Questions - OCR - A-Level Maths

Verify that when $t = 2$ the particle is at rest at the point $O$.
Calculate the acceleration of the particle when $t = 2$.

OCR M1 2008 January Q5

Moderate -0.3

5 A car is towing a trailer along a straight road using a light tow-bar which is parallel to the road. The masses of the car and the trailer are 900 kg and 250 kg respectively. The resistance to motion of the car is 600 N and the resistance to motion of the trailer is 150 N .

At one stage of the motion, the road is horizontal and the pulling force exerted on the trailer is zero.
(a) Show that the acceleration of the trailer is $- 0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
(b) Find the driving force exerted by the car.
(c) Calculate the distance required to reduce the speed of the car and trailer from $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
At another stage of the motion, the car and trailer are moving down a slope inclined at $3 ^ { \circ }$ to the horizontal. The resistances to motion of the car and trailer are unchanged. The driving force exerted by the car is 980 N . Find
(a) the acceleration of the car and trailer,
(b) the pulling force exerted on the trailer.

OCR M1 2008 January Q6

Standard +0.3

6 A block of weight 14.7 N is at rest on a horizontal floor. A force of magnitude 4.9 N is applied to the block.

The block is in limiting equilibrium when the 4.9 N force is applied horizontally. Show that the coefficient of friction is $\frac { 1 } { 3 }$.
\includegraphics[max width=\textwidth, alt={}, center]{db77a63a-6ff8-4fe5-bdd0-15afb7eb4866-3_278_657_552_785} When the force of 4.9 N is applied at an angle of $30 ^ { \circ }$ above the horizontal, as shown in the diagram, the block moves across the floor. Calculate
(a) the vertical component of the contact force between the floor and the block, and the magnitude of the frictional force,
(b) the acceleration of the block.
Calculate the magnitude of the frictional force acting on the block when the 4.9 N force acts at an angle of $30 ^ { \circ }$ to the upward vertical, justifying your answer fully.

OCR M1 2008 January Q7

Standard +0.3

7
\includegraphics[max width=\textwidth, alt={}, center]{db77a63a-6ff8-4fe5-bdd0-15afb7eb4866-4_419_419_274_735} Particles $A$ and $B$ are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are released from rest, with the string taut, and $A$ and $B$ at the same height above a horizontal floor (see diagram). In the subsequent motion, $A$ descends with acceleration $1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and strikes the floor 0.8 s after being released. It is given that $B$ never reaches the pulley.

Calculate the distance $A$ moves before it reaches the floor and the speed of $A$ immediately before it strikes the floor.
Show that $B$ rises a further 0.064 m after $A$ strikes the floor, and calculate the total length of time during which $B$ is rising.
Sketch the ( $t , v$ ) graph for the motion of $B$ from the instant it is released from rest until it reaches a position of instantaneous rest.
Before $A$ strikes the floor the tension in the string is 5.88 N . Calculate the mass of $A$ and the mass of $B$.
The pulley has mass 0.5 kg , and is held in a fixed position by a light vertical chain. Calculate the tension in the chain
(a) immediately before $A$ strikes the floor,
(b) immediately after $A$ strikes the floor. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

OCR C2 Q1

Easy -1.2

A geometric progression has first term 75 and second term - 15 .
1. Find the common ratio.
2. Find the sum to infinity.
3. Find the area of the finite region enclosed by the curve $y = 5 x - x ^ { 2 }$ and the $x$-axis.
4. During one day, a biological culure is allowed to grow under controlled conditions. At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that $t$ hours after 8 a.m., the number of bacteria present, $N$, is given by
$$N = 20000 \times ( 1.06 ) ^ { t }$$ Using this model,
find the number of bacteria present at 11 a.m.,
find, to the nearest minute, the time when the initial number of bacteria will have doubled.

OCR C2 Q4

Moderate -0.8

4.
\includegraphics[max width=\textwidth, alt={}, center]{fe47eac1-645a-46c6-a2b9-c4ad0bcaa538-1_433_844_1416_575} The diagram shows the curve with equation $y = \left( x - \log _ { 10 } x \right) ^ { 2 } , x > 0$.

Copy and complete the table below for points on the curve, giving the $y$ values to 2 decimal places.
$x$ 2 3 4 5 6
$y$ 2.89 6.36
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 2$ and $x = 6$.
Use the trapezium rule with all the values in your table to estimate the area of the shaded region.
State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area.

OCR C2 Q5

Moderate -0.3

5. (i) Given that $\sin \theta = 2 - \sqrt { 2 }$, find the value of $\cos ^ { 2 } \theta$ in the form $a + b \sqrt { 2 }$ where $a$ and $b$ are integers.
(ii) Find, in terms of $\pi$, all values of $x$ in the interval $0 \leq x < \pi$ for which $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$

OCR C2 Q6

Standard +0.3

\includegraphics[max width=\textwidth, alt={}]{fe47eac1-645a-46c6-a2b9-c4ad0bcaa538-2_337_896_694_452}

The diagram shows triangle $A B C$ in which $A C = 8 \mathrm {~cm}$ and $\angle B A C = \angle B C A = 30 ^ { \circ }$.

Find the area of triangle $A B C$ in the form $k \sqrt { 3 }$. The point $M$ is the mid-point of $A C$ and the points $N$ and $O$ lie on $A B$ and $B C$ such that $M N$ and $M O$ are arcs of circles with centres $A$ and $C$ respectively.
Show that the area of the shaded region $B N M O$ is $\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }$.

OCR C2 Q7

Moderate -0.8

7. (i) Expand $( 2 + x ) ^ { 4 }$ in ascending powers of $x$, simplifying each coefficient.
(ii) Find the integers $A , B$ and $C$ such that $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } \equiv A + B x ^ { 2 } + C x ^ { 4 }$$ (iii) Find the real values of $x$ for which $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } = 136$$

OCR C2 Q8

Moderate -0.8

(i) The gradient of a curve is given by

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0$$ Find an equation for the curve given that it passes through the point $( 2,6 )$.
(ii) Show that $$\int _ { 2 } ^ { 3 } \left( 6 \sqrt { x } - \frac { 4 } { \sqrt { x } } \right) d x = k \sqrt { 3 }$$ where $k$ is an integer to be found.

OCR C2 Q9

Standard +0.3

9. The polynomial $\mathrm { f } ( x )$ is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where $k$ is a constant.
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ) the remainder is $r$.
When $\mathrm { f } ( x )$ is divided by $( x - 3 )$ the remainder is $3 r$.

Find the value of $k$.
Find the value of $r$.
Show that $( x - 5 )$ is a factor of $\mathrm { f } ( x )$.
Show that there is only one real solution to the equation $\mathrm { f } ( x ) = 0$.

OCR M1 2009 January Q1

Moderate -0.3

1
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_227_878_269_635} A particle $P$ of mass 0.5 kg is travelling with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a smooth horizontal plane towards a stationary particle $Q$ of mass $m \mathrm {~kg}$ (see diagram). The particles collide, and immediately after the collision $P$ has speed $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $Q$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Given that both particles are moving in the same direction after the collision, calculate $m$.
Given instead that the particles are moving in opposite directions after the collision, calculate $m$.

OCR M1 2009 January Q2

Moderate -0.8

2 A trailer of mass 500 kg is attached to a car of mass 1250 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road. The resistance to motion of the trailer is 400 N and the resistance to motion of the car is 900 N . Find both the tension in the tow-bar and the driving force of the car in each of the following cases.

The car and trailer are travelling at constant speed.
The car and trailer have acceleration $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

OCR M1 2009 January Q3

Moderate -0.8

3
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_570_679_1512_731} Three horizontal forces act at the point $O$. One force has magnitude 7 N and acts along the positive $x$-axis. The second force has magnitude 9 N and acts along the positive $y$-axis. The third force has magnitude 5 N and acts at an angle of $30 ^ { \circ }$ below the negative $x$-axis (see diagram).

Find the magnitudes of the components of the 5 N force along the two axes.
Calculate the magnitude of the resultant of the three forces. Calculate also the angle the resultant makes with the positive $x$-axis.

OCR M1 2009 January Q4

Moderate -0.3

4
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_200_897_269_625} A block of mass 3 kg is placed on a horizontal surface. A force of magnitude 20 N acts downwards on the block at an angle of $30 ^ { \circ }$ to the horizontal (see diagram).

Given that the surface is smooth, calculate the acceleration of the block.
Given instead that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the surface.

OCR M1 2009 January Q5

Moderate -0.3

5 A car is travelling at $13 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a straight road when it passes a point $A$ at time $t = 0$, where $t$ is in seconds. For $0 \leqslant t \leqslant 6$, the car accelerates at $0.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Calculate the speed of the car when $t = 6$.
Calculate the displacement of the car from $A$ when $t = 6$.
Three $( t , x )$ graphs are shown below, for $0 \leqslant t \leqslant 6$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_458_1366_340} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_460_1366_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_384_461_1366_1420} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} (a) State which of these three graphs is most appropriate to represent the motion of the car.
(b) For each of the two other graphs give a reason why it is not appropriate to represent the motion of the car.

OCR M1 2009 January Q6

Moderate -0.3

6 Small parcels are being loaded onto a trolley. Initially the parcels are 2.5 m above the trolley.

A parcel is released from rest and falls vertically onto the trolley. Calculate
(a) the time taken for a parcel to fall onto the trolley,
(b) the speed of a parcel when it strikes the trolley.
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_327_723_603_751} Parcels are often damaged when loaded in the way described, so a ramp is constructed down which parcels can slide onto the trolley. The ramp makes an angle of $60 ^ { \circ }$ to the vertical, and the coefficient of friction between the ramp and a parcel is 0.2 . A parcel of mass 2 kg is released from rest at the top of the ramp (see diagram). Calculate the speed of the parcel after sliding down the ramp.

OCR M1 2009 January Q7

Standard +0.3

7
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_227_901_1352_623} Two particles $P$ and $Q$ have masses 0.7 kg and 0.3 kg respectively. $P$ and $Q$ are simultaneously projected towards each other in the same straight line on a horizontal surface with initial speeds of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively (see diagram). Before $P$ and $Q$ collide the only horizontal force acting on each particle is friction and each particle decelerates at $0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The particles coalesce when they collide.

Given that $P$ and $Q$ collide 2 s after projection, calculate the speed of each particle immediately before the collision, and the speed of the combined particle immediately after the collision.
Given instead that $P$ and $Q$ collide 3 s after projection,
(a) sketch on a single diagram the $( t , v )$ graphs for the two particles in the interval $0 \leqslant t < 3$,
(b) calculate the distance between the two particles at the instant when they are projected.

OCR C2 Q1

Moderate -0.3

Solve the equation

$$\log _ { 5 } ( 4 x + 3 ) - \log _ { 5 } ( x - 1 ) = 2$$

OCR C2 Q2

Moderate -0.8

Find the coefficient of $x ^ { 2 }$ in the expansion of

$$( 1 + x ) ( 1 - x ) ^ { 6 }$$

OCR C2 Q3

Moderate -0.8

(i) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (ii) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where $k$ is a rational constant to be found.

OCR C2 Q4

Moderate -0.3

4. The diagram shows triangle $P Q R$ in which $P Q = 7$ and $P R = 3 \sqrt { 5 }$.
Given that $\sin ( \angle Q P R ) = \frac { 2 } { 3 }$ and that $\angle Q P R$ is acute,

find the exact value of $\cos ( \angle Q P R )$ in its simplest form,
show that $Q R = 2 \sqrt { 6 }$,
find $\angle P Q R$ in degrees to 1 decimal place.

OCR C2 Q5

Moderate -0.3

5. (i) Find $$\int \left( 8 x - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x$$ The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$ and the curve passes through the point $( 1,1 )$.
(ii) Show that the equation of the curve can be written in the form $$y = \left( a x + \frac { b } { x } \right) ^ { 2 }$$ where $a$ and $b$ are integers to be found.

OCR C2 Q6

Moderate -0.3

6. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6$$ and that $x = - 3$ is a solution to the equation $\mathrm { f } ( x ) = 0$,

find the value of the constant $p$,
show that when $\mathrm { f } ( x )$ is divided by $( x - 2 )$ there is a remainder of 50 ,
find the other solutions to the equation $\mathrm { f } ( x ) = 0$, giving your answers to 2 decimal places.

OCR C2 Q7

Standard +0.3

7. The second and third terms of a geometric series are $\log _ { 3 } 4$ and $\log _ { 3 } 16$ respectively.

Find the common ratio of the series.
Show that the first term of the series is $\log _ { 3 } 2$.
Find, to 3 significant figures, the sum of the first six terms of the series.

\(x\)	2	3	4	5	6
\(y\)	2.89	6.36

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