Questions — OCR PURE (137 questions)

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OCR PURE 2066 Q2
2
  1. Express \(5 x ^ { 2 } - 20 x + 3\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
  2. State the coordinates of the minimum point of the curve \(y = 5 x ^ { 2 } - 20 x + 3\).
  3. State the equation of the normal to the curve \(y = 5 x ^ { 2 } - 20 x + 3\) at its minimum point.
OCR PURE 2066 Q3
3
  1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
  2. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is translated by 2 units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
  3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor \(\frac { 1 } { 2 }\) and, as a result, the point \(\left( \frac { 1 } { 2 } , - 4 \right)\) on the curve is transformed to the point \(P\). State the coordinates of \(P\).
OCR PURE 2066 Q4
4
  1. Find and simplify the first three terms in the expansion of \(( 2 - 5 x ) ^ { 5 }\) in ascending powers of \(x\).
  2. In the expansion of \(( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }\), the coefficient of \(x\) is 48 . Find the value of \(a\).
OCR PURE 2066 Q5
5 Points \(A , B , C\) and \(D\) have position vectors \(\mathbf { a } = \binom { 1 } { 2 } , \mathbf { b } = \binom { 3 } { 5 } , \mathbf { c } = \binom { 7 } { 4 }\) and \(\mathbf { d } = \binom { 4 } { k }\).
  1. Find the value of \(k\) for which \(D\) is the midpoint of \(A C\).
  2. Find the two values of \(k\) for which \(| \overrightarrow { A D } | = \sqrt { 13 }\).
  3. Find one value of \(k\) for which the four points form a trapezium.
OCR PURE 2066 Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-5_647_741_260_260} The diagram shows part of the curve \(y = ( 5 - x ) ( x - 1 )\) and the line \(x = a\).
Given that the total area of the regions shaded in the diagram is 19 units \({ } ^ { 2 }\), determine the exact value of \(a\).
OCR PURE 2066 Q8
8
  1. Show that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\), where \(k\) is a constant, can be expressed in the form \(x ^ { 2 } - 8 k x + 8 = 0\).
  2. Given that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\) has only one real root, find the value of this root.
OCR PURE 2066 Q9
9 Three forces \(\binom { 7 } { - 6 } \mathrm {~N} , \binom { 2 } { 5 } \mathrm {~N}\) and \(\mathbf { F N }\) act on a particle.
Given that the particle is in equilibrium under the action of these three forces, calculate \(\mathbf { F }\).
OCR PURE 2066 Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-6_670_1106_797_258} The diagram shows the velocity-time graph modelling the velocity of a car as it approaches, and drives through, a residential area. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(0 \leqslant t \leqslant 5\) is modelled by the equation \(v = p t ^ { 2 } + q t + r\), where \(p , q\) and \(r\) are constants. It is given that the acceleration of the car is zero at \(t = 5\) and the speed of the car then remains constant.
  1. Determine the values of \(p , q\) and \(r\).
  2. Calculate the distance travelled by the car from \(t = 2\) to \(t = 10\).
OCR PURE 2066 Q11
11 Two small balls \(P\) and \(Q\) have masses 3 kg and 2 kg respectively. The balls are attached to the ends of a string. \(P\) is held at rest on a rough horizontal surface. The string passes over a pulley which is fixed at the edge of the surface. \(Q\) hangs vertically below the pulley at a height of 2 m above a horizontal floor.
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-7_346_906_445_255} The system is initially at rest with the string taut. A horizontal force of magnitude 40 N acts on \(P\) as shown in the diagram.
\(P\) is released and moves directly away from the pulley. A constant frictional force of magnitude 8 N opposes the motion of \(P\). It is given that \(P\) does not leave the horizontal surface and that \(Q\) does not reach the pulley in the subsequent motion. The balls are modelled as particles, the pulley is modelled as being small and smooth, and the string is modelled as being light and inextensible.
  1. Show that the magnitude of the acceleration of each particle is \(2.48 \mathrm {~ms} ^ { - 2 }\).
  2. Find the tension in the string. When the balls have been in motion for 0.5 seconds, the string breaks.
  3. Find the additional time that elapses until \(Q\) hits the floor.
  4. Find the speed of \(Q\) as it hits the floor.
  5. Write down the magnitude of the normal reaction force acting on \(Q\) when \(Q\) has come to rest on the floor.
  6. State one improvement that could be made to the model. \section*{OCR} Oxford Cambridge and RSA
OCR PURE Q1
1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 9.5 cm . The angle \(A O B\) is \(25 ^ { \circ }\).
  1. Calculate the length of the straight line \(A B\).
  2. Find the area of the segment shaded in the diagram.
OCR PURE Q2
2 Two curves have equations \(y = \ln x\) and \(y = \frac { k } { x }\), where \(k\) is a positive constant.
  1. Sketch the curves on a single diagram.
  2. Explain how your diagram shows that the equation \(x \ln x - k = 0\) has exactly one real root.
OCR PURE Q5
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
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
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
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
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
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
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
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
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
    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
    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 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
    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 coordinates of \(C\).
      • The exact radius of the circle.
      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
    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.