Questions — OCR (4907 questions)

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OCR PURE Q5
5 marks Moderate -0.8
5 A curve has equation \(y = a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants. The curve has a stationary point at \(( - 3,2 )\).
  1. State the values of \(b\) and \(c\). When the curve is translated by \(\binom { 4 } { 0 }\) the transformed curve passes through the point \(( 3 , - 18 )\).
  2. Determine the value of \(a\).
OCR PURE Q7
13 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{7fc02f90-8f8b-4153-bba1-dc0807124e96-5_421_944_251_242} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the crosssection \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\).
  1. Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(a\) and \(b\) are integers to be determined.
  2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
  3. Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
  4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state.
OCR PURE Q8
3 marks Moderate -0.3
8 A particle is in equilibrium under the action of the following three forces: \(( 2 p \mathbf { i } - 4 \mathbf { j } ) N , ( - 3 q \mathbf { i } + 5 p \mathbf { j } ) N\) and \(( - 13 \mathbf { i } - 6 \mathbf { j } ) N\).
Find the values of p and q .
OCR PURE Q9
6 marks Standard +0.3
9 A crane lifts a car vertically. The car is inside a crate which is raised by the crane by means of a strong cable. The cable can withstand a maximum tension of 9500 N without breaking. The crate has a mass of 55 kg and the car has a mass of 830 kg .
  1. Find the maximum acceleration with which the crate and car can be raised.
  2. Show on a clearly labelled diagram the forces acting on the crate while it is in motion.
  3. Determine the magnitude of the reaction force between the crate and the car when they are ascending with maximum acceleration.
OCR PURE Q10
6 marks Standard +0.3
10 A particle \(P\) is moving in a straight line. At time \(t\) seconds \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) where \(v = ( 2 t + 1 ) ( 3 - t )\).
  1. Find the deceleration of \(P\) when \(t = 4\).
  2. State the positive value of \(t\) for which \(P\) is instantaneously at rest.
  3. Find the total distance that \(P\) travels between times \(t = 0\) and \(t = 4\).
OCR PURE Q11
10 marks Standard +0.3
11 A car starts from rest at a set of traffic lights and moves along a straight road with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). A motorcycle, travelling parallel to the car with constant speed \(16 \mathrm {~ms} ^ { - 1 }\), passes the same traffic lights exactly 1.5 seconds after the car starts to move. The time after the car starts to move is denoted by \(t\) seconds.
  1. Determine the two values of \(t\) at which the car and motorcycle are the same distance from the traffic lights. These two values of \(t\) are denoted by \(t _ { 1 }\) and \(t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
  2. Describe the relative positions of the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\).
  3. Determine the maximum distance between the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\). \section*{END OF QUESTION PAPER}
OCR PURE Q1
2 marks Moderate -0.8
1 Given that \(( x - 2 )\) is a factor of \(2 x ^ { 3 } + k x - 4\), find the value of the constant \(k\).
OCR PURE Q2
3 marks Easy -1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-03_835_545_749_244} The diagram shows the line \(y = - 2 x + 4\) and the curve \(y = x ^ { 2 } - 4\). The region \(R\) is the unshaded region together with its boundaries. Write down the inequalities that define \(R\).
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 \(k\).