Questions — OCR PURE (139 questions)

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OCR PURE Q3
5 marks Moderate -0.8
3 Sam invested in a shares scheme. The value, \(\pounds V\), of Sam's shares was reported \(t\) months after investment.
  • Exactly 6 months after investment, the value of Sam's shares was \(\pounds 2375\).
  • Exactly 1 year after investment, the value of Sam's shares was \(\pounds 2825\).
    1. Using a straight-line model, determine an equation for \(V\) in terms of \(t\).
Sam's original investment in the scheme was \(\pounds 1900\).
  • Explain whether or not this fact supports the use of the straight-line model in part (a).
    •
    OCR PURE Q4
    5 marks Moderate -0.5
    4 The quadratic polynomial \(2 x ^ { 2 } - 3\) is denoted by \(\mathrm { f } ( x )\).
    Use differentiation from first principles to determine the value of \(\mathrm { f } ^ { \prime } ( 2 )\).
    OCR PURE Q5
    9 marks Standard +0.3
    5
    1. Show that the equation \(2 \cos x \tan ^ { 2 } x = 3 ( 1 + \cos x )\) can be expressed in the form $$5 \cos ^ { 2 } x + 3 \cos x - 2 = 0$$ \section*{(b) In this question you must show detailed reasoning.} Hence solve the equation $$2 \cos 3 \theta \tan ^ { 2 } 3 \theta = 3 ( 1 + \cos 3 \theta ) ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(120 ^ { \circ }\), correct to \(\mathbf { 1 }\) decimal place where appropriate.
    OCR PURE Q6
    6 marks Moderate -0.3
    6 A curve \(C\) has an equation which satisfies \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { 2 } + 2\), for all values of \(x\).
    1. It is given that \(C\) has a single stationary point. Determine the nature of this stationary point. The diagram shows the graph of the gradient function for \(C\). \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-04_702_442_1672_242}
    2. Given that \(C\) passes through the point \(\left( - 1 , \frac { 1 } { 4 } \right)\), find the equation of \(C\) in the form \(y = \mathrm { f } ( x )\).
    OCR PURE Q7
    9 marks Standard +0.3
    7 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-05_848_1049_260_242} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 9 y + 19 = 0\) and centre \(C\).
    1. Find the following.
      The tangent to the circle at \(D\) meets the \(x\)-axis at the point \(A \left( \frac { 55 } { 4 } , 0 \right)\) and the \(y\)-axis at the point \(B ( 0 , - 11 )\).
    2. Determine the area of triangle \(O B D\).
    OCR PURE Q8
    11 marks Standard +0.8
    8 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-06_823_588_260_242} The diagram shows the curve \(y = 1 - x + \frac { 6 } { \sqrt { x } }\) and the line \(l\), which is the normal to the curve at the point (1, 6).
    1. Determine the equation of \(l\) in the form $$a x + b y = c$$ where \(a\), \(b\) and \(c\) are integers whose values are to be stated.
    2. Verify that the curve intersects the \(x\)-axis at the point where \(x = 4\).
    3. In this question you must show detailed reasoning. Determine the exact area of the shaded region enclosed between \(l\), the curve, the \(x\)-axis and the \(y\)-axis.
    OCR PURE Q9
    3 marks Moderate -0.8
    9 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-07_609_894_495_242} A body remains at rest when subjected to the horizontal and vertical forces shown in the diagram.
    Determine the value of \(F _ { 1 }\) and the value of \(F _ { 2 }\).
    OCR PURE Q10
    8 marks Standard +0.3
    10 A cyclist starts from rest and moves with constant acceleration along a straight horizontal road. The cyclist reaches a speed of \(6 \mathrm {~ms} ^ { - 1 }\) in 25 seconds. The cyclist then moves with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until the speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist then moves with constant deceleration until coming to rest. The total time for the cyclist's journey is 150 seconds.
    1. Sketch a velocity-time graph to represent the cyclist's motion.
    2. 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.
    3. Determine the value of \(T\).
    4. Determine the average speed for the cyclist's journey.
    OCR PURE Q11
    7 marks Moderate -0.3
    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 .
    1. Determine the time, in minutes, to travel between \(P\) and \(Q\). For the train's motion between \(P\) and \(Q\), determine the following.
    2. The driving force of the engine.
    3. The tension in the coupling between \(A\) and \(B\).
    OCR PURE Q12
    7 marks Standard +0.8
    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
    2 marks Easy -1.2
    1 Write the solution of the inequality \(( x - 2 ) ( x + 3 ) > 0\) using set notation.
    OCR PURE Q3
    2 marks Easy -1.2
    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
    3 marks Moderate -0.5
    4 The circle \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + k = 0\) has radius 5.
    Determine the value of \(k\).
    OCR PURE Q5
    7 marks Standard +0.3
    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.
    2. The diagram indicates that the curve \(C\) has no stationary points. State what feature of your sketch in part (a)(ii) corresponds to this.
    3. 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
    12 marks Standard +0.8
    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 \(y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3\).
    1. State the \(x\)-coordinate of point \(A\).
    2. Determine the \(x\)-coordinate of point \(B\).
    3. 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 \(^ { 2 }\).
    4. 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
    9 marks Moderate -0.3
    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 \(\overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j }\).
    1. Show that \(\cos A O B = - \frac { 2 \sqrt { 5 } } { 25 }\).
    2. Hence find the exact value of \(\sin A O B\).
    3. Determine the area of \(O A C B\).
    OCR PURE Q8
    11 marks Standard +0.8
    8
    1. 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 \(\mathrm { f } ^ { \prime } ( x )\).
    2. \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
    2 marks Moderate -0.8
    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
    3 marks Moderate -0.8
    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
    9 marks Moderate -0.3
    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 \(P\).
    1. State the acceleration of \(P\) at time \(T\).
    2. In this question you must show detailed reasoning. Determine the value of \(T\).
    3. 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).
      1. 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.
      2. Determine the magnitude of the force exerted on the pulley by the string before \(P\) strikes the plane.
      3. 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).
      4. State one factor that could account for this difference.
    OCR PURE Q1
    3 marks Moderate -0.3
    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
    8 marks Standard +0.3
    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.
    2. Given that a seat first returns to its lowest point after 150 seconds, calculate the value of \(k\).
    3. 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
    6 marks Standard +0.3
    4
    1. 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.
    2. 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 marks Standard +0.3
    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.
    1. 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 .
    2. 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
    10 marks Moderate -0.3
    6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).
    1. Show that \(A B = B C\).
    2. Show that angle \(A B C\) is not a right angle.
    3. Find the coordinates of the midpoint of \(A C\).
    4. 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.
    5. 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.
    6. Find the coordinates of this second point.