Questions — OCR (4907 questions)

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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.
OCR PURE Q7
7 marks Standard +0.3
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 \(C\).
  1. 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 \(l _ { 2 }\).
  2. Determine the inequalities that define the shaded region, including its boundaries.
OCR PURE Q9
4 marks Moderate -0.8
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 \(B\).
  1. Find the time, in seconds, for which the cyclist is decelerating.
  2. 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 .
  3. Find the time, in seconds, for which the cyclist is moving at constant velocity.
OCR PURE Q10
8 marks Standard +0.3
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 \(t = 1\).
  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 \(t _ { 1 } < t _ { 2 }\).
  2. Find the values of \(t _ { 1 }\) and \(t _ { 2 }\).
  3. Determine the total distance travelled by \(P\) between times \(t = 0\) and \(t = t _ { 2 }\).
OCR PURE Q11
13 marks Challenging +1.2
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.
  1. By considering the motion of \(P\), find the tension in the string connecting \(P\) and \(Q\).
  2. 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.
  3. Show that, according to the model, \(P\) does not reach the pulley. It is given that in fact ball \(P\) does reach the pulley.
  4. Identify one factor in the modelling that could account for this difference.
OCR Further Pure Core AS 2018 June Q1
5 marks Moderate -0.3
1
  1. Find a vector which is perpendicular to both \(\left( \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 3 \\ - 6 \\ 4 \end{array} \right)\).
  2. The cartesian equation of a line is \(\frac { x } { 2 } = y - 3 = 2 z + 4\). Express the equation of this line in vector form.
OCR Further Pure Core AS 2018 June Q2
3 marks Standard +0.3
2 In this question you must show detailed reasoning.
The cubic equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).
OCR Further Pure Core AS 2018 June Q3
9 marks Moderate -0.3
3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 2 - 3 i\) and \(z _ { 2 } = a + 4 i\) where \(a\) is a real number.
  1. Express \(z _ { 1 }\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
  2. Find \(z _ { 1 } z _ { 2 }\) in terms of \(a\), writing your answer in the form \(c + \mathrm { id }\).
  3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z _ { 1 } z _ { 2 }\) lies on the line \(y = x\), find the value of \(a\).
  4. Given instead that \(z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }\) find the value of \(a\).
OCR Further Pure Core AS 2018 June Q4
7 marks Standard +0.3
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 2 & 1 & 2 \\ 1 & - 1 & 1 \\ 2 & 2 & a \end{array} \right)\).
  1. Show that \(\operatorname { det } \mathbf { A } = 6 - 3 a\).
  2. State the value of \(a\) for which \(\mathbf { A }\) is singular.
  3. Given that \(\mathbf { A }\) is non-singular find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
OCR Further Pure Core AS 2018 June Q5
10 marks Moderate -0.3
5 In this question you must show detailed reasoning.
  1. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
  2. Hence verify that \(2 + 3\) i is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
  3. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.
OCR Further Pure Core AS 2018 June Q6
7 marks Moderate -0.5
6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.
  1. Show that \(| \mathbf { A } | = | \mathbf { B } |\).
  2. Verify that \(| \mathbf { A B } | = | \mathbf { A } \| \mathbf { B } |\).
  3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\).
OCR Further Pure Core AS 2018 June Q7
6 marks Moderate -0.3
7 Prove by induction that \(2 ^ { n + 1 } + 5 \times 9 ^ { n }\) is divisible by 7 for all integers \(n \geqslant 1\).
OCR Further Pure Core AS 2018 June Q8
13 marks Standard +0.8
8 The \(2 \times 2\) matrix A represents a transformation T which has the following properties.
  • The image of the point \(( 0,1 )\) is the point \(( 3,4 )\).
  • An object shape whose area is 7 is transformed to an image shape whose area is 35 .
  • T has a line of invariant points.
    1. Find a possible matrix for \(\mathbf { A }\).
The transformation S is represented by the matrix \(\mathbf { B }\) where \(\mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).
  • Find the equation of the line of invariant points of S .
  • Show that any line of the form \(y = x + c\) is an invariant line of S .
    •
    OCR Further Pure Core AS 2022 June Q1
    8 marks Moderate -0.8
    1
    1. Determine whether the point \(( 19 , - 12,17 )\) lies on the line \(\mathbf { r } = \left( \begin{array} { r } 4 \\ - 2 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right)\). Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = \left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 3 \\ 6 \\ 2 \end{array} \right)\).
      1. Find, in degrees, the angle between \(\mathbf { a }\) and \(\mathbf { b }\).
      2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
    OCR Further Pure Core AS 2022 June Q2
    7 marks Moderate -0.3
    2 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } a & 1 \\ - 1 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right)\) where \(a\) is a constant.
    1. Find the following matrices.
      • \(\mathbf { A } + \mathbf { B }\)
      • AB
      • \(\mathbf { A } ^ { 2 }\)
        1. Given that the determinant of \(\mathbf { A }\) is 25 find the value of \(a\).
        2. You are given instead that the following system of equations does not have a unique solution.
      $$\begin{array} { r } a x + y = - 2 \\ - x + 3 y = - 6 \end{array}$$ Determine the value of \(a\).
    OCR Further Pure Core AS 2022 June Q4
    5 marks Standard +0.3
    4 Prove that \(3 ^ { n } > 10 n\) for all integers \(n \geqslant 4\).
    OCR Further Pure Core AS 2022 June Q5
    7 marks Standard +0.3
    5 In this question you must show detailed reasoning.
    1. Use an algebraic method to find the square roots of \(- 16 + 30 \mathrm { i }\).
    2. By finding the cube of one of your answers to part (a) determine a cube root of \(\frac { - 99 + 5 i } { 4 }\). Give your answer in the form \(a + b \mathrm { i }\).
    OCR Further Pure Core AS 2022 June Q6
    11 marks Standard +0.8
    6 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \frac { 1 } { 13 } \left( \begin{array} { r r } 5 & 12 \\ 12 & - 5 \end{array} \right)\). You are given that \(\mathbf { A }\) represents the transformation T which is a reflection in a certain straight line. You are also given that this straight line, the mirror line, passes through the origin, \(O\).
    1. Explain why there must be a line of invariant points for T . State the geometric significance of this line.
    2. By considering the line of invariant points for T , determine the equation of the mirror line. Give your answer in the form \(y = m x + c\). The coordinates of the point \(P\) are \(( 1,5 )\).
    3. By considering the image of \(P\) under the transformation T , or otherwise, determine the coordinates of the point on the mirror line which is closest to \(P\).
    4. The line with equation \(y = a x + 2\) is an invariant line for T. Determine the value of \(a\).