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OCR H240/03 2019 June Q5
9 marks Standard +0.3
In this question you must show detailed reasoning.
  1. Prove that \((\cot \theta + \cosec \theta)^2 = \frac{1 + \cos \theta}{1 - \cos \theta}\). [4]
  2. Hence solve, for \(0 < \theta < 2\pi\), \(3(\cot \theta + \cosec \theta)^2 = 2 \sec \theta\). [5]
OCR H240/03 2019 June Q6
10 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows part of the curve \(y = \frac{2x - 1}{(2x + 3)(x + 1)^2}\). Find the exact area of the shaded region, giving your answer in the form \(p + q \ln r\), where \(p\) and \(q\) are positive integers and \(r\) is a positive rational number. [10]
OCR H240/03 2019 June Q7
4 marks Easy -1.2
A cyclist starting from rest accelerates uniformly at \(1.5 \text{ m s}^{-2}\) for \(4\) s and then travels at constant speed.
  1. Sketch a velocity-time graph to represent the first \(10\) seconds of the cyclist's motion. [2]
  2. Calculate the distance travelled by the cyclist in the first \(10\) seconds. [2]
OCR H240/03 2019 June Q8
10 marks Standard +0.3
A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after \(2.4\) seconds. The horizontal component of the initial velocity of \(P\) is \(\frac{5}{3}d \text{ m s}^{-1}\).
  1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground. [1]
  2. Find the vertical component of the initial velocity of \(P\). [2]
\(P\) just clears a vertical wall which is situated at a horizontal distance \(d\) m from \(O\).
  1. Find the height of the wall. [3]
The speed of \(P\) as it passes over the wall is \(16 \text{ m s}^{-1}\).
  1. Find the value of \(d\) correct to \(3\) significant figures. [4]
OCR H240/03 2019 June Q9
9 marks Standard +0.3
\includegraphics{figure_9} The diagram shows a small block \(B\), of mass \(0.2\) kg, and a particle \(P\), of mass \(0.5\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of \(\theta\) with the horizontal where \(\tan \theta = \frac{3}{4}\). The system is released from rest. In the first \(0.4\) seconds of the motion \(P\) moves \(0.3\) m down the plane and \(B\) does not reach the pulley.
  1. Find the tension in the string during the first \(0.4\) seconds of the motion. [4]
  2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]
OCR H240/03 2019 June Q10
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 \(R\) of mass \(2\) kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, the velocity \(\mathbf{v} \text{ m s}^{-1}\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf{v} = (pt^2 - 3t)\mathbf{i} + (8t + q)\mathbf{j}\), where \(p\) and \(q\) are constants and \(p < 0\).
  1. Given that when \(t = 0.5\) the magnitude of \(\mathbf{F}\) is \(20\), find the value of \(p\). [6]
When \(t = 0\), \(R\) is at the point with position vector \((2\mathbf{i} - 3\mathbf{j})\) m.
  1. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). [4]
When \(t = 1\), \(R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2\mathbf{i} - 8\mathbf{j}\).
  1. Find the value of \(q\). [3]
OCR H240/03 2019 June Q11
14 marks Standard +0.3
[diagram]
The diagram shows a ladder \(AB\), of length \(2a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30°\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2m\) is placed on the ladder at a point \(C\) where \(AC = d\). The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.
  1. Show that the normal contact force between the ladder and the wall is \(\frac{mg(a + 2d)\sqrt{3}}{4h}\). [4]
It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac{1}{3}\sqrt{3}\).
  1. Show that \(h = k(a + 2d)\), where \(k\) is a constant to be determined. [7]
  2. Hence find, in terms of \(a\), the greatest possible value of \(d\). [2]
  3. State one improvement that could be made to the model. [1]
OCR H240/03 2020 November Q1
2 marks Easy -1.8
Triangle \(ABC\) has \(AB = 8.5\) cm, \(BC = 6.2\) cm and angle \(B = 35°\). Calculate the area of the triangle. [2]
OCR H240/03 2020 November Q2
3 marks Moderate -0.8
A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]
OCR H240/03 2020 November Q3
11 marks Moderate -0.8
The functions f and g are defined for all real values of x by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).
  1. Find the range of f. [3]
  2. Give a reason why f has no inverse. [1]
  3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
  4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]
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]