Questions — OCR H240/03 (134 questions)

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OCR H240/03 2018 March Q3
4 marks Hard +2.5
A sequence of three transformations maps the curve \(y = \ln x\) to the curve \(y = \mathrm{e}^{3x} - 5\). Give details of these transformations. [4]
OCR H240/03 2018 March Q4
11 marks Standard +0.3
A curve is defined, for \(t \geqslant 0\), by the parametric equations $$x = t^2, \quad y = t^3.$$
  1. Show that the equation of the tangent at the point with parameter \(t\) is $$2y = 3tx - t^3.$$ [4]
  1. In this question you must show detailed reasoning. It is given that this tangent passes through the point \(A\left(\frac{19}{2}, -\frac{15}{8}\right)\) and it meets the \(x\)-axis at the point \(B\). Find the area of triangle \(OAB\), where \(O\) is the origin. [7]
OCR H240/03 2018 March Q5
14 marks Standard +0.8
In this question you must show detailed reasoning. \includegraphics{figure_5} The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm{f}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ The diagram shows the curve \(y = \mathrm{f}(x)\).
  1. Find the range of f. [6]
  1. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm{g}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ Given that g is a one-one function, state the least possible value of \(k\). [1]
  1. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis. [7]
OCR H240/03 2018 March Q6
10 marks Standard +0.3
  1. Determine the values of \(p\) and \(q\) for which $$x^2 - 6x + 10 \equiv (x - p)^2 + q.$$ [2]
  1. Use the substitution \(x - p = \tan u\), where \(p\) takes the value found in part (i), to evaluate $$\int_3^4 \frac{1}{x^2 - 6x + 10} \, dx.$$ [3]
  1. Determine the value of $$\int_3^4 \frac{x}{x^2 - 6x + 10} \, dx,$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants to be determined. [5]
OCR H240/03 2018 March Q7
3 marks Moderate -0.8
Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a particle are given by $$\mathbf{F}_1 = (3\mathbf{i} - 2a\mathbf{j})\text{N}, \quad \mathbf{F}_2 = (2b\mathbf{i} + 3a\mathbf{j})\text{N} \quad \text{and} \quad \mathbf{F}_3 = (-2\mathbf{i} + b\mathbf{j})\text{N}.$$ The particle is in equilibrium under the action of these three forces. Find the value of \(a\) and the value of \(b\). [3]
OCR H240/03 2018 March Q8
11 marks Standard +0.3
A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4\,\text{m}\,\text{s}^{-2}\) until reaching a velocity of \(V\,\text{m}\,\text{s}^{-1}\). The jogger then runs at a constant velocity of \(V\,\text{m}\,\text{s}^{-1}\) before decelerating at a constant rate of \(0.08\,\text{m}\,\text{s}^{-2}\) back to rest. The jogger runs a total distance of \(880\,\text{m}\) in \(250\,\text{s}\).
  1. Sketch the velocity-time graph for the jogger's journey. [2]
  2. Show that \(3V^2 - 100V + 352 = 0\). [6]
  3. Hence find the value of \(V\), giving a reason for your answer. [3]
OCR H240/03 2018 March Q9
14 marks Standard +0.8
Two particles \(A\) and \(B\) have position vectors \(\mathbf{r}_A\) metres and \(\mathbf{r}_B\) metres at time \(t\) seconds, where $$\mathbf{r}_A = t^2\mathbf{i} + (3t - 1)\mathbf{j} \quad \text{and} \quad \mathbf{r}_B = (1 - 2t^2)\mathbf{i} + (3t - 2t^2)\mathbf{j}, \quad \text{for } t \geqslant 0.$$
  1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed. [5]
  2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d^2 = 13t^4 - 10t^2 + 2.$$ [3]
  3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion. [6]
OCR H240/03 2018 March Q10
9 marks Standard +0.3
\includegraphics{figure_10} A uniform rod \(AB\), of weight \(W\) N and length \(2a\) m, rests with the end \(A\) on a rough horizontal table. A small object of weight \(2W\) N is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30°\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).
  1. Find the least possible value of the coefficient of friction between the rod and the table. [7]
  2. Given that the magnitude of the contact force at \(A\) is \(\sqrt{39}\) N, find the value of \(W\). [2]
OCR H240/03 2018 March Q11
12 marks Challenging +1.8
In this question you must show detailed reasoning. \includegraphics{figure_11} A football \(P\) is kicked with speed \(25\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(\alpha\) from a point \(A\) on horizontal ground. At the same instant a second football \(Q\) is kicked with speed \(15\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(2\alpha\) from a point \(B\) on the same horizontal ground, where \(AB = 72\) m. The footballs are modelled as particles moving freely under gravity in the same vertical plane and they collide with each other at the point \(C\) (see diagram).
  1. Calculate the height of \(C\) above the ground. [7]
  2. Find the direction of motion of \(P\) at the moment of impact. [4]
  3. Suggest one improvement that could be made to the model. [1]
OCR H240/03 2018 December Q1
3 marks Moderate -0.8
Use logarithms to solve the equation \(2^{3x-1} = 3^{x+4}\), giving your answer correct to 3 significant figures. [3]
OCR H240/03 2018 December Q2
5 marks Moderate -0.3
In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7\) is positive. Give your answer in set notation. [5]
OCR H240/03 2018 December Q3
5 marks Easy -1.8
\includegraphics{figure_3} The diagram shows a circle with centre \((a, -a)\) that passes through the origin.
  1. Write down an equation for the circle in terms of \(a\). [2]
  2. Given that the point \((1, -5)\) lies on the circle, find the exact area of the circle. [3]
OCR H240/03 2018 December Q4
6 marks Standard +0.3
The first three terms of an arithmetic series are \(9p\), \(8p - 3\), \(5p\) respectively, where \(p\) is a constant. Given that the sum of the first \(n\) terms of this series is \(-1512\), find the value of \(n\). [6]
OCR H240/03 2018 December Q5
16 marks Standard +0.3
\includegraphics{figure_5} The functions f(x) and g(x) are defined for \(x \geqslant 0\) by \(\text{f}(x) = \frac{x}{x^2 + 3}\) and \(\text{g}(x) = \text{e}^{-2x}\). The diagram shows the curves \(y = \text{f}(x)\) and \(y = \text{g}(x)\). The equation \(\text{f}(x) = \text{g}(x)\) has exactly one real root \(\alpha\).
  1. Show that \(\alpha\) satisfies the equation \(\text{h}(x) = 0\), where \(\text{h}(x) = x^2 + 3 - x\text{e}^{2x}\). [2]
  2. Hence show that a Newton-Raphson iterative formula for finding \(\alpha\) can be written in the form $$x_{n+1} = \frac{x_n^2(1 - 2\text{e}^{2x_n}) - 3}{2x_n - (1 + 2x_n)\text{e}^{2x_n}}.$$ [5]
  3. Use this iterative formula, with a suitable initial value, to find \(\alpha\) correct to 3 decimal places. Show the result of each iteration. [3]
  4. In this question you must show detailed reasoning. Find the exact value of \(x\) for which \(\text{fg}(x) = \frac{2}{13}\). [6]
OCR H240/03 2018 December Q6
15 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows the curve with parametric equations \(x = \ln(t^2 - 4)\), \(y = \frac{4}{t}\), where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).
  1. Show that the area of \(R\) is given by $$\int_a^b \frac{8}{t(t^2 - 4)} dt,$$ where \(a\) and \(b\) are constants to be determined. [4]
  2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined. [8]
  3. Find a cartesian equation of the curve in the form \(y = \text{f}(x)\). [3]
OCR H240/03 2018 December Q7
6 marks Moderate -0.8
A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 5\mathbf{j})\text{m s}^{-2}\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \((2\mathbf{i} + 4\mathbf{j})\text{m s}^{-1}\).
  1. Find the displacement vector of \(P\) at time \(t = 4\) seconds. [2]
  2. Find the speed of \(P\) at time \(t = 0\) seconds. [4]
OCR H240/03 2018 December Q8
7 marks Standard +0.3
A uniform ladder \(AB\), of weight \(150\text{N}\) and length \(4\text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 3\). A man of weight \(750\text{N}\) is standing on the ladder at a distance \(x\text{m}\) from \(A\).
  1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{75}{2}(2 + 5x)\text{N}\). [4]
The coefficient of friction between the ladder and the ground is \(\frac{1}{4}\).
  1. Find the greatest value of \(x\) for which equilibrium is possible. [3]
OCR H240/03 2018 December Q9
10 marks Moderate -0.3
A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\text{m s}^{-1}\), where \(v = 2t^4 + kt^2 - 4\). The acceleration of \(P\) when \(t = 2\) is \(28\text{m s}^{-2}\).
  1. Show that \(k = -9\). [3]
  2. Show that the velocity of \(P\) has its minimum value when \(t = 1.5\). [3]
When \(t = 1\), \(P\) is at the point \((-6.4125, 0)\).
  1. Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity. [4]
OCR H240/03 2018 December Q10
11 marks Standard +0.8
\includegraphics{figure_10} \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30°\) to the horizontal. The distance \(AB\) is \(20\text{m}\). \(M\) is a point on the plane between \(A\) and \(B\). The surface of the plane is smooth between \(A\) and \(M\), and rough between \(M\) and \(B\). A particle \(P\) is projected with speed \(4.2\text{m s}^{-1}\) from \(A\) down the line of greatest slope (see diagram). \(P\) moves down the plane and reaches \(B\) with speed \(12.6\text{m s}^{-1}\). The coefficient of friction between \(P\) and the rough part of the plane is \(\frac{\sqrt{3}}{6}\).
  1. Find the distance \(AM\). [8]
  2. Find the angle between the contact force and the downward direction of the line of greatest slope when \(P\) is in motion between \(M\) and \(B\). [3]
OCR H240/03 2018 December Q11
16 marks Standard +0.3
A ball \(B\) is projected with speed \(V\) at an angle \(\alpha\) above the horizontal from a point \(O\) on horizontal ground. The greatest height of \(B\) above \(O\) is \(H\) and the horizontal range of \(B\) is \(R\). The ball is modelled as a particle moving freely under gravity.
  1. Show that
    1. \(H = \frac{V^2}{2g}\sin^2 \alpha\), [2]
    2. \(R = \frac{V^2}{g}\sin 2\alpha\). [3]
  2. Hence show that \(16H^2 - 8R_0 H + R^2 = 0\), where \(R_0\) is the maximum range for the given speed of projection. [5]
  3. Given that \(R_0 = 200\text{m}\) and \(R = 192\text{m}\), find
    1. the two possible values of the greatest height of \(B\), [2]
    2. the corresponding values of the angle of projection. [3]
  4. State one limitation of the model that could affect your answers to part (iii). [1]
OCR H240/03 2017 Specimen Q1
7 marks Moderate -0.3
  1. If \(|x| = 3\), find the possible values of \(|2x - 1|\). [3]
  2. Find the set of values of \(x\) for which \(|2x - 1| > x + 1\). Give your answer in set notation. [4]
OCR H240/03 2017 Specimen Q2
4 marks Easy -1.2
  1. Use the trapezium rule, with four strips each of width 0.25, to find an approximate value for \(\int_0^1 \frac{1}{\sqrt{1+x^2}} dx\). [3]
  2. Explain how the trapezium rule might be used to give a better approximation to the integral given in part (a). [1]
OCR H240/03 2017 Specimen Q3
4 marks Standard +0.8
In this question you must show detailed reasoning. Given that \(5\sin 2x = 3\cos x\), where \(0° < x < 90°\), find the exact value of \(\sin x\). [4]
OCR H240/03 2017 Specimen Q4
4 marks Standard +0.3
For a small angle \(\theta\), where \(\theta\) is in radians, show that \(1 + \cos \theta - 3\cos^2 \theta \approx -1 + \frac{3}{2}\theta^2\). [4]
OCR H240/03 2017 Specimen Q5
8 marks Standard +0.3
  1. Find the first three terms in the expansion of \((1 + px)^{\frac{1}{3}}\) in ascending powers of \(x\). [3]
  2. The expansion of \((1 + qx)(1 + px)^{\frac{1}{3}}\) is \(1 + x - \frac{2}{9}x^2 + ...\) Find the possible values of the constants \(p\) and \(q\). [5]