Questions - OCR - A-Level Maths

OCR FP1 AS 2021 June Q1

3 marks Standard +0.3

1 In this question you must show detailed reasoning.
The cubic equation $2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$. By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$.

OCR FP1 AS 2021 June Q2

7 marks Moderate -0.8

2 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r c } 2 & 1 & 2 \\ 1 & - 1 & 1 \\ 2 & 2 & a \end{array} \right)$.

Show that $\operatorname { det } \mathbf { A } = 6 - 3 a$.
State the value of $a$ for which $\mathbf { A }$ is singular.
Given that $\mathbf { A }$ is non-singular find $\mathbf { A } ^ { - 1 }$ in terms of $a$.

OCR FP1 AS 2021 June Q3

7 marks Standard +0.3

3 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)$ where $t$ is a constant.

Show that $| \mathrm { A } | = | \mathrm { B } |$.
Verify that $| \mathrm { AB } | = | \mathrm { A } | | \mathrm { B } |$.
Given that $| \mathbf { A B } | = - 1$ explain what this means about the constant $t$. The $2 \times 2$ matrix $A$ represents a transformation $T$ which has the following properties.
The transformation $S$ is represented by the matrix $B$ where $B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)$.
(b) Find the equation of the line of invariant points of S .
(c) Show that any line of the form $y = x + c$ is an invariant line of S .

OCR FP1 AS 2021 June Q1

6 marks Moderate -0.3

1 In this question you must show detailed reasoning.
Use an algebraic method to find the square roots of $- 77 - 36 \mathrm { i }$.

OCR FP1 AS 2021 June Q2

13 marks Moderate -0.3

2 In this question you must show detailed reasoning.
The complex number $7 - 4 \mathrm { i }$ is denoted by $z$.

Giving your answers in the form $a + b \mathrm { i }$, where $a$ and $b$ are rational numbers, find the following.
1. $3 z - 4 z ^ { * }$
2. $( z + 1 - 3 \mathrm { i } ) ^ { 2 }$
3. $\frac { z + 1 } { z - 1 }$
Express $z$ in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
The complex number $\omega$ is such that $z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )$. Find the following.

OCR FP1 AS 2021 June Q3

7 marks Challenging +1.8

3 In this question you must show detailed reasoning.
The cubic equation $5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0$ has roots $\alpha , \beta$ and $\gamma$.
Find a cubic equation with integer coefficients whose roots are $\alpha + \beta , \beta + \gamma$ and $\gamma + \alpha$.

OCR FP1 AS 2021 June Q4

5 marks Standard +0.3

4 Prove that $n ! > 2 ^ { 2 n }$ for all integers $n \geqslant 9$.

OCR FP1 AS 2021 June Q2

6 marks Standard +0.3

2 You are given the system of equations $$\begin{array} { r } a ^ { 2 } x - 2 y = 1 \\ x + b ^ { 2 } y = 3 \end{array}$$ where $a$ and $b$ are real numbers.

Use a matrix method to find $x$ and $y$ in terms of $a$ and $b$.
Explain why the method used in part (a) works for all values of $a$ and $b$.

OCR FP1 AS 2021 June Q3

6 marks Standard +0.3

3 The equations of two intersecting lines are $\mathbf { r } = \left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) \quad \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right) + \mu \left( \begin{array} { c } - 3 \\ 1 \\ - 1 \end{array} \right)$ where $a$ is a constant.

Find a vector, $\mathbf { b }$, which is perpendicular to both lines.
Show that b. $\left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) =$ b. $\left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right)$.
Hence, or otherwise, find the value of $a$.

OCR FP1 AS 2021 June Q4

8 marks Standard +0.8

4 Two loci, $C _ { 1 }$ and $C _ { 2 }$, are defined by $$\begin{aligned} & C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\} \\ & C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\} \end{aligned}$$ where $d$ is a real number.

Find, in terms of $d$, the complex number which is represented on an Argand diagram by the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.
[0pt] [You may assume that $C _ { 1 } \cap C _ { 2 } \neq \varnothing$.]
Explain why the solution found in part (a) is not valid when $d = 3$.

OCR FS1 AS 2021 June Q1

8 marks Moderate -0.8

1 The probability distribution for the discrete random variable $W$ is given in the table.

$w$	1	2	3	4
$\mathrm { P } ( W = w )$	0.25	0.36	$x$	$x ^ { 2 }$

Show that $\operatorname { Var } ( W ) = 0.8571$.
Find $\operatorname { Var } ( 3 W + 6 )$.

OCR FS1 AS 2021 June Q2

6 marks Moderate -0.8

2 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by $X$.

State a further assumption needed for $X$ to be well modelled by a Poisson distribution. Assume now that $X$ can be well modelled by the distribution $\operatorname { Po } ( 0.7 )$.
Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution $\mathrm { Po } ( 1.6 )$.

Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4 . Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings $X$, to take a mathematics aptitude test, with scores $Y$. The results are summarised as follows. $$n = 20 , \Sigma x = 3600 , \Sigma x ^ { 2 } = 660500 , \Sigma y = 1440 , \Sigma y ^ { 2 } = 105280 , \Sigma x y = 260990$$

Calculate the value of Pearson's product-moment correlation coefficient $r$.
State an assumption needed to be able to carry out a significance test on the value of $r$.
Assume now that the assumption in part (b) is valid. Test at the $5 \%$ significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
Calculate the value of Pearson's product-moment correlation coefficient $r$.
State an assumption needed to be able to carry out a significance test on the value of $r$.
Assume now that the assumption in part (b) is valid. Test at the $5 \%$ significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.

There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\mathrm { ELO } \text { rating } = 8 \times \mathrm { BCF } \text { rating } + 650 .$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion. An environmentalist measures the mean concentration, $c$ milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, $m$ pounds, of fish of a certain species found in those rivers. The results are given in the table.

Question

Answer

Marks

AO

Guidance

1

(a)

\(\begin{aligned}

0.25 + 0.36 + x + x ^ { 2 } = 1

x ^ { 2 } + x - 0.39 = 0

x = 0.3 \text { (or } - 1.3 \text { ) }

x \text { cannot be negative }

\mathrm { E } ( W ) = 2.23

\mathrm { E } \left( W ^ { 2 } \right) = \Sigma w ^ { 2 } \mathrm { p } ( w ) \quad [ = 5.83 ]

\text { Subtract } [ \mathrm { E } ( W ) ] ^ { 2 } \text { to get } \mathbf { 0 . 8 5 7 1 } \end{aligned}\)

\(\begin{gathered} \text { M1 }

\text { A1 }

\text { B1ft }

\text { B1 }

\text { M1 }

\text { A1 }

{ [ 7 ] } \end{gathered}\)

3.1a

1.1b

2.3

1.1b

1.1

2.1

Equation using $\Sigma p = 1$

Correct simplified quadratic Correctly obtain $x = 0.3$

Explicitly reject other solution

2.23 or exact equivalent only Use $\Sigma w ^ { 2 } \mathrm { p } ( w )$

Correctly obtain given answer, www

Can be implied

Method needed ft on their quadratic Allow for $\mathrm { E } ( W ) ^ { 2 } = 4.9729$

Need 2.23 or 4.9729 and 5.83 or full numerical $\Sigma w ^ { 2 } \mathrm { p } ( w )$

1

(b)

$9 \times 0.8571 = 7.7139$

B1

[1]

1.1b

Allow 7.71 or 7.714

2

(a)

Flaws must occur at constant average rate (uniform rate)

B1

[1]

1.2

Context (e.g. "flaws") needed

Extra answers, e.g. "singly": B0

Not "constant rate" or "average constant rate".

2

(b)

$\operatorname { Po(2.1)~or~ } e ^ { - \lambda } \frac { \lambda ^ { 3 } } { 3 ! }$

M1

A1

[2]

1.1

1.1b

Po(2.1) stated or implied, or formula with $\lambda = 2.1$ stated Awrt 0.189

2

(c)

Po(3)

$1 - \mathrm { P } ( \leq 3 )$

M1

A1

[3]

1.1

1.1b

$\operatorname { Po } ( 2 \times 0.7 + 1.6 )$ stated or implied

Allow $1 - \mathrm { P } ( \leq 4 ) = 0.1847$, or from wrong $\lambda$

Awrt 0.353

Or all combinations $\leq 3$

$1 -$ above, not just $= 3$

Question

Answer

Marks

AO

Guidance

3

(a)

0.4(00)

B2

[2]

1.1

1.1b

SC: if B0, give SC B1 for two of $S _ { x x } = 12500 , S _ { y y } = 1600 , S _ { x y } = 1790$ and $S _ { x y } / \sqrt { } \left( S _ { x x } S _ { y y } \right)$

Also allow SC B1 for equivalent methods using Covariance \

SDs

3

(b)

Data needs to have a bivariate normal distribution

B1

[1]

1.2

Needs "bivariate normal" or clear equivalent. Not just "both normally distributed"

Allow "scatter diagram forms ellipse"

3

(c)

$\mathrm { H } _ { 0 }$ : higher maths scores are not associated with higher BCF grading; $\mathrm { H } _ { 1 }$ : positively associated

CV 0.3783

$0.400 > 0.3783$ so reject $\mathrm { H } _ { 0 }$

Significant evidence that higher maths scores are associated with higher BCF grading

B1

M1ft

A1ft

[4]

2.5

1.1b

2.2b

3.5a

Needs context and clearly onetailed $O R \rho$ used and defined Not "evidence that ..."

Allow 0.378

Reject/do not reject $\mathrm { H } _ { 0 }$

Contextualised, not too definite Needn't say "positive" if $\mathrm { H } _ { 1 } \mathrm { OK }$

SC 2-tail: B0; 0.4438, or 0.3783 B1; then M1A0

$\mathrm { H } _ { 0 } : \rho = 0 , \mathrm { H } _ { 1 } : \rho > 0$ where $\rho$ is population pmcc (not $r$ )

FT on their $r$, but not CV

Not "scores are associated

...". FT on their $r$ only

3

(d)

It makes no difference as this is a linear transformation

B1

[1]

2.2a

Need both "unchanged" oe and reason, need "linear" or exact equivalent

"oe" includes "their 0.4"

4

(a)

Neither

B1

[1]

2.5

OE

Not "neither is independent of the other"

4

(b)

$c = 2.848 - 0.1567 m$

B1

[3]

1.1

Correct $a$, awrt 2.85

Correct $b$, awrt 0.157

Letters correct from correct method

(If both wrongly rounded, e.g. $c = 2.84 - 0.156 m$, give B2)

$\mathrm { SC } : m$ on $c$ :

$m = 15.65 - 4.832 c$ : B2

$y = 15.65 - 4.832 x$ : B1

$c = 15.65 - 4.832 m : \mathrm { B } 1$

If B0B0, give B1 for correct letters from valid working

Question

Answer

Marks

AO

Guidance

4

(c)

$a$ unchanged, $b$ multiplied by 2.2 (allow " $a$ unchanged, $b$ increases", etc)

B1 [1]

2.2a

oe, e.g. $c = 2.848 - 0.345 m$; $m = 7.114 - 2.196 c$

SC: $m$ on $c$ in (b): Both divided by 2.2 B1

4

(d)

Draw approximate line of best fit

Draw at least one vertical from line to point

Say that "Best fit" line minimises the sum of squares of these distances

M1

A1

[3]

1.1

2.4

Needs M2 and "minimises" and "sums of squares" oe

SC: Horizontal(s):

full marks (indept of (b))

OCR FS1 AS 2021 June Q1

6 marks Standard +0.3

1 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution $\mathrm { Po } ( 120 )$.

Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
State a necessary assumption for the validity of your calculation in part (b).

OCR FS1 AS 2021 June Q2

7 marks Standard +0.3

2 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

Find the probability that all the men are next to each other.
Find the probability that no two men are next to one another.

OCR FS1 AS 2021 June Q3

5 marks Moderate -0.3

3 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.

For all sixteen candidates, the value of the product moment correlation coefficient $r$ for the marks on the two papers was 0.701 correct to 3 significant figures. Test whether there is evidence, at the $5 \%$ significance level, of association between the marks on the two papers.
A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by $n = 10 , \Sigma x = 750 , \Sigma y = 690 , \Sigma x ^ { 2 } = 57690 , \Sigma y ^ { 2 } = 49676 , \Sigma x y = 50829$.
1. Calculate the value of $r$ for these 10 candidates.
2. What do the two values of $r$, in parts (a) and (b)(i), tell you about the scores of the sixteen candidates? A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by $p$.
  1. Sasha selects 10 beads at random, with replacement. Write down an expression, in terms of $p$, for the variance of the number of green beads Sasha selects. Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the $X$ th bead that Freda selects.
  2. Assume that $p = 0.3$. Find
    1. $\mathrm { P } ( X \geqslant 5 )$,
    2. $\operatorname { Var } ( X )$.
3. In fact, on the basis of a large number of observations of $X$, it is found that $\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )$. Estimate the value of $p$.

OCR FS1 AS 2021 June Q1

5 marks Moderate -0.3

1 Five observations of bivariate data $( x , y )$ are given in the table.

$x$	7	8	12	6	4
$y$	20	16	7	17	23

Find the value of Pearson's product-moment correlation coefficient.
State what your answer to part (a) tells you about a scatter diagram representing the data.
A new variable $a$ is defined by $a = 3 x + 4$. Dee says "The value of Pearson's product-moment correlation coefficient between $a$ and $y$ will not be the same as the answer to part (a)." State with a reason whether you agree with Dee. An investor obtains data about the profits of 8 randomly chosen investment accounts over two one-year periods. The profit in the first year for each account is $p \%$ and the profit in the second year for each account is $q \%$. The results are shown in the table and in the scatter diagram.
Account A B C D E F G H
$p$ 1.6 2.1 2.4 2.7 2.8 3.3 5.2 8.4
$q$ 1.6 2.3 2.2 2.2 3.1 2.9 7.6 4.8
$n = 8 \quad \Sigma p = 28.5 \quad \Sigma q = 26.7 \quad \Sigma p ^ { 2 } = 136.35 \quad \Sigma q ^ { 2 } = 116.35 \quad \Sigma p q = 116.70$ \includegraphics[max width=\textwidth, alt={}, center]{4c7546b9-03ee-47a1-915f-41e2b4ca19c0-03_762_1248_906_260}
1. State which, if either, of the variables $p$ and $q$ is independent.
2. Calculate the equation of the regression line of $q$ on $p$.
3. 1. Use the regression line to estimate the value of $q$ for an investment account for which $p = 2.5$.
  2. Give two reasons why this estimate could be considered reliable.
4. Comment on the reliability of using the regression line to predict the value of $q$ when $p = 7.0$.

OCR FS1 AS 2021 June Q3

12 marks Standard +0.3

3 At a cinema there are three film sessions each Saturday, "early", "middle" and "late". The numbers of the audience, in different age groups, at the three showings on a randomly chosen Saturday are given in Table 1. \begin{table}[h] \end{table}

Question

Solution

Marks

AOs

Guidance

1

(a)

-0.954 BC

B2 [2]

1.1 1.1

SC: If B0, give B1 if two of 7.04, 29.0[4], -13.6[4] (or 35.2, 145[.2], -68.2) seen

1

(b)

Points lie close to a straight line Line has negative gradient

B1 B1 [2]

2.2b 1.1

Must refer to line, not just "negative correlation"

1

(c)

No, it will be the same as $x \rightarrow a$ is a linear transformation

B1 [1]

2.2a

OE. Either "same" with correct reason, or "disagree" with correct reason. Allow any clear valid technical term

2

(a)

Neither

B1 [1]

1.2

2

(b)

$q = 1.13 + 0.620 p$

B1B1 B1 [3]

1.1,1.1 1.1

0.62(0) correct; both numbers correct Fully correct answer including letters

2

(c)

(i)

2.68

B1ft [1]

1.1

awrt 2.68, ft on their (b) if letters correct

2

(c)

(ii)

2.5 is within data range, and points (here) are close to line/well correlated

B1 B1 [2]

2.2b 2.2b

At least one reason, allow "no because points not close to line" Full argument, two reasons needed

2

(d)

Not much data here/points scattered/ possible outliers

So not very reliable

M1 A1 [2]

2.3 1.1

Reason for not very reliable (not "extrapolation") Full argument and conclusion, not too assertive (not wholly unreliable!)

3

(a)

Expected frequency for Middle/25 to 60 is 4.4 which is < 5 so must combine cells

B1*ft depB1 [2]

2.4 3.5b

Correctly obtain this $F _ { E }$, ft on addition errors " < 5" explicit and correct deduction

3

(b)

Early	Middle	Late
29.4	23.1	31.5
26.6	20.9	28.5

Early	Middle	Late
0.9918	0.4160	2.2937
1.0962	0.4598	2.5351

B1

1.1

Both, allow 28.4 for 28.5

awrt 2.29, but allow 2.3 In range [2.53, 2.54]

Question

Solution

Marks

AOs

Guidance

3

(c)

$\mathrm { H } _ { 0 }$ : no association between session and age group. $\mathrm { H } _ { 1 }$ : some association

$\Sigma X ^ { 2 } = 7.793$

$v = 2 , \chi ^ { 2 } ( 2 ) _ { \text {crit } } = 5.991$

Reject $\mathrm { H } _ { 0 }$.

Significant evidence of association between session attended and age group.

B1

M1ft

A1ft [5]

1.1

2.2b

Both. Allow "independent" etc

Correct value of $X ^ { 2 }$, awrt 7.79 (allow even if wrong in (b))

Correct CV and comparison

Correct first conclusion, FT on their TS only

Contextualised, not too assertive

3

(d)

The two biggest contributions to $\chi ^ { 2 }$ are both for the late session ... ... when the proportion of younger people is higher, and of older people is lower, than the null hypothesis would suggest.

M1ft

A1ft

[2]

1.1

2.4

Refer to biggest contribution(s), FT on their answers to (b), needs "reject $\mathrm { H } _ { 0 }$ "

Full answer, referring to at least one cell (ignore comments on next highest cells)

\multirow[t]{2}{*}{4}

\multirow{2}{*}{}

\multirow{2}{*}{OR:}

$\frac { { } ^ { 2 m } C _ { 2 } \times m } { { } ^ { 3 m } C _ { 3 } }$

$= \frac { 2 m ( 2 m - 1 ) } { 2 } \times m \div \frac { 3 m ( 3 m - 1 ) ( 3 m - 2 ) } { 6 }$

$= \frac { 2 m ( 2 m - 1 ) } { ( 3 m - 1 ) ( 3 m - 2 ) }$ $\frac { 2 m ( 2 m - 1 ) } { ( 3 m - 1 ) ( 3 m - 2 ) } = \frac { 28 } { 55 }$

$\Rightarrow 16 m ^ { 2 } - 71 m + 28 = 0$

$m = 4$ BC

Reject $m = \frac { 7 } { 16 }$ as $m$ is an integer

M1

A1

M1

A1

M1

A1

[7]

3.1b

2.1

3.1a

2.1

1.1

3.2a

Use ${ } ^ { 2 m } C _ { 2 }$ and $m$
Divide by ${ } ^ { 3 m } C _ { 3 }$
Correct expression in terms of $m$ (allow with $m$ not cancelled yet)
Equate to $\frac { 28 } { 55 }$ \	simplify to three-term quadratic
Correct simplified quadratic, or (quadratic) $\times m , = 0$, aef Solve to get both 4 and $\frac { 7 } { 16 }$
Explicitly reject $m = \frac { 7 } { 16 }$

$\frac { 2 m ( 2 m - 1 ) \times m \times 3 ! } { 3 m ( 3 m - 1 ) ( 3 m - 2 ) \times 2 }$ then as above

Multiplication method can get full marks, but if no 3 or 3 !, max

M1M0A0 M1A0M0A0

OCR FS1 AS 2021 June Q1

8 marks Moderate -0.3

1 Every time a spinner is spun, the probability that it shows the number 4 is 0.2 , independently of all other spins.

A pupil spins the spinner repeatedly until it shows the number 4 . Find the mean of the number of spins required.
Calculate the probability that the number of spins required is between 3 and 10 inclusive.
Each pupil in a class of 30 spins the spinner until it shows the number 4 . Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by $X$. Determine the variance of $X$.

OCR FS1 AS 2021 June Q2

9 marks Moderate -0.5

2 After a holiday organised for a group, the company organising the holiday obtained scores out of 10 for six different aspects of the holiday. The company obtained responses from 100 couples and 100 single travellers. The total scores for each of the aspects are given in the following table.

Question

Answer

Mark

AO

Guidance

1

(a)

$\frac { 1 } { 0.2 } = 5$

M1 A1 [2]

3.3 1.1

Geometric distribution soi 5 (or $5.00 \ldots$ ) only

1

(b)

$0.8 ^ { 2 } - 0.8 ^ { 10 }$ $= \mathbf { 0 . 5 3 3 } \quad ( 0.5326258 \ldots )$

M1 A1 [2]

1.1 3.4

Allow for powers 2, 3, 4 and 9, 10, 11 .

Awrt 0.533, www. [5201424/976562]

Or $0.2 \left( 0.8 ^ { 2 } + \ldots . + 0.8 ^ { 9 } \right) , \pm 1$ term at either end [0.506, 0.378, 0.275, 0.405, 0.302, 0.554, 0.426, 0.324]

1

(c)

$\mathrm { P } ( \geq 10 ) = 0.8 ^ { 9 }$

$= 0.1342 \ldots$

B(30, 0.1342...)

Variance $= n p q$ = 3.486...

M1

A1

M1

A1ft [4]

3.1b

1.1

3.1b

1.1

Or $0.8 ^ { 10 }$. Can be implied by correct $p$

[0.10737... is M1A0 here]

Stated or implied, their $0.8 ^ { 9 }$ or $0.8 ^ { 10 }$

In range [3.48, 3.49]

SC: 0.134(2) oe not properly shown: B2 for correct final answer.

SC: 2.875 from $0.8 ^ { 10 }$ : M1A0M1A1ft

Question

Answer

Mark

AO

Guidance

2

(a)

Test is for rankings/rankings arbitrary/not bivariate normal etc

B1 [1]

2.4

OE

2

(b)

$\mathrm { H } _ { 0 } : \rho _ { s } = 0 , \mathrm { H } _ { 1 } : \rho _ { s } > 0$, where $\rho _ { s }$ is the population rank correlation coefficient

Ranks 543612

512643

$\Sigma d ^ { 2 } = 20$

$r _ { s } = 1 - \frac { 6 \times 20 } { 6 \times 35 }$

$= 3 / 7$ or $0.42857 \ldots$

<0.9429

B1

M1

A1

B1

1.1

Allow $\rho _ { s }$ not defined; allow $\rho$.

Allow: $\mathrm { H } _ { 0 }$ : no association between rankings.

$\mathrm { H } _ { 1 }$ : positive association (but not $\mathrm { H } _ { 1 }$ : association)

Do not reject $\mathrm { H } _ { 0 }$

Insufficient evidence of association between ranking given by the two categories

M1ft

A1ft

[7]

1.1

2.2b

FT on their $\Sigma d ^ { 2 }$ only

2

(c)

Not dependent on any distributional assumptions

B1

[1]

1.2

Oe (cf. Specification, 5.08f)

Question

Answer

Mark

AO

Guidance

3

(a)

Failures occur to no fixed pattern/are not predictable

B1 [1]

1.1

OE. NOT "independent"

3

(b)

Failures occur independently of one another and at constant average rate

B1

[2]

1.1

Not recoverable from (a) if independence not restated here; must be contextualised

Ignore "singly". Allow "uniform" rate, not "constant rate" or "constant probability"; must be contextualised

3

(c)

Variance (1.6384) $\approx$ mean

So suggests that it is likely to be well modelled

M1

A1

[2]

1.1

3.5a

Compare variance (or SD). Allow square/square-root confusion

Correct comparison and conclusion, 1.64 or better seen

3

(d)

$\mathrm { e } ^ { - 1.61 }$

B1

[1]

3.4

Exact needed, allow even if $0 !$ or $1.61 ^ { 0 }$ or both left in

3

(e)

1	$\geq 2$
0.322	0.478

B1

[2]

3.4

1.1

One correct e.g. 0.3218

Other correct e.g. 0.4783

3

(f)

$\mathrm { P } ( F = 1 )$ will be smaller as single failures are less likely

B1*

depB1

[2]

3.5c

3.3

OE. Partial answer: B1

OCR Further Mechanics 2021 June Q2

10 marks Standard +0.3

2 A solenoid is a device formed by winding a wire tightly around a hollow cylinder so that the wire forms (approximately) circular loops along the cylinder (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{91098ecb-fb4a-44aa-9e59-6c6fe3704966-02_164_697_1484_233} When the wire carries an electrical current a magnetic field is created inside the solenoid which can cause a particle which is moving inside the solenoid to accelerate. A student is carrying out experiments on particles moving inside solenoids. His professor suggests that, for a particle of mass $m$ moving with speed $v$ inside a solenoid of length $h$, the acceleration $a$ of the particle can be modelled by a relationship of the form $a = k m ^ { \alpha } v ^ { \beta } h ^ { \gamma }$, where $k$ is a constant. The professor tells the student that $[ k ] = \mathrm { MLT } ^ { - 1 }$.

Use dimensional analysis to find $\alpha , \beta$ and $\gamma$.
The mass of an electron is $9.11 \times 10 ^ { - 31 } \mathrm {~kg}$ and the mass of a proton is $1.67 \times 10 ^ { - 27 } \mathrm {~kg}$. For an electron and a proton moving inside the same solenoid with the same speed, use the model to find the ratio of the acceleration of the electron to the acceleration of the proton.
The professor tells the student that $a$ also depends on the number of turns or loops of wire, $N$, that the solenoid has. Explain why dimensional analysis cannot be used to determine the dependence of $a$ on $N$.

OCR Further Mechanics 2021 June Q3

9 marks Standard +0.8

3 A right circular cone $C$ of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of $C$. The other end of the string is attached to a particle $P$ of mass 2.5 kg . $P$ moves in a horizontal circle with constant speed and in contact with the smooth curved surface of $C$. The extension of the string is 1.5 m .

Find the tension in the string.
Find the speed of $P$.

OCR Further Mechanics 2021 June Q4

9 marks Challenging +1.8

4 Two particles $A$ and $B$, of masses $m \mathrm {~kg}$ and 1 kg respectively, are connected by a light inextensible string of length $d \mathrm {~m}$ and placed at rest on a smooth horizontal plane a distance of $\frac { 1 } { 2 } d \mathrm {~m}$ apart. $B$ is then projected horizontally with speed $v \mathrm {~ms} ^ { - 1 }$ in a direction perpendicular to $A B$.

Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, $I \mathrm { Ns }$, is given by $I = \frac { \sqrt { 3 } m v } { 2 ( 1 + m ) }$.
Find, in terms of $m$ and $v$, the kinetic energy of $B$ at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
In the case where $m$ is very large, describe, with justification, the approximate motion of $B$ after the string becomes taut.

OCR Further Mechanics 2021 June Q1

13 marks Standard +0.3

1 A particle $Q$ of mass $m \mathrm {~kg}$ is acted on by a single force so that it moves with constant acceleration $\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }$. Initially $Q$ is at the point $O$ and is moving with velocity $\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }$. After $Q$ has been moving for 5 seconds it reaches the point $A$.

Use the equation $\mathbf { v } . \mathbf { v } = \mathbf { u } . \mathbf { u } + 2 \mathbf { a } . \mathbf { x }$ to show that at $A$ the kinetic energy of $Q$ is 37 m J .
1. Show that the power initially generated by the force is - 8 mW W.
2. The power in part (b)(i) is negative. Explain what this means about the initial motion of $Q$.
1. Find the time at which the power generated by the force is instantaneously zero.
2. Find the minimum kinetic energy of $Q$ in terms of $m$.

OCR Further Mechanics 2021 June Q2

19 marks Standard +0.8

2 A particle $P$ of mass 4.5 kg is free to move along the $x$-axis. In a model of the motion it is assumed that $P$ is acted on by two forces:

a constant force of magnitude $f \mathrm {~N}$ in the positive $x$ direction;
a resistance to motion, $R \mathrm {~N}$, whose magnitude is proportional to the speed of $P$.

At time $t$ seconds the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$. When $t = 0 , P$ is at the origin $O$ and is moving in the positive direction with speed $u \mathrm {~ms} ^ { - 1 }$, and when $v = 5 , R = 2$. \begin{enumerate}[label=(\alph*)] \item Show that, according to the model, $\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 10 f - 4 v } { 45 }$. \item

By solving the differential equation in part (a), show that $v = \frac { 1 } { 2 } \left( 5 f - ( 5 f - 2 u ) \mathrm { e } ^ { - \frac { 4 } { 45 } t } \right)$.
Describe briefly how, according to the model, the speed of $P$ varies over time in each of the following cases.
The flat surface of a smooth solid hemisphere of radius $r$ is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by $\gamma$. $O$ is the centre of the flat surface of the hemisphere. A particle $P$ is held at a point on the surface of the hemisphere such that the angle between $O P$ and the upward vertical through $O$ is $\alpha$, where $\cos \alpha = \frac { 3 } { 4 }$. $P$ is then released from rest. $F$ is the point on the plane where $P$ first hits the plane (see diagram).
1. Find an exact expression for the distance $O F$. The acceleration due to gravity on and near the surface of the planet Earth is roughly $6 \gamma$.
2. Explain whether $O F$ would increase, decrease or remain unchanged if the action were repeated on the planet Earth.

OCR Further Mechanics 2021 June Q1

6 marks Standard +0.8

1 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.

\(w\)	1	2	3	4
\(\mathrm { P } ( W = w )\)	0.25	0.36	\(x\)	\(x ^ { 2 }\)

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Account	A	B	C	D	E	F	G	H
\(p\)	1.6	2.1	2.4	2.7	2.8	3.3	5.2	8.4
\(q\)	1.6	2.3	2.2	2.2	3.1	2.9	7.6	4.8

\(x\)	7	8	12	6	4
\(y\)	20	16	7	17	23