Questions - A-Level Maths

OCR Further Mechanics 2017 Specimen Q1

9 marks Standard +0.8

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

5 marks Standard +0.3

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 FM1 AS 2017 Specimen Q2

7 marks Standard +0.3

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

15 marks Standard +0.3

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
1. the total kinetic energy of the spheres before the collision,
2. the total kinetic energy of the spheres after the collision.
3. Given that the total kinetic energy of the spheres after the collision is $\lambda$ times the total kinetic energy before the collision, show that $$\lambda = \frac { 27 e ^ { 2 } + 25 } { 52 }$$
4. 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).
  1. Find the tension in the $\operatorname { rod } P Q$.
  2. Find $\omega$.
  3. Find the speed of $P$.
  4. Find the tension in the $\operatorname { rod } A P$.
  5. Hence find the magnitude of the force in rod $B P$. Decide whether this rod is under tension or compression.

OCR MEI FP2 2006 June Q5

18 marks Challenging +1.2

5 A curve has parametric equations $$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$ where $k$ is a positive constant.

For the case $k = 1$, use your graphical calculator to sketch the curve. Describe its main features.
Sketch the curve for a value of $k$ between 0 and 1 . Describe briefly how the main features differ from those for the case $k = 1$.
For the case $k = 2$ :
(A) sketch the curve;
(B) find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$;
(C) show that the width of each loop, measured parallel to the $x$-axis, is $$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$
Use your calculator to find, correct to one decimal place, the value of $k$ for which successive loops just touch each other.

CAIE S1 2021 November Q1

5 marks Easy -1.8

1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.

	Piano	Guitar	Drums
Male	25	44	11
Female	42	38	20

A student at the college is chosen at random.

Find the probability that the student plays the guitar.
Find the probability that the student is male given that the student plays the drums.
Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.

CAIE S1 2021 November Q2

5 marks Moderate -0.3

2 A group of 6 people is to be chosen from 4 men and 11 women.

In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
Two of the 11 women are sisters Jane and Kate.
In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?

CAIE S1 2021 November Q3

7 marks Moderate -0.8

3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.

Show that the probability that exactly one of the marbles is yellow is $\frac { 5 } { 14 }$.
The random variable $X$ is the number of yellow marbles selected.
Draw up the probability distribution table for $X$.
Find $\mathrm { E } ( X )$.

CAIE S1 2021 November Q4

6 marks Standard +0.3

4

In how many different ways can the 9 letters of the word TELESCOPE be arranged?
In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?

CAIE S1 2021 November Q5

7 marks Moderate -0.8

5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.

Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
Find the probability that the first wet day in October is 8 October.
For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.

CAIE S1 2021 November Q6

10 marks Moderate -0.8

6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.

Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
$20 \%$ of employees take longer than $t$ minutes to complete the task. Find the value of $t$.
Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.

CAIE S1 2021 November Q8

Easy -1.8

8	MATHEMATICS	9709/52
0	Paper 5 Probability \	Statistics 1	October/November 2021
$\infty$		1 hour 15 minutes
	You must answer on the question paper.
	You will need: List of formulae (MF19)

\section*{INSTRUCTIONS}

Answer all questions.
Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
Write your name, centre number and candidate number in the boxes at the top of the page.
Write your answer to each question in the space provided.
Do not use an erasable pen or correction fluid.
Do not write on any bar codes.
If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown.
You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

\section*{INFORMATION}

The total mark for this paper is 50.
The number of marks for each question or part question is shown in brackets [ ].

1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.

	Piano	Guitar	Drums
Male	25	44	11
Female	42	38	20

A student at the college is chosen at random.

Find the probability that the student plays the guitar.
Find the probability that the student is male given that the student plays the drums.
Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
2 A group of 6 people is to be chosen from 4 men and 11 women.
1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
  Two of the 11 women are sisters Jane and Kate.
2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
  3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
3. Show that the probability that exactly one of the marbles is yellow is $\frac { 5 } { 14 }$.
  The random variable $X$ is the number of yellow marbles selected.
4. Draw up the probability distribution table for $X$.
5. Find $\mathrm { E } ( X )$.
  4
6. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
7. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
  5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.
8. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
9. Find the probability that the first wet day in October is 8 October.
10. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.
  6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.
11. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
12. $20 \%$ of employees take longer than $t$ minutes to complete the task. Find the value of $t$.
13. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.
  7 The distances, $x \mathrm {~m}$, travelled to school by 140 children were recorded. The results are summarised in the table below.
  Distance, $x \mathrm {~m}$ $x \leqslant 200$ $x \leqslant 300$ $x \leqslant 500$ $x \leqslant 900$ $x \leqslant 1200$ $x \leqslant 1600$
  Cumulative frequency 16 46 88 122 134 140
14. On the grid, draw a cumulative frequency graph to represent these results. \includegraphics[max width=\textwidth, alt={}, center]{93ff111b-0267-4b4b-a41c-64c3307115af-10_1593_1593_701_306}
15. Use your graph to estimate the interquartile range of the distances.
16. Calculate estimates of the mean and standard deviation of the distances.
  If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

Edexcel C1 Q1

Easy -1.3

(a) Write down the value of $8 ^ { \frac { 1 } { 3 } }$.
(b) Find the value of $8 ^ { - \frac { 2 } { 3 } }$.
Given that $y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0$,
(a) find $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
(b) find $\int y \mathrm {~d} x$.

Edexcel C1 Q2

Moderate -0.8

2. The sequence of positive numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1$$

Find $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
Write down the value of $u _ { 20 }$.

Edexcel C1 Q4

Moderate -0.8

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-006_689_920_292_511}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve passes through the origin $O$ and through the point $( 6,0 )$. The maximum point on the curve is $( 3,5 )$. On separate diagrams, sketch the curve with equation

$y = 3 \mathrm { f } ( x )$,
$y = \mathrm { f } ( x + 2 )$. On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the $x$-axis.

Edexcel C1 Q5

Moderate -0.5

5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$

Edexcel C1 Q8

Moderate -0.8

8. The line $l _ { 1 }$ passes through the point $( 9 , - 4 )$ and has gradient $\frac { 1 } { 3 }$.

Find an equation for $l _ { 1 }$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The line $l _ { 2 }$ passes through the origin $O$ and has gradient - 2 . The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.
Calculate the coordinates of $P$. Given that $l _ { 1 }$ crosses the $y$-axis at the point $C$,
calculate the exact area of $\triangle O C P$. \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882} \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}

Edexcel C1 Q9

Moderate -0.8

9. An arithmetic series has first term $a$ and common difference $d$.

Prove that the sum of the first $n$ terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of $n$ months. His monthly repayments form an arithmetic sequence. He repays $\pounds 149$ in the first month, $\pounds 147$ in the second month, $\pounds 145$ in the third month, and so on. He makes his final repayment in the $n$th month, where $n > 21$.
Find the amount Sean repays in the 21st month. Over the $n$ months, he repays a total of $\pounds 5000$.
Form an equation in $n$, and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
Solve the equation in part (c).
State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

Edexcel C3 Q1

8 marks Standard +0.3

Given that $\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1$, show that $1 + \tan ^ { 2 } \theta \equiv \sec ^ { 2 } \theta$.
Solve, for $0 \leq \theta < 360 ^ { \circ }$, the equation $$2 \tan ^ { 2 } \theta + \sec \theta = 1 ,$$ giving your answers to 1 decimal place.

Edexcel C3 Q2

10 marks Moderate -0.3

2. (a) Differentiate with respect to $x$

$3 \sin ^ { 2 } x + \sec 2 x$,
$\{ x + \ln ( 2 x ) \} ^ { 3 }$. Given that $y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , x \neq 1$,
(b) show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }$.

Edexcel C3 Q3

12 marks Standard +0.3

3. The function $f$ is defined by $$f : x \mapsto \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$

Show that $\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1$.
Find $\mathrm { f } ^ { - 1 } ( x )$. The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } + 5 , \quad x \in \mathbb { R } .$$ (b) Solve $\mathrm { fg } ( x ) = \frac { 1 } { 4 }$.

Edexcel C3 Q4

10 marks Standard +0.3

4. $$\mathrm { f } ( x ) = 3 \mathrm { e } ^ { x } - \frac { 1 } { 2 } \ln x - 2 , \quad x > 0 .$$

Differentiate to find $\mathrm { f } ^ { \prime } ( x )$. The curve with equation $y = \mathrm { f } ( x )$ has a turning point at $P$. The $x$-coordinate of $P$ is $\alpha$.
Show that $\alpha = \frac { 1 } { 6 } \mathrm { e } ^ { - \alpha }$. The iterative formula $$x _ { n + 1 } = \frac { 1 } { 6 } \mathrm { e } ^ { - x _ { n } } , \quad x _ { 0 } = 1$$ is used to find an approximate value for $\alpha$.
Calculate the values of $x _ { 1 } , x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving your answers to 4 decimal places.
By considering the change of sign of $\mathrm { f } ^ { \prime } ( x )$ in a suitable interval, prove that $\alpha = 0.1443$ correct to 4 decimal places.

Edexcel C3 Q5

8 marks Standard +0.3

5.

Using the identity $\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$, prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$
Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$
Express $4 \cos \theta + 6 \sin \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
Hence, for $0 \leq \theta < \pi$, solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
Hence, for $0 \leq \theta < \pi$, solve \includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.

Edexcel C3 Q6

15 marks Standard +0.3

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_371_643_338_1852}

\end{figure} Figure 1 shows part of the graph of $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The graph consists of two line segments that meet at the point $( 1 , a ) , a < 0$. One line meets the $x$-axis at $( 3,0 )$. The other line meets the $x$-axis at $( - 1,0 )$ and the $y$-axis at $( 0 , b ) , b < 0$. In separate diagrams, sketch the graph with equation

$y = \mathrm { f } ( x + 1 )$,
$y = \mathrm { f } ( | x | )$. Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
Given that $\mathrm { f } ( x ) = | x - 1 | - 2$, find
the value of $a$ and the value of $b$,
the value of $x$ for which $\mathrm { f } ( x ) = 5 x$.

Edexcel C3 Q7

11 marks Standard +0.8

7. A particular species of orchid is being studied. The population $p$ at time $t$ years after the study started is assumed to be $$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

show that $a = 0.12$,
use the equation with $a = 0.12$ to predict the number of years before the population of orchids reaches 1850 .
Show that $p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }$.
Hence show that the population cannot exceed 2800.

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OCR Further Mechanics 2017 Specimen Q1

OCR Further Mechanics 2017 Specimen Q3

OCR FM1 AS 2017 Specimen Q2

OCR FM1 AS 2017 Specimen Q5

OCR MEI FP2 2006 June Q5

CAIE S1 2021 November Q1

CAIE S1 2021 November Q2

CAIE S1 2021 November Q3

CAIE S1 2021 November Q4

CAIE S1 2021 November Q5

CAIE S1 2021 November Q6

CAIE S1 2021 November Q8

Edexcel C1 Q1

Edexcel C1 Q2

Edexcel C1 Q4

Edexcel C1 Q5

Edexcel C1 Q8

Edexcel C1 Q9

Edexcel C3 Q1

Edexcel C3 Q2

Edexcel C3 Q3

Edexcel C3 Q4

Edexcel C3 Q5

Edexcel C3 Q6

Edexcel C3 Q7

Distance, \(x \mathrm {~m}\)	\(x \leqslant 200\)	\(x \leqslant 300\)	\(x \leqslant 500\)	\(x \leqslant 900\)	\(x \leqslant 1200\)	\(x \leqslant 1600\)
Cumulative frequency	16	46	88	122	134	140