Questions - OCR MEI - A-Level Maths

OCR MEI M1 Q1

1 A train consists of a locomotive pulling 17 identical trucks. The mass of the locomotive is 120 tonnes and the mass of each truck is 40 tonnes. The locomotive gives a driving force of 121000 N . The resistance to motion on each truck is $R \mathrm {~N}$ and the resistance on the locomotive is $5 R \mathrm {~N}$.
Initially the train is travelling on a straight horizontal track and its acceleration is $0.11 \mathrm {~ms} ^ { - 2 }$.

Show that $R = 1500$.
Find the tensions in the couplings between
(A) the last two trucks,
(B) the locomotive and the first truck. The train now comes to a place where the track goes up a straight, uniform slope at an angle $\alpha$ with the horizontal, where $\sin \alpha = \frac { 1 } { 80 }$. The driving force and the resistance forces remain the same as before.
Find the magnitude and direction of the acceleration of the train. The train then comes to a straight uniform downward slope at an angle $\beta$ to the horizontal.
The driver of the train reduces the driving force to zero and the resistance forces remain the same as before. The train then travels at a constant speed down the slope.
Find the value of $\beta$.

OCR MEI M1 Q2

2 In this question the value of $g$ should be taken as $10 \mathrm {~m \mathrm {~s} ^ { 2 }$.} As shown in Fig. 8, particles A and B are projected towards one another. Each particle has an initial speed of $10 \mathrm {~m} \mathrm {~s} ^ { 1 }$ vertically and $20 \mathrm {~m} \mathrm {~s} { } ^ { 1 }$ horizontally. Initially A and B are 70 m apart horizontally and B is 15 m higher than A . Both particles are projected over horizontal ground. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{362d5995-bd39-4b07-b6a4-63eb1dd3e69d-2_461_1114_464_505} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Show that, $t$ seconds after projection, the height in metres of each particle above its point of projection is $10 t - 5 t ^ { 2 }$.
Calculate the horizontal range of A . Deduce that A hits the horizontal ground between the initial positions of A and B .
Calculate the horizontal distance travelled by B before reaching the ground.
Show that the paths of the particles cross but that the particles do not collide if they are projected at the same time. In fact, particle A is projected 2 seconds after particle B .
Verify that the particles collide 0.75 seconds after A is projected.

OCR MEI FP2 2006 June Q5

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.

OCR MEI C1 2006 June Q1

1 The volume of a cone is given by the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$. Make $r$ the subject of this formula.

OCR MEI C1 2006 June Q2

2 One root of the equation $x ^ { 3 } + a x ^ { 2 } + 7 = 0$ is $x = - 2$. Find the value of $a$.

OCR MEI C1 2006 June Q3

3 A line has equation $3 x + 2 y = 6$. Find the equation of the line parallel to this which passes through the point $( 2,10 )$.

OCR MEI C1 2006 June Q4

4 In each of the following cases choose one of the statements $$\mathrm { P } \Rightarrow \mathrm { Q } \quad \mathrm { P } \Leftrightarrow \mathrm { Q } \quad \mathrm { P } \Leftarrow \mathrm { Q }$$ to describe the complete relationship between P and Q .

P: $x ^ { 2 } + x - 2 = 0$ Q: $x = 1$
$\mathrm { P } : y ^ { 3 } > 1$ Q: $y > 1$

OCR MEI C1 2006 June Q5

5 Find the coordinates of the point of intersection of the lines $y = 3 x + 1$ and $x + 3 y = 6$.

OCR MEI C1 2006 June Q6

6 Solve the inequality $x ^ { 2 } + 2 x < 3$.

OCR MEI C1 2006 June Q7

7

Simplify $6 \sqrt { 2 } \times 5 \sqrt { 3 } - \sqrt { 24 }$.
Express $( 2 - 3 \sqrt { 5 } ) ^ { 2 }$ in the form $a + b \sqrt { 5 }$, where $a$ and $b$ are integers.

OCR MEI C1 2006 June Q8

8 Calculate ${ } ^ { 6 } \mathrm { C } _ { 3 }$.
Find the coefficient of $x ^ { 3 }$ in the expansion of $( 1 - 2 x ) ^ { 6 }$.

OCR MEI C1 2006 June Q9

9 Simplify the following.

$\frac { 16 ^ { \frac { 1 } { 2 } } } { 81 ^ { \frac { 3 } { 4 } } }$
$\frac { 12 \left( a ^ { 3 } b ^ { 2 } c \right) ^ { 4 } } { 4 a ^ { 2 } c ^ { 6 } }$

OCR MEI C1 2006 June Q10

10 Find the coordinates of the points of intersection of the circle $x ^ { 2 } + y ^ { 2 } = 25$ and the line $y = 3 x$. Give your answers in surd form.

OCR MEI C1 2006 June Q12

12 You are given that $\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12$.

Show that $x = - 2$ is a root of $\mathrm { f } ( x ) = 0$.
Divide $\mathrm { f } ( x )$ by $x + 6$.
Express $\mathrm { f } ( x )$ in fully factorised form.
Sketch the graph of $y = \mathrm { f } ( x )$.
Solve the equation $\mathrm { f } ( x ) = 12$.

OCR MEI C1 2006 June Q13

13 Answer the whole of this question on the insert provided.
The insert shows the graph of $y = \frac { 1 } { x } , x \neq 0$.

Use the graph to find approximate roots of the equation $\frac { 1 } { x } = 2 x + 3$, showing your method clearly.
Rearrange the equation $\frac { 1 } { x } = 2 x + 3$ to form a quadratic equation. Solve the resulting equation, leaving your answers in the form $\frac { p \pm \sqrt { q } } { r }$.
Draw the graph of $y = \frac { 1 } { x } + 2 , x \neq 0$, on the grid used for part (i).
Write down the values of $x$ which satisfy the equation $\frac { 1 } { x } + 2 = 2 x + 3$.

OCR MEI C2 2008 June Q1

1 Express $\frac { 7 \pi } { 6 }$ radians in degrees.

OCR MEI C2 2008 June Q3

3 State the transformation which maps the graph of $y = x ^ { 2 } + 5$ onto the graph of $y = 3 x ^ { 2 } + 15$.

OCR MEI C2 2008 June Q4

4 Use calculus to find the set of values of $x$ for which $\mathrm { f } ( x ) = 12 x - x ^ { 3 }$ is an increasing function.

OCR MEI C2 2008 June Q5

5 In Fig. 5, A and B are the points on the curve $y = 2 ^ { x }$ with $x$-coordinates 3 and 3.1 respectively. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-2_700_728_1197_705} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure}

Find the gradient of the chord AB . Give your answer correct to 2 decimal places.
Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to $y = 2 ^ { x }$ at A .

OCR MEI C2 2008 June Q6

6 A curve has gradient given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x }$. Find the equation of the curve, given that it passes through the point $( 9,105 )$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-3_420_522_264_810} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure} A sector of a circle of radius 6 cm has angle 1.6 radians, as shown in Fig. 7.
Find the area of the sector.
Hence find the area of the shaded segment.

OCR MEI C2 2008 June Q8

5 marks

8 The 11th term of an arithmetic progression is 1 . The sum of the first 10 terms is 120 . Find the 4th term. [5]

OCR MEI C2 2008 June Q9

9 Use logarithms to solve the equation $5 ^ { x } = 235$, giving your answer correct to 2 decimal places.

OCR MEI C2 2008 June Q10

10 Showing your method, solve the equation $2 \sin ^ { 2 } \theta = \cos \theta + 2$ for values of $\theta$ between $0 ^ { \circ }$ and $360 ^ { \circ }$.

OCR MEI C2 2008 June Q11

11 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-4_1022_942_356_603} \captionsetup{labelformat=empty} \caption{Fig. 11}

\end{figure} Fig. 11 shows a sketch of the cubic curve $y = \mathrm { f } ( x )$. The values of $x$ where it crosses the $x$-axis are - 5 , - 2 and 2 , and it crosses the $y$-axis at $( 0 , - 20 )$.

Express $\mathrm { f } ( x )$ in factorised form.
Show that the equation of the curve may be written as $y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20$.
Use calculus to show that, correct to 1 decimal place, the $x$-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
State, correct to 1 decimal place, the coordinates of the maximum point on the curve $y = \mathrm { f } ( 2 x )$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-5_689_1006_269_568} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough.
A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation $y = 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15$, for $0 \leqslant x \leqslant 0.5$. Calculate $\int _ { 0 } ^ { 0.5 } \left( 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15 \right) \mathrm { d } x$ and state what this represents.
Hence find the volume of water in the trough as given by this model.

OCR MEI C2 2008 June Q13

13 The percentage of the adult population visiting the cinema in Great Britain has tended to increase since the 1980s. The table shows the results of surveys in various years.

Year

$1986 / 87$

$1991 / 92$

$1996 / 97$

$1999 / 00$

$2000 / 01$

$2001 / 02$

Percentage of the

adult population

visiting the cinema

31

44

54

56

55

57

Source: Department of National Statistics, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
This growth may be modelled by an equation of the form $$P = a t ^ { b } ,$$ where $P$ is the percentage of the adult population visiting the cinema, $t$ is the number of years after the year 1985/86 and $a$ and $b$ are constants to be determined.

Show that, according to this model, the graph of $\log _ { 10 } P$ against $\log _ { 10 } t$ should be a straight line of gradient $b$. State, in terms of $a$, the intercept on the vertical axis. \section*{Answer part (ii) of this question on the insert provided.}
Complete the table of values on the insert, and plot $\log _ { 10 } P$ against $\log _ { 10 } t$. Draw by eye a line of best fit for the data.
Use your graph to find the equation for $P$ in terms of $t$.
Predict the percentage of the adult population visiting the cinema in the year 2007/2008 (i.e. when $t = 22$ ), according to this model.

Questions — OCR MEI (4301 questions)

Browse by module

Browse by board