Questions — OCR (4907 questions)

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OCR H240/02 2018 December Q14
11 marks Standard +0.8
Mr Jones has 3 tins of beans and 2 tins of pears. His daughter has removed the labels for a school project, and the tins are identical in appearance. Mr Jones opens tins in turn until he has opened at least 1 tin of beans and at least 1 tin of pears. He does not open any remaining tins.
  1. Draw a tree diagram to illustrate this situation, labelling each branch with its associated probability. [3]
  2. Find the probability that Mr Jones opens exactly 3 tins. [3]
  3. It is given that the last tin Mr Jones opens is a tin of pears. Find the probability that he opens exactly 3 tins. [5]
OCR H240/02 2018 December Q15
9 marks Moderate -0.3
A fair dice is thrown 1000 times and the number, \(X\), of throws on which the score is 6 is noted.
    1. State the distribution of \(X\). [1]
    2. Explain why a normal distribution would be an appropriate approximation to the distribution of \(X\). [1]
  2. Use a normal distribution to find two positive integer values, \(a\) and \(b\), such that \(\text{P}(a < X < b) \approx 0.4\). [5]
  3. For your two values of \(a\) and \(b\), use the distribution of part (a)(i) to find the value of \(\text{P}(a < X < b)\), correct to 3 significant figures. [2]
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 H240/01 2017 Specimen Q1
4 marks Moderate -0.5
Solve the simultaneous equations. \(x^2 + 8x + y^2 = 84\) \(x - y = 10\) [4]
OCR H240/01 2017 Specimen Q2
5 marks Moderate -0.8
The points A, B and C have position vectors \(\mathbf{3i - 4j + 2k}\), \(\mathbf{-i + 6k}\) and \(\mathbf{7i - 4j - 2k}\) respectively. M is the midpoint of BC.
  1. Show that the magnitude of \(\overrightarrow{OM}\) is equal to \(\sqrt{17}\). [2]
Point D is such that \(\overrightarrow{BC} = \overrightarrow{AD}\).
  1. Show that position vector of the point D is \(\mathbf{1i - 8j - 6k}\). [3]
OCR H240/01 2017 Specimen Q3
3 marks Moderate -0.8
The diagram below shows the graph of \(y = f(x)\). \includegraphics{figure_3}
  1. On the diagram in the Printed Answer Booklet, draw the graph of \(y = f(\frac{1}{2}x)\). [1]
  2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = f(x - 2) + 1\). [2]
OCR H240/01 2017 Specimen Q4
7 marks Moderate -0.3
The diagram shows a sector \(AOB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_4} The angle \(AOB\) is \(\theta\) radians. The arc length \(AB\) is 15 cm and the area of the sector is 45 cm\(^2\).
  1. Find the values of \(r\) and \(\theta\). [4]
  2. Find the area of the segment bounded by the arc \(AB\) and the chord \(AB\). [3]
OCR H240/01 2017 Specimen Q5
4 marks Moderate -0.3
In this question you must show detailed reasoning. Use logarithms to solve the equation \(3^{2x+1} = 4^{100}\), giving your answer correct to 3 significant figures. [4]
OCR H240/01 2017 Specimen Q6
3 marks Moderate -0.5
Prove by contradiction that there is no greatest even positive integer. [3]
OCR H240/01 2017 Specimen Q7
10 marks Moderate -0.8
Business A made a £5000 profit during its first year. In each subsequent year, the profit increased by £1500 so that the profit was £6500 during the second year, £8000 during the third year and so on. Business B made a £5000 profit during its first year. In each subsequent year, the profit was 90% of the previous year's profit.
  1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form. [2]
  2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form. [3]
  3. Find how many years it will take for the total profit of business A to reach £385 000. [3]
  4. Comment on the profits made by each business in the long term. [2]
OCR H240/01 2017 Specimen Q8
6 marks Standard +0.3
  1. Show that \(\frac{2\tan\theta}{1 + \tan^2\theta} = \sin 2\theta\). [3]
  1. In this question you must show detailed reasoning. Solve \(\frac{2\tan\theta}{1 + \tan^2\theta} = 3\cos 2\theta\) for \(0 \leq \theta \leq \pi\). [3]
OCR H240/01 2017 Specimen Q9
9 marks Standard +0.3
The equation \(x^3 - x^2 - 5x + 10 = 0\) has exactly one real root \(\alpha\).
  1. Show that the Newton-Raphson iterative formula for finding this root can be written as $$x_{n+1} = \frac{2x_n^3 - x_n^2 - 10}{3x_n^2 - 2x_n - 5}.$$ [3]
  2. Apply the iterative formula in part (a) with initial value \(x_1 = -3\) to find \(x_2, x_3, x_4\) correct to 4 significant figures. [1]
  3. Use a change of sign method to show that \(\alpha = -2.533\) is correct to 4 significant figures. [3]
  4. Explain why the Newton-Raphson method with initial value \(x_1 = -1\) would not converge to \(\alpha\). [2]
OCR H240/01 2017 Specimen Q10
8 marks Standard +0.3
A curve has equation \(x = (y + 5)\ln(2y - 7)\).
  1. Find \(\frac{dx}{dy}\) in terms of y. [3]
  2. Find the gradient of the curve where it crosses the y-axis. [5]
OCR H240/01 2017 Specimen Q11
9 marks Moderate -0.3
For all real values of \(x\), the functions f and g are defined by \(f(x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.
  1. Find fg\((x)\). Determine the range of fg\((x)\) in terms of \(a\). [4]
  2. If fg\((2) = 144\), find the value of \(a\). [3]
  3. Determine whether the function fg has an inverse. [2]
OCR H240/01 2017 Specimen Q12
11 marks Standard +0.8
The parametric equations of a curve are given by \(x = 2\cos\theta\) and \(y = 3\sin\theta\) for \(0 \leq \theta < 2\pi\).
  1. Find \(\frac{dy}{dx}\) in terms of \(\theta\). [2]
The tangents to the curve at the points P and Q pass through the point (2, 6).
  1. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2\sin\theta + \cos\theta = 1\). [4]
  2. Find the values of \(\theta\) at the points P and Q. [5]