Questions - OCR PURE - A-Level Maths

Sketch a velocity-time graph to represent the cyclist's motion.
Find the acceleration during the first 25 seconds of the cyclist's motion. The cyclist takes $T$ seconds to decelerate from $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ until coming to rest.
Determine the value of $T$.
Determine the average speed for the cyclist's journey.

OCR PURE Q11

11
\includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-08_451_1340_251_244} A train consists of an engine $A$ of mass 50000 kg and a carriage $B$ of mass 20000 kg . The engine and carriage are connected by a rigid coupling. The coupling is modelled as light and horizontal. The resistances to motion acting on $A$ and $B$ are 9000 N and 1250 N respectively (see diagram).
The train passes through station $P$ with speed $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves along a straight horizontal track with constant acceleration $0.01 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ towards station $Q$. The distance between $P$ and $Q$ is 12.95 km .

Determine the time, in minutes, to travel between $P$ and $Q$. For the train's motion between $P$ and $Q$, determine the following.
The driving force of the engine.
The tension in the coupling between $A$ and $B$.

OCR PURE Q12

12
\includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-09_647_935_260_242} A particle $P$ moves in a straight line. At time $t$ seconds, where $t \geqslant 0$, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It is given that $v = - 3 t ^ { 2 } + 24 t + k$, where $k$ is a positive constant. The diagram shows the velocity-time graph for the motion of $P$.
$P$ attains its maximum velocity at time $T$ seconds. Given that the distance travelled by $P$ between times $t = 1$ and $t = T$ is 297 m , determine the time when $P$ is instantaneously at rest. \section*{END OF QUESTION PAPER}

OCR PURE Q1

1 Write the solution of the inequality $( x - 2 ) ( x + 3 ) > 0$ using set notation.

OCR PURE Q3

3 Give a counter example to disprove the following statement.
If $x$ and $y$ are both irrational then $x + y$ is irrational.

OCR PURE Q4

4 The circle $x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + k = 0$ has radius 5.
Determine the value of $k$.

OCR PURE Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-04_700_727_260_242} The diagram shows a curve $C$ for which $y$ is inversely proportional to $x$. The curve passes through the point $\left( 1 , - \frac { 1 } { 2 } \right)$.

1. Determine the equation of the gradient function for the curve $C$.
2. Sketch this gradient function on the axes in the Printed Answer Booklet.
The diagram indicates that the curve $C$ has no stationary points. State what feature of your sketch in part (a)(ii) corresponds to this.
The curve $C$ is translated by the vector $\binom { - 2 } { 0 }$. Find the equation of the curve after it has been translated.

OCR PURE Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-05_538_531_264_246} The shape $A B C$ shown in the diagram is a student's design for the sail of a small boat.
The curve $A C$ has equation $y = 2 \log _ { 2 } x$ and the curve $B C$ has equation $y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3$.

State the $x$-coordinate of point $A$.
Determine the $x$-coordinate of point $B$.
By solving an equation involving logarithms, show that the $x$-coordinate of point $C$ is 2 . It is given that, correct to 3 significant figures, the area of the sail is 0.656 units $^ { 2 }$.
Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines.

OCR PURE Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-06_648_586_255_244} The diagram shows the parallelogram $O A C B$ where $\overrightarrow { O A } = 2 \mathbf { i } + 4 \mathbf { j }$ and $\overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j }$.

Show that $\cos A O B = - \frac { 2 \sqrt { 5 } } { 25 }$.
Hence find the exact value of $\sin A O B$.
Determine the area of $O A C B$.

OCR PURE Q8

The quadratic polynomial $a x ^ { 2 } + b x$, where $a$ and $b$ are constants, is denoted by $\mathrm { f } ( x )$.
Use differentiation from first principles to determine, in terms of $a , b$ and $x$, an expression for $\mathrm { f } ^ { \prime } ( x )$.
\includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-07_565_1043_516_317} $$y = a x ^ { 2 } + b x$$ The diagram shows the quadratic curve $y = a x ^ { 2 } + b x$, where $a$ and $b$ are constants. The shaded region is enclosed by the curve, the $x$-axis and the lines $x = 1$ and $x = 4$. The tangent to the curve at $x = 4$ intersects the $x$-axis at the point with coordinates $( k , 0 )$.
Given that the area of the shaded region is 9 units ${ } ^ { 2 }$, and the gradient of this tangent is $- \frac { 3 } { 4 }$, determine the value of $k$.

OCR PURE Q9

9 Two forces $( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F N }$ act on a particle $P$ of mass 4 kg .
Given that the acceleration of $P$ is $( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 2 }$, calculate $\mathbf { F }$.

OCR PURE Q10

10 A small ball $B$ is projected vertically upwards from a point 2 m above horizontal ground. $B$ is projected with initial speed $3.5 \mathrm {~ms} ^ { - 1 }$, and takes $t$ seconds to reach the ground. Find the value of $t$.

OCR PURE Q11

11
\includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-08_586_672_1231_242} A particle $P$ moves along the $x$-axis. At time $t$ seconds, where $t \geqslant 0$, the velocity of $P$ in the positive $x$-direction is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It is given that $v = t ( t - 3 ) ( 8 - t )$.
$P$ attains its maximum velocity at time $T$ seconds. The diagram shows part of the velocity-time graph for the motion of $P$.

State the acceleration of $P$ at time $T$.
In this question you must show detailed reasoning. Determine the value of $T$.
Find the total distance that $P$ travels between times $t = 0$ and $t = T$.
\includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-09_524_410_251_242} Particles $P$ and $Q$, of masses 4 kg and 6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in equilibrium with $P$ hanging 1.75 m above a horizontal plane and $Q$ resting on the plane. Both parts of the string below the pulley are vertical (see diagram).
Find the magnitude of the normal reaction force acting on $Q$. The mass of $P$ is doubled, and the system is released from rest. You may assume that in the subsequent motion $Q$ does not reach the pulley.
Determine the magnitude of the force exerted on the pulley by the string before $P$ strikes the plane.
Determine the total distance travelled by $Q$ between the instant when the system is released and the instant when $Q$ first comes momentarily to rest. When this motion is observed in practice, it is found that the total distance travelled by $Q$ between the instant when the system is released and the instant when $Q$ first comes momentarily to rest is less than the answer calculated in part (c).
State one factor that could account for this difference.

OCR PURE Q1

1 The quadratic equation $k x ^ { 2 } + 3 x + k = 0$ has no real roots. Determine the set of possible values of $k$.

OCR PURE Q3

3 A Ferris wheel at a fairground rotates in a vertical plane. The height above the ground of a seat on the wheel is $h$ metres at time $t$ seconds after the seat is at its lowest point. The height is given by the equation $h = 15 - 14 \cos ( k t ) ^ { \circ }$, where $k$ is a positive constant.

1. Write down the greatest height of a seat above the ground.
2. Write down the least height of a seat above the ground.
Given that a seat first returns to its lowest point after 150 seconds, calculate the value of $k$.
Determine for how long a seat is 20 metres or more above the ground during one complete revolution. Give your answer correct to the nearest tenth of a second.

OCR PURE Q4

Find and simplify the first three terms in the expansion, in ascending powers of $x$, of $\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }$, where $k$ is a constant.
In the expansion of $( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }$, the constant term is equal to the coefficient of $x ^ { 2 }$. Determine the exact value of $k$, given that $k$ is positive.

OCR PURE Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-4_591_547_262_242} The diagram shows the graphs of $y = 2 ^ { 3 x }$ and $y = 2 ^ { 3 x + 2 }$. The graph of $y = 2 ^ { 3 x }$ can be transformed to the graph of $y = 2 ^ { 3 x + 2 }$ by means of a stretch.

Give details of the stretch. The point $A$ lies on $y = 2 ^ { 3 x }$ and the point $B$ lies on $y = 2 ^ { 3 x + 2 }$. The line segment $A B$ is parallel to the $y$-axis and the difference between the $y$-coordinates of $A$ and $B$ is 36 .
Determine the $x$-coordinate of $A$. Give your answer in the form $m \log _ { 2 } n$ where $m$ and $n$ are constants to be determined.

OCR PURE Q6

6 The vertices of triangle $A B C$ are $A ( - 3,1 ) , B ( 5,0 )$ and $C ( 9,7 )$.

Show that $A B = B C$.
Show that angle $A B C$ is not a right angle.
Find the coordinates of the midpoint of $A C$.
Determine the equation of the line of symmetry of the triangle, giving your answer in the form $p x + q y = r$, where $p , q$ and $r$ are integers to be determined.
Write down an equation of the circle with centre $A$ which passes through $B$. This circle intersects the line of symmetry of the triangle at $B$ and at a second point.
Find the coordinates of this second point.

OCR PURE Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve $C$ with equation $y = 4 x ^ { 2 } - 10 x + 7$ and two straight lines, $l _ { 1 }$ and $l _ { 2 }$. The line $l _ { 1 }$ is the normal to $C$ at the point $\left( \frac { 1 } { 2 } , 3 \right)$. The line $l _ { 2 }$ is the normal to $C$ at the minimum point of $C$.

Determine the equation of $l _ { 1 }$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be determined. The shaded region shown in the diagram is bounded by $C , l _ { 1 }$ and $l _ { 2 }$.
Determine the inequalities that define the shaded region, including its boundaries.

OCR PURE Q9

9 A cyclist travels along a straight horizontal road between house $A$ and house $B$. The cyclist starts from rest at $A$ and moves with constant acceleration for 20 seconds, reaching a velocity of $15 \mathrm {~ms} ^ { - 1 }$. The cyclist then moves at this constant velocity before decelerating at $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, coming to rest at $B$.

Find the time, in seconds, for which the cyclist is decelerating.
Sketch a velocity-time graph for the motion of the cyclist between $A$ and $B$. [Your sketch need not be drawn to scale; numerical values need not be shown.] The total distance between $A$ and $B$ is 1950 m .
Find the time, in seconds, for which the cyclist is moving at constant velocity.

OCR PURE Q10

10 A particle $P$ is moving in a straight line. At time $t$ seconds, where $t \geqslant 0 , P$ has velocity $v \mathrm {~ms} ^ { - 1 }$ and acceleration $a \mathrm {~ms} ^ { - 2 }$ where $a = 4 t - 9$. It is given that $v = 2$ when $t = 1$.

Find an expression for $v$ in terms of $t$. The particle $P$ is instantaneously at rest when $t = t _ { 1 }$ and $t = t _ { 2 }$, where $t _ { 1 } < t _ { 2 }$.
Find the values of $t _ { 1 }$ and $t _ { 2 }$.
Determine the total distance travelled by $P$ between times $t = 0$ and $t = t _ { 2 }$.

OCR PURE Q11

11 Two balls $P$ and $Q$ have masses 0.6 kg and 0.4 kg respectively. The balls are attached to the ends of a string. The string passes over a pulley which is fixed at the edge of a rough horizontal surface. Ball $P$ is held at rest on the surface 2 m from the pulley. Ball $Q$ hangs vertically below the pulley. Ball $Q$ is attached to a third ball $R$ of mass $m \mathrm {~kg}$ by another string and $R$ hangs vertically below $Q$ (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-7_419_945_493_246} The system is released from rest with the strings taut. Ball $P$ moves towards the pulley with acceleration $3.5 \mathrm {~ms} ^ { - 2 }$ and a constant frictional force of magnitude 4.5 N opposes the motion of $P$. The balls are modelled as particles, the pulley is modelled as being small and smooth, and the strings are modelled as being light and inextensible.

By considering the motion of $P$, find the tension in the string connecting $P$ and $Q$.
Hence determine the value of $m$. Give your answer correct to $\mathbf { 3 }$ significant figures. When the balls have been in motion for 0.4 seconds the string connecting $Q$ and $R$ breaks.
Show that, according to the model, $P$ does not reach the pulley. It is given that in fact ball $P$ does reach the pulley.
Identify one factor in the modelling that could account for this difference.

OCR PURE Q7

$\overrightarrow { A C }$
$\overrightarrow { O P }$
(ii) Hence prove that the diagonals of a parallelogram bisect one another.

OCR PURE 2066 Q6

Show that the equation $6 \cos ^ { 2 } \theta = \tan \theta \cos \theta + 4$
can be expressed in the form $6 \sin ^ { 2 } \theta + \sin \theta - 2 = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-4_446_1150_1119_338} The diagram shows parts of the curves $y = 6 \cos ^ { 2 } \theta$ and $y = \tan \theta \cos \theta + 4$, where $\theta$ is in degrees. Solve the inequality $6 \cos ^ { 2 } \theta > \tan \theta \cos \theta + 4$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.

Questions — OCR PURE (137 questions)

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