 3 \sin 2 \theta$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
\includegraphics[max width=\textwidth, alt={}, center]{8315a796-0e7d-464f-8604-9fe3ab7af359-2_470_657_913_319}

The greatest value of $r$ on the curve occurs at the point $P$.
1. Show that $\theta = \frac { 1 } { 4 } \pi$ at the point $P$.
2. Find the value of $r$ at the point $P$.
3. Mark the point $P$ on the copy of the graph in the Printed Answer Booklet.
In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve.

OCR Further Pure Core 1 2018 December Q3

3 You are given that $\mathrm { f } ( x ) = \ln ( 2 + x )$.

Determine the exact value of $\mathrm { f } ^ { \prime } ( 0 )$.
Show that $\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }$.
Hence write down the first three terms of the Maclaurin series for $\mathrm { f } ( x )$.

OCR Further Pure Core 1 2018 December Q4

4 In this question you must show detailed reasoning.
You are given that $z = \sqrt { 3 } + \mathrm { i }$.
$n$ is the smallest positive whole number such that $z ^ { n }$ is a positive whole number.

Determine the value of $n$.
Find the value of $z ^ { n }$.

OCR Further Pure Core 1 2018 December Q5

5 You are given that $\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1
2 & 5 & 2
3 & - 2 & - 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1
- 8 & 4 & 0
19 & - 8 & - 1 \end{array} \right)$.

Find $\mathbf { A B }$.
Hence write down $\mathbf { A } ^ { - 1 }$.
You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0
2 x + 5 y + 2 z = 1
3 x - 2 y - z = 4 \end{array}$$
1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
2. Find this unique solution.

OCR Further Pure Core 1 2018 December Q6

6 Prove by induction that, for all positive integers $n , 7 ^ { n } + 3 ^ { n - 1 }$ is a multiple of 4 .

OCR Further Pure Core 1 2018 December Q7

Determine an expression for $\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$ giving your answer in the form $\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )$.
Find the value of $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$.

OCR Further Pure Core 1 2018 December Q8

Given that $u = \tanh x$, use the definition of $\tanh x$ in terms of exponentials to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
Solve the equation $4 \tanh ^ { 2 } x + \tanh x - 3 = 0$, giving the solution in the form $a \ln b$ where $a$ and $b$ are rational numbers to be determined.
Explain why the equation in part (b) has only one root.

OCR Further Pure Core 1 2018 December Q9

9 In this question you must show detailed reasoning. Find $\int _ { - 1 } ^ { 11 } \frac { 1 } { \sqrt { x ^ { 2 } + 6 x + 13 } } \mathrm {~d} x$ giving your answer in the form $\ln ( p + q \sqrt { 2 } )$ where $p$ and $q$ are integers to be determined.

OCR Further Pure Core 1 2018 December Q10

10 In a predator-prey environment the population, at time $t$ years, of predators is $x$ and prey is $y$. The populations of predators and prey are measured in hundreds. The populations are modelled by the following simultaneous differential equations. $$\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x$$

Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x$.
1. Find the general solution for $x$.
2. Find the equivalent general solution for $y$. Initially there are 100 predators and 300 prey.
Find the particular solutions for $x$ and $y$.
Determine whether the model predicts that the predators will die out before the prey.

OCR Further Pure Core 2 2018 December Q1

The Argand diagram below shows the two points which represent two complex numbers, $z _ { 1 }$ and $z _ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{e792797e-6d20-4fc3-9733-43db10f764d7-2_371_689_429_360} On the copy of the diagram in the Printed Answer Booklet
- draw an appropriate shape to illustrate the geometrical effect of adding $z _ { 1 }$ and $z _ { 2 }$,
- indicate with a cross $( \times )$ the location of the point representing the complex number $z _ { 1 } + z _ { 2 }$.
- You are given that $\arg z _ { 3 } = \frac { 1 } { 4 } \pi$ and $\arg z _ { 4 } = \frac { 3 } { 8 } \pi$.
In each part, sketch and label the points representing the numbers $z _ { 3 } , z _ { 4 }$ and $z _ { 3 } z _ { 4 }$ on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
(i) $\left| z _ { 3 } \right| = 1.5$ and $\left| z _ { 4 } \right| = 1.2$
(ii) $\left| z _ { 3 } \right| = 0.7$ and $\left| z _ { 4 } \right| = 0.5$

OCR Further Pure Core 2 2018 December Q2

2 In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the $x$-axis. $\mathbf { A }$ is the matrix which represents S .

Write down $\mathbf { A }$.
By considering the transformation represented by $\mathbf { A } ^ { - 1 }$, determine the matrix $\mathbf { A } ^ { - 1 }$. Matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { c c } 0 & - 1
- 1 & 0 \end{array} \right)$. T is the transformation represented by $\mathbf { B }$.
Describe T.
Determine the matrix which represents the transformation S followed by T .
Demonstrate, by direct calculation, that $( \mathbf { B A } ) ^ { - 1 } = \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 }$.

OCR Further Pure Core 2 2018 December Q3

3 In this question you must show detailed reasoning. Solve the equation $2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0$ giving each answer in the form $\ln ( p + q \sqrt { r } )$ where $p$ and $q$ are rational numbers, and $r$ is an integer, whose values are to be determined.

OCR Further Pure Core 2 2018 December Q4

4 You are given that the matrix $\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0
0 & 0 & 1 \end{array} \right)$, where $a$ is a positive constant, represents the transformation R which is a reflection in 3-D.

State the plane of reflection of R .
Determine the value of $a$.
With reference to R explain why $\mathbf { A } ^ { 2 } = \mathbf { I }$, the $3 \times 3$ identity matrix.

OCR Further Pure Core 2 2018 December Q5

Find the shortest distance between the point ( $- 6,4$ ) and the line $y = - 0.75 x + 7$. Two lines, $l _ { 1 }$ and $l _ { 2 }$, are given by
$l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 4
3
- 2 \end{array} \right) + \lambda \left( \begin{array} { c } 2
1
- 4 \end{array} \right)$ and $l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 11
- 1
5 \end{array} \right) + \mu \left( \begin{array} { c } 3
- 1
1 \end{array} \right)$.
Find the shortest distance between $l _ { 1 }$ and $l _ { 2 }$.
Hence determine the geometrical arrangement of $l _ { 1 }$ and $l _ { 2 }$.

OCR Further Pure Core 2 2018 December Q6

6 Three matrices, $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$, are given by $\mathbf { A } = \left( \begin{array} { c c } 1 & 2
a & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c } 2 & - 1
4 & 1 \end{array} \right)$ and $\mathbf { C } = \left( \begin{array} { c c } 5 & 0
- 2 & 2 \end{array} \right)$ where $a$ is a

Using $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$ in that order demonstrate explicitly the associativity property of matrix multiplication.
Use $\mathbf { A }$ and $\mathbf { C }$ to disprove by counterexample the proposition 'Matrix multiplication is commutative'. For a certain value of $a , \mathbf { A } \binom { x } { y } = 3 \binom { x } { y }$.
Find
- $y$ in terms of $x$,
- the value of $a$.
  $7 C$ is the locus of numbers, $z$, for which $\operatorname { Im } \left( \frac { z + 7 \mathrm { i } } { z - 24 } \right) = \frac { 1 } { 4 }$.
  By writing $z = x + \mathrm { i } y$ give a complete description of the shape of $C$ on an Argand diagram.

OCR Further Pure Core 2 2018 December Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{e792797e-6d20-4fc3-9733-43db10f764d7-4_677_1182_587_440} The figure shows part of the graph of $y = ( x - 3 ) \sqrt { \ln x }$. The portion of the graph below the $x$-axis is rotated by $2 \pi$ radians around the $x$-axis to form a solid of revolution, $S$. Determine the exact volume of $S$.

OCR Further Pure Core 2 2018 December Q9

By using Euler's formula show that $\cosh ( \mathrm { i } z ) = \cos z$.
Hence, find, in logarithmic form, a root of the equation $\cos z = 2$. [You may assume that $\cos z = 2$ has complex roots.]

OCR Further Pure Core 2 2018 December Q10

10 A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, $t$ seconds, is measured by the angle at the hinge, $\theta$, which the plane of the door makes with the plane of the equilibrium position. See the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e792797e-6d20-4fc3-9733-43db10f764d7-5_367_1116_625_246} In an initial model of the motion of a certain swing door it is suggested that $\theta$ satisfies the following differential equation.
$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$

1. Write down the general solution to (\textit{).
2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic. In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes
  $4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm { d } t } + 25 \theta = 0 \quad ( \dagger )$
  where $\lambda$ is a positive constant.
In the case where $\lambda = 16$ the door is held open at an angle of 0.9 radians and then released from rest at time $t = 0$.
1. Find, in a real form, the general solution of ( $\dagger$ ).
2. Find the particular solution of ( $\dagger$ ).
3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( $\dagger$ ) improves the model.
Find the value of $\lambda$ for which the door is critically damped.

OCR Further Statistics 2018 December Q1

1 The performance of a piece of music is being recorded. The piece consists of three sections, $A , B$ and $C$. The times, in seconds, taken to perform the three sections are normally distributed random variables with the following means and standard deviations.

Section	Mean	Standard deviation
$A$	264	13
$B$	173	9
$C$	264	13

Assume first that the times for the three sections are independent. Find the probability that the total length of the performance is greater than 720.0 seconds.
In fact sections $A$ and $C$ are musically identical, and the recording is made by using a single performance of section $A$ twice, together with a performance of section $B$. In this case find the probability that the total length of the performance is greater than 720.0 seconds.

OCR Further Statistics 2018 December Q2

2 In a fairground game a competitor scores $0,1,2$ or 3 with probabilities given in the following table, where $a$ and $b$ are constants.

Score	0	1	2	3
Probability	$a$	$b$	$b$	$b$

The competitor's expected score is 0.9 .

Show that $b = 0.15$.
Find the variance of the score.
The competitor has to pay $\pounds 2.50$ to take part, and wins a prize of $\pounds 2 X$, where $X$ is the score achieved. Find the expectation of the competitor's loss.

OCR Further Statistics 2018 December Q3

Alex places 20 black counters and 8 white counters into a bag. She removes 8 counters at random without replacement. Find the probability that the bag now contains exactly 5 white counters.
Bill arranges 8 blue counters and 4 green counters in a random order in a straight line. Find the probability that exactly three of the green counters are next to one another.

OCR Further Statistics 2018 December Q4

4 Leyla investigates the number of shoppers who visit a shop between 10.30 am and 11 am on Saturday mornings. She makes the following assumptions.

Shoppers visit the shop independently of one another.
The average rate at which shoppers visit the shop between these times is constant.
1. State an appropriate distribution with which Leyla could model the number of shoppers who visit the shop between these times.

Leyla uses this distribution, with mean 14, as her model.

Calculate the probability that, between 10.35 am and 10.50 am on a randomly chosen Saturday, at least 10 shoppers visit the shop. Leyla chooses 25 Saturdays at random.

Find the expected number of Saturdays, out of 25, on which there are no visitors to the shop between 10.35 am and 10.50 am .

In fact on 5 of these Saturdays there were no visitors to the shop between 10.35 am and 10.50 am . Use this fact to comment briefly on the validity of the model that Leyla has used.

OCR Further Statistics 2018 December Q5

5 The birth rate, $x$ per thousand members of the population, and the life expectancy at birth, $y$ years, in 14 randomly selected African countries are given in the table.

Country	$x$	$y$	Country	$x$	$y$
Benin	4.8	59.2	Mozambique	5.4	54.63
Cameroon	4.7	54.87	Nigeria	5.7	52.29
Congo	4.9	61.42	Senegal	5.1	65.81
Gambia	5.7	59.83	Somalia	6.5	54.88
Liberia	4.7	60.25	Sudan	4.4	63.08
Malawi	5.1	60.97	Uganda	5.8	57.25
Mauretania	4.6	62.77	Zambia	5.4	58.75

$n = 14 , \sum x = 72.8 , \sum y = 826 , \sum x ^ { 2 } = 392.96 , \sum y ^ { 2 } = 48924.54 , \sum x y = 4279.16$

Calculate Pearson's product-moment correlation coefficient $r$ for the data.
State what would be the effect on the value of $r$ if the birth rate were given per hundred and not per thousand.
Explain what the sign of $r$ tells you about the relationship between life expectancy and birth rate for these countries.
Test at the $5 \%$ significance level whether there is correlation between birth rate and life expectancy at birth in African countries.
A researcher wants to estimate the life expectancy at birth in Zimbabwe, where the birth rate is 3.9 per thousand. Explain whether a reliable estimate could be obtained using the regression line of $y$ on $x$ for the given data.

OCR Further Statistics 2018 December Q6

6 The reaction times, in milliseconds, of all adult males in a standard experiment have a symmetrical distribution with mean and median both equal to 700 and standard deviation 125. The reaction times of a random sample of 6 international athletes are measured and the results are as follows:
$\begin{array} { l l l l l l } 702 & 631 & 540 & 714 & 575 & 480 \end{array}$ It is required to test whether international athletes have a mean reaction time which is less than 700.

Assume first that the reaction times of international athletes have the distribution $\mathrm { N } \left( \mu , 125 ^ { 2 } \right)$. Test at the $5 \%$ significance level whether $\mu < 700$.
Now assume only that the distribution of the data is symmetrical, but not necessarily normal.
1. State with a reason why a Wilcoxon test is preferable to a sign test.
2. Use an appropriate Wilcoxon test at the $5 \%$ significance level to test whether the median reaction time of international athletes is less than 700 .
Explain why the significance tests in part (a) and part (b)(ii) could produce different results.

Score	0	1	2	3
Probability	\(a\)	\(b\)	\(b\)	\(b\)

Country	\(x\)	\(y\)	Country	\(x\)	\(y\)
Benin	4.8	59.2	Mozambique	5.4	54.63
Cameroon	4.7	54.87	Nigeria	5.7	52.29
Congo	4.9	61.42	Senegal	5.1	65.81
Gambia	5.7	59.83	Somalia	6.5	54.88
Liberia	4.7	60.25	Sudan	4.4	63.08
Malawi	5.1	60.97	Uganda	5.8	57.25
Mauretania	4.6	62.77	Zambia	5.4	58.75

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OCR Further Pure Core 1 2018 December Q1

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