Questions - OCR - A-Level Maths

OCR Further Pure Core 2 2017 Specimen Q3

Find $\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)$.
What does the sum in part (i) tend to as $n \rightarrow \infty$ ? Justify your answer.

OCR Further Pure Core 2 2017 Specimen Q4

4 Express $\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }$ in partial fractions.

OCR Further Pure Core 2 2017 Specimen Q5

5 In this question you must show detailed reasoning. Evaluate $\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x$.
[0pt] [You may use the result $\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0$.]

OCR Further Pure Core 2 2017 Specimen Q6

6 The equation of a plane $\Pi$ is $x - 2 y - z = 30$.

Find the acute angle between the line $\mathbf { r } = \left( \begin{array} { c } 3
2
- 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5
3
2 \end{array} \right)$ and $\Pi$.
Determine the geometrical relationship between the line $\mathbf { r } = \left( \begin{array} { l } 1
4
2 \end{array} \right) + \mu \left( \begin{array} { r } 3
- 1
5 \end{array} \right)$ and $\Pi$.

OCR Further Pure Core 2 2017 Specimen Q7

Use the Maclaurin series for $\sin x$ to work out the series expansion of $\sin x \sin 2 x \sin 4 x$ up to and including the term in $x ^ { 3 }$.
Hence find, in exact surd form, an approximation to the least positive root of the equation $2 \sin x \sin 2 x \sin 4 x = x$.

OCR Further Statistics 2017 Specimen Q2

2 The mass $J \mathrm {~kg}$ of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass Kkg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04.

Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg .
Find the probability that the mass of one bag of King Edward potatoes is more than $75 \%$ of the mass of one bag of Jersey potatoes.

OCR Further Statistics 2017 Specimen Q5

5 The number of goals scored by the home team in a randomly chosen hockey match is denoted by $X$.

In order for $X$ to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for $X$ to be modelled by a Poisson distribution in this context. Assume now that $X$ can be modelled by the distribution $\operatorname { Po } ( 1.9 )$.
(a) Write down an expression for $\mathrm { P } ( X = r )$.
(b) Hence find $\mathrm { P } ( X = 3 )$.
Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean $\lambda$ between 1.31 and 1.32 . Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match.

OCR Further Statistics 2017 Specimen Q6

6 A bag contains 3 green counters, 3 blue counters and $w$ white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn. The total number of counters selected, including the white counter, is denoted by $X$.

In the case when $w = 2$,
(a) write down the distribution of $X$,
(b) find $P ( 3 < X \leq 7 )$.
In the case when $\mathrm { E } ( X ) = 2$, determine the value of $w$.
In the case when $w = 2$ and $X = 6$, find the probability that the first five counters drawn alternate in colour.

OCR Further Statistics 2017 Specimen Q7

7 Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, $x$ metres, of this sample can be summarised by the following. $$\begin{aligned} n & = 49
\Sigma x & = 74.48
\Sigma x ^ { 2 } & = 120.8896 \end{aligned}$$ Test, at the 5\% significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .

OCR Further Mechanics 2017 Specimen Q1

1 A body, $P$, of mass 2 kg moves under the action of a single force $\mathbf { F } \mathrm { N }$. At time $t \mathrm {~s}$, the velocity of the body is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, where $$\mathbf { v } = \left( t ^ { 2 } - 3 \right) \mathbf { i } + \frac { 5 } { 2 t + 1 } \mathbf { j } \text { for } t \geq 2 .$$

Obtain $\mathbf { F }$ in terms of $t$.
Calculate the rate at which the force $\mathbf { F }$ is working at $t = 4$.
By considering the change in kinetic energy of $P$, calculate the work done by the force $\mathbf { F }$ during the time interval $2 \leq t \leq 4$.

OCR Further Mechanics 2017 Specimen Q3

3 A body, $Q$, of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of $Q$. Initially the speed of $Q$ is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t \mathrm {~s}$, the magnitude $F N$ of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4 .$$

Calculate the impulse of the force over the time interval.
Hence find the speed of $Q$ when $t = 4$.

OCR Further Discrete 2017 Specimen Q1

1 Fiona is a mobile hairdresser. One day she needs to visit five clients, A to E, starting and finishing at her own house at F . She wants to find a suitable route that does not involve her driving too far.

Which standard network problem does Fiona need to solve? The shortest distances between clients, in km, are given in the matrix below.
A B C D E
A - 12 8 6 4
B 12 - 10 8 10
C 8 10 - 13 10
D 6 8 13 - 10
E 4 10 10 10 -
Use the copy of the matrix in the Printed Answer Booklet to construct a minimum spanning tree for these five client locations.
State the algorithm you have used, show the order in which you build your tree and give its total weight. Draw your minimum spanning tree. The distance from Fiona's house to each client, in km, is given in the table below.
A B C D E
F 2 11 9 7 5
Use this information together with your answer to part (ii) to find a lower bound for the length of Fiona's route.
(a) Find all the cycles that result from using the nearest neighbour method, starting at F .
(b) Use these to find an upper bound for the length of Fiona's route.
Fiona wants to drive less than 35 km . Using the information in your answers to parts (iii) and (iv) explain whether or not a route exists which is less than 35 km in length.

OCR Further Discrete 2017 Specimen Q2

2 Kirstie has bought a house that she is planning to renovate. She has broken the project into a list of activities and constructed an activity network, using activity on arc.

Activity
$A$	Structural survey
$B$	Replace damp course
$C$	Scaffolding
$D$	Repair brickwork
$E$	Repair roof
$F$	Check electrics
$G$	Replaster walls

Activity
$H$	Planning
$I$	Build extension
$J$	Remodel internal layout
$K$	Kitchens and bathrooms
$L$	Decoration and furnishing
$M$	Landscape garden

\includegraphics[max width=\textwidth, alt={}, center]{27438ff9-40d5-415e-b054-2007ea4dd6b8-03_876_1739_1037_212}

Construct a cascade chart for the project, showing the float for each non-critical activity.
Calculate the float for remodelling the internal layout stating how much of this is independent float and how much is interfering float. Kirstie needs to supervise the project. This means that she cannot allow more than three activities to happen on any day.
Describe how Kirstie should organise the activities so that the project is completed in the minimum project completion time and no more than three activities happen on any day.

OCR Further Discrete 2017 Specimen Q4

4 The table shows the pay-off matrix for player $A$ in a two-person zero-sum game between $A$ and $B$. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Player $A$}

	Strategy $X$	Strategy $Y$	Strategy $Z$
Strategy $P$	4	5	- 4
Strategy $Q$	3	- 1	2
Strategy $R$	4	0	2

\end{table}

Find the play-safe strategy for player $A$ and the play-safe strategy for player $B$. Use the values of the play-safe strategies to determine whether the game is stable or unstable.
If player $B$ knows that player $A$ will use their play-safe strategy, which strategy should player $B$ use?
Suppose that the value in the cell where both players use their play-safe strategies can be changed, but all other entries are unchanged. Show that there is no way to change this value that would make the game stable.
Suppose, instead, that the value in one cell can be changed, but all other entries are unchanged, so that the game becomes stable. Identify a suitable cell and write down a new pay-off value for that cell which would make the game stable.
Show that the zero-sum game with the new pay-off value found in part (iv) has a Nash equilibrium and explain what this means for the players.

OCR Further Additional Pure 2017 Specimen Q1

1 A curve is given by $x = t ^ { 2 } - 2 \ln t , y = 4 t$ for $t > 0$. When the arc of the curve between the points where $t = 1$ and $t = 4$ is rotated through $2 \pi$ radians about the $x$-axis, a surface of revolution is formed with surface area $A$.
Given that $A = k \pi$, where $k$ is an integer, write down an integral which gives $A$ and find the value of $k$.

OCR Further Additional Pure 2017 Specimen Q5

5 In this question you must show detailed reasoning.
It is given that $I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta$ for $n \geq 0$.

Prove that $I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$ for $n \geq 2$.
Evaluate $I _ { 1 }$ and use the reduction formula to determine the exact value of $\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta$.

OCR Further Additional Pure 2017 Specimen Q6

6 A surface $S$ has equation $z = \mathrm { f } ( x , y )$, where $\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y$. It is given that $S$ has a single stationary point, $P$.

Determine the coordinates, and the nature, of $P$.
Find the equation of the tangent plane to $S$ at the point $Q ( 1,2,38 )$.

OCR Further Additional Pure 2017 Specimen Q7

7 In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, $t$ years after their introduction, is denoted by $N _ { t }$. The initial number of breeding pairs is given by $N _ { 0 }$. An initial discrete population model is proposed for $N _ { t }$. $$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$

(a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .
(b) Show that $N _ { t + 1 } - N _ { t } < 150 - N _ { t }$ when $N _ { t }$ lies between 0 and 150 .
(c) Hence determine the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when $N _ { 0 } \in ( 0,150 )$. An alternative discrete population model is proposed for $N _ { t }$. $$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
Given that $N _ { 0 } = 8$, find the value of $N _ { 4 }$ for each of the two models and give a reason why Model II may be considered more suitable.

OCR Further Additional Pure 2017 Specimen Q8

8 The set $X$ consists of all $2 \times 2$ matrices of the form $\left( \begin{array} { r r } x & - y
y & x \end{array} \right)$, where $x$ and $y$ are real numbers which are not both zero.

(a) The matrices $\left( \begin{array} { c c } a & - b
b & a \end{array} \right)$ and $\left( \begin{array} { c c } c & - d
d & c \end{array} \right)$ are both elements of $X$. Show that $\left( \begin{array} { c c } a & - b
b & a \end{array} \right) \left( \begin{array} { c c } c & - d
d & c \end{array} \right) = \left( \begin{array} { c c } p & - q
q & p \end{array} \right)$ for some real numbers $p$ and $q$ to be found in terms of $a , b , c$ and $d$.
(b) Prove by contradiction that $p$ and $q$ are not both zero.
Prove that $X$, under matrix multiplication, forms a group $G$.
[0pt] [You may use the result that matrix multiplication is associative.]
Determine a subgroup of $G$ of order 17.

OCR FM1 AS 2017 Specimen Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-02_810_743_831_644} A smooth wire is shaped into a circle of centre $O$ and radius 0.8 m . The wire is fixed in a vertical plane. A small bead $P$ of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of $4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $O P$ makes an angle $\theta$ with the upward vertical the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram).

Show that $v ^ { 2 } = 33.32 - 15.68 \cos \theta$.
Prove that the bead is never at rest.
Find the maximum value of $v$.
Write down the dimension of density. The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is $0.4 \mathrm {~m} ^ { 2 }$ and the density of the oil is $920 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$ then the period of oscillation of the pump is 0.7 s .
A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, $\rho$, the acceleration due to gravity, $g$, and the surface area, $A$, of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, $T$, is given by $T = C \rho ^ { \alpha } g ^ { \beta } A ^ { \gamma }$ where $C$ is a dimensionless constant.
Use dimensional analysis to find the values of $\alpha , \beta$ and $\gamma$.
Hence give the value of $C$ to 3 significant figures.
Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only $\rho , g$ and $A$. A car of mass 1250 kg experiences a resistance to its motion of magnitude $k v ^ { 2 } \mathrm {~N}$, where $k$ is a constant and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of $P \mathrm {~W}$. At a point $A$ on the road the car's speed is $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and it has an acceleration of magnitude $0.54 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At a point $B$ on the road the car's speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and it has an acceleration of magnitude $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find the values of $k$ and $P$. The power is increased to 15 kW .
Calculate the maximum steady speed of the car on a straight horizontal road.

OCR FM1 AS 2017 Specimen Q5

60 marks

5
\includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-04_221_1233_367_328} The masses of two spheres $A$ and $B$ are $3 m \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively. The spheres are moving towards each other with constant speeds $2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively along the same straight line towards each other on a smooth horizontal surface (see diagram). The two spheres collide and the coefficient of restitution between the spheres is $e$. After colliding, $A$ and $B$ both move in the same direction with speeds $v \mathrm {~ms} ^ { - 1 }$ and $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$, respectively.

Find an expression for $v$ in terms of $e$ and $u$.
Write down unsimplified expressions in terms of $e$ and $u$ for
(a) the total kinetic energy of the spheres before the collision,
(b) the total kinetic energy of the spheres after the collision.
Given that the total kinetic energy of the spheres after the collision is $\lambda$ times the total kinetic energy before the collision, show that $$\lambda = \frac { 27 e ^ { 2 } + 25 } { 52 }$$
Comment on the cases when
(a) $\lambda = 1$,
(b) $\lambda = \frac { 25 } { 52 }$.
\includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-05_789_981_324_543} The fixed points $A$, $B$ and $C$ are in a vertical line with $A$ above $B$ and $B$ above $C$. A particle $P$ of mass 2.5 kg is joined to $A$, to $B$ and to a particle $Q$ of mass 2 kg , by three light rods where the length of rod $A P$ is 1.5 m and the length of rod $P Q$ is 0.75 m . Particle $P$ moves in a horizontal circle with centre $B$. Particle $Q$ moves in a horizontal circle with centre $C$ at the same constant angular speed $\omega$ as $P$, in such a way that $A , B , P$ and $Q$ are coplanar. The rod $A P$ makes an angle of $60 ^ { \circ }$ with the downward vertical, rod $P Q$ makes an angle of $30 ^ { \circ }$ with the downward vertical and rod $B P$ is horizontal (see diagram).
Find the tension in the $\operatorname { rod } P Q$.
Find $\omega$.
Find the speed of $P$.
Find the tension in the $\operatorname { rod } A P$.
Hence find the magnitude of the force in rod $B P$. Decide whether this rod is under tension or compression.

OCR C2 Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-02_579_895_817_625} A sector $O A B$ of a circle of radius $r \mathrm {~cm}$ has angle $\theta$ radians. The length of the arc of the sector is 12 cm and the area of the sector is $36 \mathrm {~cm} ^ { 2 }$ (see diagram).

Write down two equations involving $r$ and $\theta$.
Hence show that $r = 6$, and state the value of $\theta$.
Find the area of the segment bounded by the arc $A B$ and the chord $A B$.

OCR C2 Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-03_314_1089_267_529} In the diagram, $A B C D$ is a quadrilateral in which $A D$ is parallel to $B C$. It is given that $A B = 9 , B C = 6$, $C A = 5$ and $C D = 15$.

Show that $\cos B C A = - \frac { 1 } { 3 }$, and hence find the value of $\sin B C A$.
Find the angle $A D C$ correct to the nearest $0.1 ^ { \circ }$.

OCR C2 Q7

Evaluate $\log _ { 5 } 15 + \log _ { 5 } 20 - \log _ { 5 } 12$.
Given that $y = 3 \times 10 ^ { 2 x }$, show that $x = a \log _ { 10 } ($ by $)$, where the values of the constants $a$ and $b$ are to be found.

	Strategy \(X\)	Strategy \(Y\)	Strategy \(Z\)
Strategy \(P\)	4	5	- 4
Strategy \(Q\)	3	- 1	2
Strategy \(R\)	4	0	2

Activity
\(A\)	Structural survey
\(B\)	Replace damp course
\(C\)	Scaffolding
\(D\)	Repair brickwork
\(E\)	Repair roof
\(F\)	Check electrics
\(G\)	Replaster walls

Activity
\(H\)	Planning
\(I\)	Build extension
\(J\)	Remodel internal layout
\(K\)	Kitchens and bathrooms
\(L\)	Decoration and furnishing
\(M\)	Landscape garden

Questions — OCR (4619 questions)

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OCR Further Pure Core 2 2017 Specimen Q2

OCR Further Pure Core 2 2017 Specimen Q3

OCR Further Pure Core 2 2017 Specimen Q4

OCR Further Pure Core 2 2017 Specimen Q5

OCR Further Pure Core 2 2017 Specimen Q6

OCR Further Pure Core 2 2017 Specimen Q7

OCR Further Statistics 2017 Specimen Q2

OCR Further Statistics 2017 Specimen Q5

OCR Further Statistics 2017 Specimen Q6

OCR Further Statistics 2017 Specimen Q7

OCR Further Mechanics 2017 Specimen Q1

OCR Further Mechanics 2017 Specimen Q3

OCR Further Discrete 2017 Specimen Q1

OCR Further Discrete 2017 Specimen Q2

OCR Further Discrete 2017 Specimen Q4

OCR Further Additional Pure 2017 Specimen Q1

OCR Further Additional Pure 2017 Specimen Q5

OCR Further Additional Pure 2017 Specimen Q6

OCR Further Additional Pure 2017 Specimen Q7

OCR Further Additional Pure 2017 Specimen Q8

OCR FM1 AS 2017 Specimen Q2

OCR FM1 AS 2017 Specimen Q5

OCR C2 Q2

OCR C2 Q4

OCR C2 Q7