Questions - OCR - A-Level Maths

OCR Further Pure Core 1 2021 June Q3

5 marks Challenging +1.2

3 By expanding $\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 3 }$, where $z = \mathrm { e } ^ { \mathrm { i } \theta }$, show that $4 \cos ^ { 3 } 2 \theta = \cos 6 \theta + 3 \cos 2 \theta$.

OCR Further Pure Core 1 2021 June Q4

5 marks Standard +0.8

4 The equations of two non-intersecting lines, $l _ { 1 }$ and $l _ { 2 }$, are $l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 1 \\ - 2 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { c } 1 \\ - 1 \\ 4 \end{array} \right)$.
Find the shortest distance between lines $l _ { 1 }$ and $l _ { 2 }$.

OCR Further Pure Core 1 2021 June Q5

5 marks Standard +0.3

5 Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9 .

OCR Further Pure Core 1 2021 June Q6

9 marks Challenging +1.2

6 You are given that the cubic equation $2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0$, where $p$ and $q$ are real numbers, has a complex root $\alpha = 1 + i \sqrt { 2 }$.

Write down a second complex root, $\beta$.
Determine the third root, $\gamma$.
Find the value of $p$ and the value of $q$.
Show that if $n$ is an integer then $\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }$ where $\tan \theta = \sqrt { 2 }$.

OCR Further Pure Core 1 2021 June Q7

8 marks Standard +0.8

7 A curve has cartesian equation $x ^ { 3 } + y ^ { 3 } = 2 x y$. $C$ is the portion of the curve for which $x \geqslant 0$ and $y \geqslant 0$. The equation of $C$ in polar form is given by $r = \mathrm { f } ( \theta )$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.

Find $f ( \theta )$.
Find an expression for $\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)$, giving your answer in terms of $\sin \theta$ and $\cos \theta$.
Hence find the line of symmetry of $C$.
Find the value of $r$ when $\theta = \frac { 1 } { 4 } \pi$.
By finding values of $\theta$ when $r = 0$, show that $C$ has a loop.

OCR Further Pure Core 1 2021 June Q1

3 marks Standard +0.8

1 Find an expression for $1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + 3 \times 4 ^ { 2 } + \ldots + n ( n + 1 ) ^ { 2 }$ in terms of $n$. Give your answer in fully factorised form.

OCR Further Pure Core 1 2021 June Q2

5 marks Moderate -0.3

2
You are given the matrix $\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)$.

Find $\mathbf { A } ^ { 4 }$.
Describe the transformation that A represents. The matrix $\mathbf { B }$ represents a reflection in the plane $x = 0$.
Write down the matrix $B$. The point $P$ has coordinates $( 2,3,4 )$. The point $P ^ { \prime }$ is the image of $P$ under the transformation represented by $\mathbf { B }$.
Find the coordinates of $P ^ { \prime }$.

OCR Further Pure Core 1 2021 June Q3

10 marks Challenging +1.2

3

Using exponentials, show that $\cosh 2 u \equiv 2 \sinh ^ { 2 } u + 1$.
By differentiating both sides of the identity in part (a) with respect to $u$, show that $\sinh 2 u \equiv 2 \sinh u \cosh u$.
Use the substitution $x = \sinh ^ { 2 } u$ to find $\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x$. Give your answer in the form $a \sinh ^ { - 1 } b \sqrt { x } + \mathrm { f } ( x )$ where $a$ and $b$ are integers and $\mathrm { f } ( x )$ is a function to be determined.
Hence determine the exact area of the region between the curve $y = \sqrt { \frac { x } { x + 1 } }$, the $x$-axis, the line $x = 1$ and the line $x = 2$. Give your answer in the form $p + q \ln r$ where $p , q$ and $r$ are numbers to be determined.

OCR Further Pure Core 1 2021 June Q4

13 marks Standard +0.3

4 A particle of mass 0.5 kg is initially at point $O$. It moves from rest along the $x$-axis under the influence of two forces $F _ { 1 } \mathrm {~N}$ and $F _ { 2 } \mathrm {~N}$ which act parallel to the $x$-axis. At time $t$ seconds the velocity of the particle is $v \mathrm {~ms} ^ { - 1 }$. $F _ { 1 }$ is acting in the direction of motion of the particle and $F _ { 2 }$ is resisting motion.
In an initial model

$F _ { 1 }$ is proportional to $t$ with constant of proportionality $\lambda > 0$,
$F _ { 2 }$ is proportional to $v$ with constant of proportionality $\mu > 0$.
1. Show that the motion of the particle can be modelled by the following differential equation.

$$\frac { 1 \mathrm {~d} v } { 2 \mathrm {~d} t } = \lambda t - \mu v$$

Solve the differential equation in part (a), giving the particular solution for $v$ in terms of $t$, $\lambda$ and $\mu$. You are now given that $\lambda = 2$ and $\mu = 1$.

Find a formula for an approximation for $v$ in terms of $t$ when $t$ is large. In a refined model

OCR Further Pure Core 1 2021 June Q5

6 marks Challenging +1.2

5
Show that $\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }$ where $a , b$ and $c$ are integers to be determined.

OCR Further Pure Core 2 2021 June Q1

6 marks Moderate -0.8

1

The Argand diagram below shows the two points which represent two complex numbers, $z _ { 1 }$ and $z _ { 2 }$. \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-02_321_592_276_347} On the copy of the diagram in the Resource Materials.
- 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 Resource Materials. You should join each point to the origin with a straight line.
1. $\left| z _ { 3 } \right| = 1.5$ and $\left| z _ { 4 } \right| = 1.2$
2. $\left| z _ { 3 } \right| = 0.7$ and $\left| z _ { 4 } \right| = 0.5$

OCR Further Pure Core 2 2021 June Q2

6 marks Standard +0.8

2 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. 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.
1. By using Euler's formula show that $\cosh ( \mathrm { iz } ) = \cos z$.
2. Hence, find, in logarithmic form, a root of the equation $\cos z = 2$. [You may assume that $\cos z = 2$ has complex roots.] 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]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255} In an initial model of the motion of a certain swing door it is suggested that $\theta$ satisfies the following differential equation. $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$
3. 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 $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$ where $\lambda$ is a positive constant.
4. 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.
  4. Find the value of $\lambda$ for which the door is critically damped.

OCR Further Pure Core 2 2021 June Q2

5 marks Moderate -0.3

2 A 2-D transformation $T$ is a shear which leaves the $y$-axis invariant and which transforms the object point $( 2,1 )$ to the image point $( 2,9 )$. $A$ is the matrix which represents the transformation $T$.

Find A .
By considering the determinant of A , explain why the area of a shape is invariant under T .

OCR Further Pure Core 2 2021 June Q3

11 marks Standard +0.3

3 A particle of mass 2 kg moves along the $x$-axis. At time $t$ seconds the velocity of the particle is $v \mathrm {~ms} ^ { - 1 }$. The particle is subject to two forces.

One acts in the positive $x$-direction with magnitude $\frac { 1 } { 2 } t \mathrm {~N}$.
One acts in the negative $x$-direction with magnitude $v \mathrm {~N}$.
1. Show that the motion of the particle can be modelled by the differential equation

$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when $t = 0$.

Find $v$ in terms of $t$.

Find the velocity of the particle when $t = 2$. When $t = 2$ the force acting in the positive $x$-direction is replaced by a constant force of magnitude $\frac { 1 } { 2 } \mathrm {~N}$ in the same direction.

Refine the differential equation given in part (a) to model the motion for $t \geqslant 2$.

Use the refined model from part (d) to find an exact expression for $v$ in terms of $t$ for $t \geqslant 2$.

OCR Further Pure Core 2 2021 June Q4

8 marks Challenging +1.8

4 In this question you must show detailed reasoning.

By writing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$ show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
Hence show that $\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }$.
1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln \left( \frac { 1 } { 2 } + \cos x \right)$.
2. By considering the root of the equation $\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0$ deduce that $\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$.

OCR Further Pure Core 2 2021 June Q1

6 marks Challenging +1.2

1 In this question you must show detailed reasoning.
The roots of the equation $3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0$ are $\alpha , \beta$ and $\gamma$.

Find a cubic equation with integer coefficients whose roots are $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$.
Find the exact value of $\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }$.

OCR Further Pure Core 2 2021 June Q2

6 marks Standard +0.3

2 In this question you must show detailed reasoning.

Use partial fractions to show that $\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }$.
Write down the value of $\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)$.

OCR Further Pure Core 2 2021 June Q3

6 marks Challenging +1.8

3 The equation of a curve in polar coordinates is $r = \ln ( 1 + \sin \theta )$ for $\alpha \leqslant \theta \leqslant \beta$ where $\alpha$ and $\beta$ are non-negative angles. The curve consists of a single closed loop through the pole.

By solving the equation $r = 0$, determine the smallest possible values of $\alpha$ and $\beta$.
Find the area enclosed by the curve, giving your answer to 4 significant figures.
Hence, by considering the value of $r$ at $\theta = \frac { \alpha + \beta } { 2 }$, show that the loop is not circular.

OCR Further Pure Core 2 2021 June Q4

9 marks Standard +0.8

4 In this question you must show detailed reasoning.
The complex number $- 4 + i \sqrt { 48 }$ is denoted by $z$.

Determine the cube roots of $z$, giving the roots in exponential form. The points which represent the cube roots of $z$ are denoted by $A , B$ and $C$ and these form a triangle in an Argand diagram.
Write down the angles that any lines of symmetry of triangle $A B C$ make with the positive real axis, justifying your answer.

OCR Further Pure Core 2 2021 June Q5

10 marks Standard +0.3

5 Let $\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )$.

1. Determine $f ^ { \prime \prime } ( x )$.
2. Determine the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$.
3. By considering the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( \mathrm { x } )$, find an approximation to $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer correct to 6 decimal places.
By writing $\mathrm { f } ( x )$ as $\sin ^ { - 1 } ( x ) \times 1$, determine the value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer in exact form.

OCR Further Statistics 2021 June Q1

5 marks Standard +0.3

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. \end{table}

Question

Answer

Mark

AO

Guidance

\multirow[t]{3}{*}{1}

\multirow[t]{3}{*}{(a)}

$A + B + C \sim \mathrm {~N} ( 701 , \ldots$

.. 419)

M1

1.1a

Normal, mean $\mu _ { A } + \mu _ { B } + \mu _ { C }$

\multirow{3}{*}{}

A1

1.1

Variance 419

$\mathrm { P } ( > 720 ) = 0.176649$

A1

1.1

Answer, 0.177 or better, www

\multirow[t]{2}{*}{1}

\multirow[t]{2}{*}{(b)}

$2 A + B \sim \mathrm {~N} ( 701,757 )$

M1

1.1a

Normal, same mean, $4 \sigma _ { A } { } ^ { 2 } + \sigma _ { B } { } ^ { 2 }$

\multirow{2}{*}{}

$\mathrm { P } ( > 720 ) = 0.244919$

A1 [2]

1.1

Answer, art 0.245

\multirow{2}{*}{2}

\multirow{2}{*}{(a)}

$\frac { { } ^ { 8 } C _ { 3 } \times { } ^ { 20 } C _ { 5 } } { { } ^ { 28 } C _ { 8 } }$

M1 A1

3.1b 1.1

(Product of two ${ } ^ { n } C _ { r }$ ) ÷ ${ } ^ { n } C _ { r }$ At least two ${ } ^ { n } C _ { r }$ correct

\multirow[t]{2}{*}{Or $\frac { 8 } { 28 } \times \frac { 7 } { 27 } \times \frac { 6 } { 26 } \times \frac { 20 } { 25 } \times \ldots \times \frac { 16 } { 21 } \times { } ^ { 8 } C _ { 3 } = 0.27934 \ldots$}

$\frac { 56 \times 15504 } { 3108105 } = 0.27934 \ldots$

A1 [3]

1.1

Any exact form or awrt 0.279

2

(b)

× B × B × B × B × B × B × B × B x

GGG in one $\mathrm { x } , \mathrm { G }$ in another: $9 \times 8$ $\div \frac { 12 ! } { 8 ! \times 4 ! }$ $= \frac { 72 } { 495 } = \frac { 8 } { 55 } \text { or } 0.145 \ldots$

M1 A1

3.1b 2.1

Or e.g. $\frac { 10 ! } { 8 ! } - 2 \times 9$

Divide by ${ } _ { 12 } \mathrm { C } _ { 4 }$ oe

Or, e.g. find ${ } _ { 12 } \mathrm { C } _ { 4 }$ - (\# (all separate) +\#(all together) $+ \# ( 2,1,1 ) \times 3 +$ \#(2,2))

M1

1.1

A1

1.1

[4]

Question

Answer

Mark

AO

Guidance

\multirow{7}{*}{3}

\multirow{7}{*}{(a)}

$\mathrm { H } _ { 0 } : \mu = 700$

B2

1.1

One error, e.g. no or wrong

Ignore failure to define $\mu$

$\mathrm { H } _ { 1 } : \mu < 700$ where $\mu$ is the mean reaction

1.1

letter, $\neq$, etc : B1

here

$\bar { x } = 607$

M1

3.3

Find sample mean

$z = - 1.822$ or $p = 0.0342$ or $\mathrm { CV } = 616.05 \ldots$

A1

3.4

Correct $z , p$ or CV

$z < - 1.645$ or $p < 0.05$ or $607 < \mathrm { CV }$

A1

1.1

Correct comparison

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

M1ft

1.1

Correct first conclusion

Needs correct method, like-

Significant evidence that mean reaction times

A1ft

2.2b

Context, not too definite (e.g. not "international athletes' reaction times are shorter"

ft on their $z , p$ or CV

3

(b)

(i)

Uses more information (e.g. magnitudes of differences)

B1 [1]

2.4

\multirow{5}{*}{3}

\multirow{5}{*}{(b)}

\multirow{5}{*}{(ii)}

$\mathrm { H } _ { 0 } : m = 700 , \mathrm { H } _ { 1 } : m < 700$ where $m$ is the median reaction time for all international athletes

B1

2.5

Same as in (i) but different letter or "median" stated

$W _ { - } = 18$

$W _ { + } = 3$ so $T = 3$

For both, and $T$ correct

$n = 6 , \mathrm { CV } = 2$

A1

1.1

Correct CV

Do not reject $\mathrm { H } _ { 0 }$. Insufficient evidence that median reaction times of international athletes are shorter

A1ft [6]

2.2b

In context, not too definite

FT on their $T$

3

(c)

They use different assumptions

B1 [1]

2.3

Not "one is more accurate"

Question

Answer

Mark

AO

Guidance

4

(a)

\(\begin{aligned}

\int _ { 0 } ^ { a } x \frac { 2 x } { a ^ { 2 } } d x = 4

{ \left[ \frac { 2 x ^ { 3 } } { 3 a ^ { 2 } } \right] = 4 }

\frac { 2 } { 3 } a = 4 \Rightarrow a = 6 \end{aligned}\)

M1

B1

A1 [3]

3.1a

1.1

2.2a

4

(b)

$\mathrm { F } ( x ) = \frac { x ^ { 2 } } { 36 }$
Let the CDF of $M$ be $\mathrm { H } ( m )$. Then $\mathrm { H } ( m ) = \mathrm { P } ($ all observations less than $m )$ $= [ \mathrm { P } ( X \leqslant m ) ] ^ { 5 }$ $= \left[ \frac { m ^ { 2 } } { 36 } \right] ^ { 5 }$
\(\mathrm { H } ( m ) = \begin{cases} 0	m < 0 ,
\frac { m ^ { 10 } } { 60466176 }	0 \leq m \leq 6 ,
1	m > 6 . \end{cases}\)

M1 A1ft

M1

A1

[8]

1.1
1.1
2.1
3.1a
2.2a
2.1
2.1
1.2

Find $\mathrm { F } ( x ) ; = \frac { x ^ { 2 } } { a ^ { 2 } }$

Correct basis for CDF of $m$

Correct function, any letter Range $0 \leq m \leq 6$

Letter not $x$, and 0, 1 present

ft on their $a$

Allow

OCR Further Statistics 2021 June Q1

4 marks Moderate -0.8

1
The continuous random variable $X$ has the distribution $\mathrm { N } ( \mu , 30 )$. The mean of a random sample of 8 observations of $X$ is 53.1 . Determine a $95 \%$ confidence interval for $\mu$. You should give the end points of the interval correct to 4 significant figures.

OCR Further Statistics 2021 June Q2

12 marks Standard +0.3

2 A book collector compared the prices of some books, $\pounds x$, when new in 1972 and the prices of copies of the same books, $\pounds y$, on a second-hand website in 2018.
The results are shown in Table 1 and are summarised below the table. \begin{table}[h]

Book	A	B	C	D	E	F	G	H	I	J	K	L
$x$	0.95	0.65	0.70	0.90	0.55	1.40	1.50	0.50	1.15	0.35	0.20	0.35
$y$	6.06	7.00	2.00	5.87	4.00	5.36	7.19	2.50	3.00	8.29	1.37	2.00

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} $$n = 12 , \Sigma x = 9.20 , \Sigma y = 54.64 , \Sigma x ^ { 2 } = 8.9950 , \Sigma y ^ { 2 } = 310.4572 , \Sigma x y = 46.0545$$

It is given that the value of Pearson's product-moment correlation coefficient for the data is 0.381 , correct to 3 significant figures.
1. State what this information tells you about a scatter diagram illustrating the data.
2. Test at the $5 \%$ significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018.
The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J. Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books.

OCR Further Statistics 2021 June Q3

11 marks Standard +0.3

3 The numbers of CD players sold in a shop on three consecutive weekends were 7,6 and 2 . It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by $X$.

How appropriate is the Poisson distribution as a model for $X$ ? Now assume that a Poisson distribution with mean 5 is an appropriate model for $X$.
Find
1. $\mathrm { P } ( X = 6 )$,
2. $\mathrm { P } ( X \geqslant 8 )$. The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution $\operatorname { Po } ( 7.2 )$.
Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive.
State an assumption needed for your answer to part (c) to be valid.
Give a reason why the assumption in part (d) may not be valid in practice.

OCR Further Statistics 2021 June Q4

15 marks Standard +0.8

4 The continuous random variable $X$ has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where $n$ and $k$ are constants and $n$ is an integer greater than 1 .

Find $k$ in terms of $n$.
1. When $n = 4$, find the cumulative distribution function of $X$.
2. Hence determine $\mathrm { P } ( X > 7 \mid X > 5 )$ when $n = 4$.
Determine the values of $n$ for which $\operatorname { Var } ( X )$ is not defined.

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Book	A	B	C	D	E	F	G	H	I	J	K	L
\(x\)	0.95	0.65	0.70	0.90	0.55	1.40	1.50	0.50	1.15	0.35	0.20	0.35
\(y\)	6.06	7.00	2.00	5.87	4.00	5.36	7.19	2.50	3.00	8.29	1.37	2.00