Questions — OCR (4907 questions)

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OCR H240/03 2021 November Q8
11 marks Standard +0.3
\includegraphics{figure_8} The diagram shows the curve \(M\) with equation \(y = xe^{-2x}\).
  1. Show that \(M\) has a point of inflection at the point \(P\) where \(x = 1\). [5]
The line \(L\) passes through the origin \(O\) and the point \(P\). The shaded region \(R\) is enclosed by the curve \(M\) and the line \(L\).
  1. Show that the area of \(R\) is given by $$\frac{1}{4}(a + be^{-2}),$$ where \(a\) and \(b\) are integers to be determined. [6]
OCR H240/03 2021 November Q9
3 marks Moderate -0.8
There are three checkpoints, \(A\), \(B\) and \(C\), in that order, on a straight horizontal road. A car travels along the road, in the direction from \(A\) to \(C\), with constant acceleration. The car takes 20 s to travel from \(B\) to \(C\). The speed of the car at \(B\) is 14 m s\(^{-1}\) and the speed of the car at \(C\) is 18 m s\(^{-1}\).
  1. Find the acceleration of the car. [1]
It is given that the distance between \(A\) and \(B\) is 330 m.
  1. Determine the speed of the car at \(A\). [2]
OCR H240/03 2021 November Q10
6 marks Standard +0.3
\includegraphics{figure_10} A block \(D\) of weight 50 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 15 N is applied to \(D\) at an angle \(\theta\) to the horizontal (see diagram).
  1. Complete the diagram in the Printed Answer Booklet showing all the forces acting on \(D\). [1]
It is given that \(D\) remains at rest and the coefficient of friction between \(D\) and the surface is 0.2.
  1. Show that $$15\cos\theta - 3\sin\theta \leqslant 10.$$ [5]
OCR H240/03 2021 November Q11
10 marks Standard +0.3
\includegraphics{figure_11} A golfer hits a ball from a point \(A\) with a speed of 25 m s\(^{-1}\) at an angle of 15° above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be 10 m s\(^{-2}\). The ball first lands at a point \(B\) which is 4 m below the level of \(A\) (see diagram).
  1. Determine the time taken for the ball to travel from \(A\) to \(B\). [3]
  2. Determine the horizontal distance of \(B\) from \(A\). [2]
  3. Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball. [4]
The horizontal distance from \(A\) to \(B\) is found to be greater than the answer to part (b).
  1. State one factor that could account for this difference. [1]
OCR H240/03 2021 November Q12
7 marks Standard +0.3
[diagram]
A beam, \(AB\), has length 4 m and mass 20 kg. The beam is suspended horizontally by two vertical ropes. One rope is attached to the beam at \(C\), where \(AC = 0.5\) m. The other rope is attached to the beam at \(D\), where \(DB = 0.7\) m (see diagram). The beam is modelled as a non-uniform rod and the ropes as light inextensible strings. It is given that the tension in the rope at \(C\) is three times the tension in the rope at \(D\).
  1. Determine the distance of the centre of mass of the beam from \(A\). [5]
A particle of mass \(m\) kg is now placed on the beam at a point where the magnitude of the moment of the particle's weight about \(C\) is 3.5\(mg\) N m. The beam remains horizontal and in equilibrium.
  1. Determine the largest possible value of \(m\). [2]
OCR H240/03 2021 November Q13
13 marks Standard +0.8
In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force \((-8\mathbf{i} - 54\mathbf{j})\) N and a variable horizontal force \((4t\mathbf{i} + 6(2t - 1)^2\mathbf{j})\) N.
  1. Determine the value of \(t\) when the forces acting on \(P\) are in equilibrium. [2]
It is given that \(P\) is at rest when \(t = 0\).
  1. Determine the speed of \(P\) at the instant when \(P\) is moving due north. [6]
  2. Determine the distance between the positions of \(P\) when \(t = 0\) and \(t = 3\). [5]
OCR H240/03 2021 November Q14
11 marks Challenging +1.2
\includegraphics{figure_14} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg. The other end of the string is attached to a second particle \(B\) of mass 3 kg. Particle \(A\) is in contact with a smooth plane inclined at 30° to the horizontal and particle \(B\) is in contact with a rough horizontal plane. A second light inextensible string is attached to \(B\). The other end of this second string is attached to a third particle \(C\) of mass 4 kg. Particle \(C\) is in contact with a smooth plane \(\Pi\) inclined at an angle of 60° to the horizontal. Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from \(A\) to the pulley, and from \(C\) to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram). The coefficient of friction between \(B\) and the horizontal plane is \(\mu\). The system is released from rest and in the subsequent motion \(C\) moves down \(\Pi\) with acceleration \(a\) m s\(^{-2}\).
  1. By considering an equation involving \(\mu\), \(a\) and \(g\) show that \(a < \frac{1}{9}g(2\sqrt{3} - 1)\). [7]
  2. Given that \(a = \frac{1}{5}g\), determine the magnitude of the contact force between \(B\) and the horizontal plane. Give your answer correct to 3 significant figures. [4]
OCR H240/03 2022 June Q1
3 marks Easy -1.8
Solve the equation \(|2x - 3| = 9\). [3]
OCR H240/03 2022 June Q2
5 marks Easy -1.2
  1. Give full details of the single transformation that transforms the graph of \(y = x^3\) to the graph of \(y = x^3 - 8\). [2]
The function f is defined by \(\mathrm{f}(x) = x^3 - 8\).
  1. Find an expression for \(\mathrm{f}^{-1}(x)\). [2]
  2. State how the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\) are related geometrically. [1]
OCR H240/03 2022 June Q3
4 marks Moderate -0.8
The points \(P\) and \(Q\) have coordinates \((2, -5)\) and \((3, 1)\) respectively. Determine the equation of the circle that has \(PQ\) as a diameter. Give your answer in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
OCR H240/03 2022 June Q4
8 marks Standard +0.8
The positive integers \(x\), \(y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference \(-4\). Also, \(x\), \(\frac{15}{y}\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.
  1. Show that \(y\) satisfies the equation \(y^4 - 16y^2 - 225 = 0\). [4]
  2. Hence determine the sum to infinity of the geometric progression. [4]
OCR H240/03 2022 June Q5
14 marks Standard +0.3
In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the curve with equation \(y = \frac{2x - 3}{4x^2 + 1}\). The tangent to the curve at the point \(P\) has gradient 2.
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$4x^3 + 3x - 3 = 0.$$ [5]
  2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.5 and 1. [2]
  3. Show that the iteration $$x_{n+1} = \frac{3 - 4x_n^3}{3}$$ cannot converge to the \(x\)-coordinate of \(P\) whatever starting value is used. [2]
  4. Use the Newton-Raphson method, with initial value 0.5, to determine the coordinates of \(P\) correct to 5 decimal places. [5]
OCR H240/03 2022 June Q6
8 marks Standard +0.3
In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curves \(y = \sqrt{2x + 9}\) and \(y = 4\mathrm{e}^{-2x} - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis. Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined. [8]
OCR H240/03 2022 June Q7
8 marks Standard +0.8
In this question you must show detailed reasoning.
  1. Show that the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\) can be expressed in the form $$m \tan^2 \theta - 4 \tan \theta + (m + 3) = 0.$$ [3]
  2. It is given that there is only one value of \(\theta\), for \(0 < \theta < \pi\), satisfying the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\). Given also that \(m\) is a negative integer, find this value of \(\theta\), correct to 3 significant figures. [5]
OCR H240/03 2022 June Q8
2 marks Easy -1.3
\includegraphics{figure_8} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of \(15°\) with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N, find the tension in the rope. [2]
OCR H240/03 2022 June Q9
6 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows a velocity-time graph representing the motion of two cars \(A\) and \(B\) which are both travelling along a horizontal straight road. At time \(t = 0\), car \(B\), which is travelling with constant speed \(12 \mathrm{m s}^{-1}\), is overtaken by car \(A\) which has initial speed \(20 \mathrm{m s}^{-1}\). From \(t = 0\) car \(A\) travels with constant deceleration for 30 seconds. When \(t = 30\) the speed of car \(A\) is \(8 \mathrm{m s}^{-1}\) and the car maintains this speed in subsequent motion.
  1. Calculate the deceleration of car \(A\). [2]
  2. Determine the value of \(t\) when \(B\) overtakes \(A\). [4]
OCR H240/03 2022 June Q10
8 marks Standard +0.3
\includegraphics{figure_10} A rectangular block \(B\) is at rest on a horizontal surface. A particle \(P\) of mass 2.5 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle \(Q\) of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between \(P\) and the pulley is horizontal (see diagram). The particles are released from rest with the string taut. It is given that \(B\) remains in equilibrium while \(P\) moves on the upper surface of \(B\). The tension in the string while \(P\) moves on \(B\) is 16.8 N.
  1. Find the acceleration of \(Q\) while \(P\) and \(B\) are in contact. [2]
  2. Determine the coefficient of friction between \(P\) and \(B\). [3]
  3. Given that the coefficient of friction between \(B\) and the horizontal surface is \(\frac{5}{49}\), determine the least possible value for the mass of \(B\). [3]
OCR H240/03 2022 June Q11
7 marks Challenging +1.2
\includegraphics{figure_11} A uniform rod \(AB\) of mass 4 kg and length 3 m rests in a vertical plane with \(A\) on rough horizontal ground. A particle of mass 1 kg is attached to the rod at \(B\). The rod makes an angle of \(60°\) with the horizontal and is held in limiting equilibrium by a light inextensible string \(CD\). \(D\) is a fixed point vertically above \(A\) and \(CD\) makes an angle of \(60°\) with the vertical. The distance \(AC\) is \(x\) m (see diagram).
  1. Find, in terms of \(g\) and \(x\), the tension in the string. [3]
The coefficient of friction between the rod and the ground is \(\frac{9\sqrt{3}}{35}\).
  1. Determine the value of \(x\). [4]
OCR H240/03 2022 June Q12
13 marks Standard +0.3
In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geq 0\), the velocity \(\mathbf{v} \mathrm{m s}^{-1}\) of \(P\), relative to a fixed origin \(O\), is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$
  1. Show that \(P\) is never stationary. [2]
  2. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\) at time \(t\). [1]
The mass of \(P\) is 0.5 kg.
  1. Determine the magnitude of \(\mathbf{F}\) when \(P\) is moving in the direction of the vector \(-2\mathbf{i} + \mathbf{j}\). Give your answer correct to 3 significant figures. [5]
When \(t = 1\), \(P\) is at the point with position vector \(\frac{1}{6}\mathbf{j}\).
  1. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\). [5]
OCR H240/03 2022 June Q13
14 marks Standard +0.3
A small ball \(B\) moves in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertically upwards axis \(Oy\). \(B\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U \mathrm{m s}^{-1}\) and \(V \mathrm{m s}^{-1}\), respectively. The units of \(x\) and \(y\) are metres. \(B\) is modelled as a particle moving freely under gravity.
  1. Show that the path of \(B\) has equation \(2U^2 y = 2UVx - gx^2\). [3]
During its motion, \(B\) just clears a vertical wall of height \(\frac{1}{3}a\) m at a horizontal distance \(a\) m from \(O\). \(B\) strikes the ground at a horizontal distance \(3a\) m beyond the wall.
  1. Determine the angle of projection of \(B\). Give your answer in degrees correct to 3 significant figures. [5]
  2. Given that the speed of projection of \(B\) is \(54.6 \mathrm{m s}^{-1}\), determine the value of \(a\). [2]
  3. Hence find the maximum height of \(B\) above the ground during its motion. [3]
  4. State one refinement of the model, other than including air resistance, that would make it more realistic. [1]
OCR H240/03 2023 June Q1
3 marks Easy -1.2
Using logarithms, solve the equation $$4^{2x+1} = 5^x,$$ giving your answer correct to 3 significant figures. [3]
OCR H240/03 2023 June Q2
5 marks Moderate -0.3
  1. Express \(3 \sin x - 4 \cos x\) in the form \(R \sin(x - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). Give the value of \(\alpha\) correct to 4 significant figures. [3]
  2. Hence solve the equation \(3 \sin x - 4 \cos x = 2\) for \(0° < x < 90°\), giving your answer correct to 3 significant figures. [2]
OCR H240/03 2023 June Q3
8 marks Moderate -0.3
The cubic polynomial \(\text{f}(x)\) is defined by \(\text{f}(x) = x^3 + px + q\), where \(p\) and \(q\) are constants.
    1. Given that \(\text{f}'(2) = 13\), find the value of \(p\). [2]
    2. Given also that \((x - 2)\) is a factor of \(\text{f}(x)\), find the value of \(q\). [2]
    The curve \(y = \text{f}(x)\) is translated by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\).
  2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]
OCR H240/03 2023 June Q4
7 marks Standard +0.3
A circle \(C\) has equation \(x^2 + y^2 - 6x + 10y + k = 0\).
  1. Find the set of possible values of \(k\). [2]
  2. It is given that \(k = -46\). Determine the coordinates of the two points on \(C\) at which the gradient of the tangent is \(\frac{1}{2}\). [5]
OCR H240/03 2023 June Q5
9 marks Standard +0.8
A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below. \includegraphics{figure_5} The emblem is modelled by the region between the \(x\)-axis and the curve with parametric equations $$x = 1 + 0.2t - \cos t, \quad y = k \sin^2 t,$$ where \(k\) is a positive constant and \(0 \leq t \leq \pi\). Lengths are in metres and the area of the emblem must be \(1 \text{m}^2\).
  1. Show that \(k \int_0^\pi (0.2 + \sin t - 0.2 \cos^2 t - \sin t \cos^2 t) dt = 1\). [3]
  2. Determine the exact value of \(k\). [6]