Questions H240/03 (134 questions)

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OCR H240/03 2020 November Q4
11 marks Standard +0.3
A curve has equation \(y = 2\ln(k - 3x) + x^2 - 3x\), where \(k\) is a positive constant.
  1. Given that the curve has a point of inflection where \(x = 1\), show that \(k = 6\). [5] It is also given that the curve intersects the \(x\)-axis at exactly one point.
  2. Show by calculation that the \(x\)-coordinate of this point lies between 0.5 and 1.5. [2]
  3. Use the Newton-Raphson method, with initial value \(x_0 = 1\), to find the \(x\)-coordinate of the point where the curve intersects the \(x\)-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places. [3]
  4. By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places. [1]
OCR H240/03 2020 November Q5
12 marks Standard +0.8
\includegraphics{figure_5} The diagram shows the curve \(C\) with parametric equations \(x = \frac{3}{t}\), \(y = t^2 e^{-2t}\), where \(t > 0\). The maximum point on \(C\) is denoted by \(P\).
  1. Determine the exact coordinates of \(P\). [4] The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
  2. Show that the area of \(R\) is given by $$\int_a^b 3te^{-2t} dt,$$ where \(a\) and \(b\) are constants to be determined. [3]
  3. Hence determine the exact area of \(R\). [5]
OCR H240/03 2020 November Q6
11 marks Challenging +1.2
In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curve with equation \(4xy = 2(x^2 + 4y^2) - 9x\).
  1. Show that \(\frac{dy}{dx} = \frac{4x - 4y - 9}{4x - 16y}\). [3] At the point \(P\) on the curve the tangent to the curve is parallel to the \(y\)-axis and at the point \(Q\) on the curve the tangent to the curve is parallel to the \(x\)-axis.
  2. Show that the distance \(PQ\) is \(k\sqrt{5}\), where \(k\) is a rational number to be determined. [8]
OCR H240/03 2020 November Q7
6 marks Moderate -0.8
A particle \(P\) moves with constant acceleration \((-4\mathbf{i} + 2\mathbf{j})\) ms\(^{-2}\). At time \(t = 0\) seconds, \(P\) is moving with velocity \((7\mathbf{i} + 6\mathbf{j})\) ms\(^{-1}\).
  1. Determine the speed of \(P\) when \(t = 3\). [4]
  2. Determine the change in displacement of \(P\) between \(t = 0\) and \(t = 3\). [2]
OCR H240/03 2020 November Q8
7 marks Moderate -0.3
A car is travelling on a straight horizontal road. The velocity of the car, \(v\) ms\(^{-1}\), at time \(t\) seconds as it travels past three points, \(P\), \(Q\) and \(R\), is modelled by the equation \(v = at^2 + bt + c\), where \(a\), \(b\) and \(c\) are constants. The car passes \(P\) at time \(t = 0\) with velocity \(8\) ms\(^{-1}\).
  1. State the value of \(c\). [1] The car passes \(Q\) at time \(t = 5\) and at that instant its deceleration is \(0.12\) ms\(^{-2}\). The car passes \(R\) at time \(t = 18\) with velocity \(2.96\) ms\(^{-1}\).
  2. Determine the values of \(a\) and \(b\). [4]
  3. Find, to the nearest metre, the distance between points \(P\) and \(R\). [2]
OCR H240/03 2020 November Q9
13 marks Standard +0.3
\includegraphics{figure_9} 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 2.5 kg. Particle \(A\) is in contact with a rough plane inclined at \(\theta\) to the horizontal, where \(\cos \theta = \frac{4}{5}\). The string is taut and passes over a small smooth pulley \(P\) at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Particle \(B\) hangs freely below \(P\) at a distance 1.5 m above horizontal ground, as shown in the diagram. The coefficient of friction between \(A\) and the plane is \(\mu\). The system is released from rest and in the subsequent motion \(B\) hits the ground before \(A\) reaches \(P\). The speed of \(B\) at the instant that it hits the ground is \(1.2\) ms\(^{-1}\).
  1. For the motion before \(B\) hits the ground, show that the acceleration of \(B\) is \(0.48\) ms\(^{-2}\). [1]
  2. For the motion before \(B\) hits the ground, show that the tension in the string is \(23.3\) N. [3]
  3. Determine the value of \(\mu\). [5] After \(B\) hits the ground, \(A\) continues to travel up the plane before coming to instantaneous rest before it reaches \(P\).
  4. Determine the distance that \(A\) travels from the instant that \(B\) hits the ground until \(A\) comes to instantaneous rest. [4]
OCR H240/03 2020 November Q10
11 marks Standard +0.3
\includegraphics{figure_10} The diagram shows a wall-mounted light. It consists of a rod \(AB\) of mass 0.25 kg and length 0.8 m which is freely hinged to a vertical wall at \(A\), and a lamp of mass 0.5 kg fixed at \(B\). The system is held in equilibrium by a chain \(CD\) whose end \(C\) is attached to the midpoint of \(AB\). The end \(D\) is fixed to the wall a distance 0.4 m vertically above \(A\). The rod \(AB\) makes an angle of \(60°\) with the downward vertical. The chain is modelled as a light inextensible string, the rod is modelled as uniform and the lamp is modelled as a particle.
  1. By taking moments about \(A\), determine the tension in the chain. [4]
    1. Determine the magnitude of the force exerted on the rod at \(A\). [4]
    2. Calculate the direction of the force exerted on the rod at \(A\). [2]
  3. Suggest one improvement that could be made to the model to make it more realistic. [1]
OCR H240/03 2020 November Q11
13 marks Standard +0.3
\includegraphics{figure_11} A particle \(P\) moves freely under gravity in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertical axis \(Oy\). \(P\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U\) and \(V\), respectively. \(P\) returns to the ground at a point \(C\).
  1. Determine, in terms of \(U\), \(V\) and \(g\), the distance \(OC\). [4] \includegraphics{figure_11b} \(P\) passes through two points \(A\) and \(B\), each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram.
  2. Write down the horizontal and vertical components of the velocity of \(P\) at \(A\). [2]
  3. Hence determine an expression for \(d\) in terms of \(U\), \(V\), \(g\) and \(h\). [3]
  4. Given that the direction of motion of \(P\) as it passes through \(A\) is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{1}{2}\), determine an expression for \(V\) in terms of \(g\), \(d\) and \(h\). [4]
OCR H240/03 2021 November Q1
3 marks Easy -1.8
Show in a sketch the region of the \(x\)-\(y\) plane within which all three of the following inequalities hold. \(y \geqslant x^2\), \(x + y \leqslant 2\), \(x \geqslant 0\). You should indicate the region for which the inequalities hold by labelling the region \(R\). [3]
OCR H240/03 2021 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} The diagram shows triangle \(ABC\) in which angle \(A\) is \(60°\) and the lengths of \(AB\) and \(AC\) are \((4 + h)\) cm and \((4 - h)\) cm respectively.
  1. Show that the length of \(BC\) is \(p\) cm where $$p^2 = 16 + 3h^2.$$ [2]
  2. Hence show that, when \(h\) is small, \(p \approx 4 + \lambda h^2 + \mu h^4\), where \(\lambda\) and \(\mu\) are rational numbers whose values are to be determined. [4]
OCR H240/03 2021 November Q3
5 marks Standard +0.3
An arithmetic progression has first term \(2\) and common difference \(d\), where \(d \neq 0\). The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression. By determining \(d\), show that the arithmetic progression is an increasing sequence. [5]
OCR H240/03 2021 November Q4
5 marks Moderate -0.3
  1. Sketch, on a single diagram, the following graphs.
    [2]
  2. Hence explain why the equation \(x|x - 1| = k\) has exactly one real root for any negative value of \(k\). [1]
  3. Determine the real root of the equation \(x|x - 1| = -6\). [2]
OCR H240/03 2021 November Q5
6 marks Moderate -0.3
A particle \(P\) moves along a straight line in such a way that at time \(t\) seconds \(P\) has velocity \(v\) m s\(^{-1}\), where \(v = 12\cos t + 5\sin t\).
  1. Express \(v\) in the form \(R\cos(t - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the value of \(\alpha\) correct to 4 significant figures. [3]
  2. Hence find the two smallest positive values of \(t\) for which \(P\) is moving, in either direction, with a speed of 3 m s\(^{-1}\). [3]
OCR H240/03 2021 November Q6
6 marks Standard +0.3
The equation \(6\arcsin(2x - 1) - x^2 = 0\) has exactly one real root.
  1. Show by calculation that the root lies between 0.5 and 0.6. [2]
In order to find the root, the iterative formula \(x_{n+1} = p + q\sin(rx_n^2)\), with initial value \(x_0 = 0.5\), is to be used.
  1. Determine the values of the constants \(p\), \(q\) and \(r\). [2]
  2. Hence find the root correct to 4 significant figures. Show the result of each step of the iteration process. [2]
OCR H240/03 2021 November Q7
8 marks Standard +0.3
A curve \(C\) in the \(x\)-\(y\) plane has the property that the gradient of the tangent at the point \(P(x, y)\) is three times the gradient of the line joining the point \((3, 2)\) to \(P\).
  1. Express this property in the form of a differential equation. [2]
It is given that \(C\) passes through the point \((4, 3)\) and that \(x > 3\) and \(y > 2\) at all points on \(C\).
  1. Determine the equation of \(C\) giving your answer in the form \(y = f(x)\). [4]
The curve \(C\) may be obtained by a transformation of part of the curve \(y = x^3\).
  1. Describe fully this transformation. [2]
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]