Questions H240/03 (134 questions)

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OCR H240/03 2018 June Q1
3 marks Easy -1.2
1 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 x - 2 y - 7 = 0\).
Find
  1. the coordinates of \(C\),
  2. the radius of the circle.
OCR H240/03 2018 June Q2
3 marks Moderate -0.8
2 Solve the equation \(| 2 x - 1 | = | x + 3 |\).
OCR H240/03 2018 June Q4
8 marks Moderate -0.3
4 In this question you must show detailed reasoning.
The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + 2 .$$
  1. Write down expressions for
    1. \(\mathrm { fg } ( x )\),
    2. \(\operatorname { gf } ( x )\).
    3. Hence find the values of \(x\) for which \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24\).
OCR H240/03 2018 June Q5
13 marks Standard +0.3
5
  1. Use the trapezium rule, with two strips of equal width, to show that $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x \approx \frac { 11 } { 4 } - \sqrt { 2 }$$
  2. Use the substitution \(x = u ^ { 2 }\) to find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x$$
  3. Using your answers to parts (i) and (ii), show that $$\ln 2 \approx k + \frac { \sqrt { 2 } } { 4 }$$ where \(k\) is a rational number to be determined.
OCR H240/03 2018 June Q6
8 marks Standard +0.3
6 It is given that the angle \(\theta\) satisfies the equation \(\sin \left( 2 \theta + \frac { 1 } { 4 } \pi \right) = 3 \cos \left( 2 \theta + \frac { 1 } { 4 } \pi \right)\).
  1. Show that \(\tan 2 \theta = \frac { 1 } { 2 }\).
  2. Hence find, in surd form, the exact value of \(\tan \theta\), given that \(\theta\) is an obtuse angle.
OCR H240/03 2018 June Q7
9 marks Standard +0.3
7 The gradient of the curve \(y = \mathrm { f } ( x )\) is given by the differential equation $$( 2 x - 1 ) ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y ^ { 2 } = 0$$ and the curve passes through the point \(( 1,1 )\). By solving this differential equation show that $$f ( x ) = \frac { a x ^ { 2 } - a x + 1 } { b x ^ { 2 } - b x + 1 }$$ where \(a\) and \(b\) are integers to be determined.
OCR H240/03 2018 June Q8
6 marks Moderate -0.8
8 In this question \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) denote unit vectors which are horizontal and vertically upwards respectively.
A particle of mass 5 kg , initially at rest at the point with position vector \(\binom { 2 } { 45 } \mathrm {~m}\), is acted on by gravity and also by two forces \(\binom { 15 } { - 8 } \mathrm {~N}\) and \(\binom { - 7 } { - 2 } \mathrm {~N}\).
  1. Find the acceleration vector of the particle.
  2. Find the position vector of the particle after 10 seconds.
OCR H240/03 2018 June Q9
9 marks Standard +0.3
9 A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-6_181_1271_1101_395} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find
  1. the magnitude of the reaction of the support on the plank at \(D\),
  2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
  3. Find the weight of the stone block.
  4. Explain the limitation of modelling
    1. the stone block as a particle,
    2. the plank as a rigid rod.
OCR H240/03 2018 June Q10
11 marks Standard +0.3
10 Three forces, of magnitudes \(4 \mathrm {~N} , 6 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-7_604_601_306_724} The forces are in equilibrium.
  1. Show that \(\theta = 41.4 ^ { \circ }\), correct to 3 significant figures.
  2. Hence find the value of \(P\). The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.
  3. Find
    1. the magnitude of the resultant of the two remaining forces,
    2. the direction of the resultant of the two remaining forces.
OCR H240/03 2018 June Q11
10 marks Moderate -0.3
11 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a car at time \(t \mathrm {~s}\), during the first 20 s of its journey, is given by \(v = k t + 0.03 t ^ { 2 }\), where \(k\) is a constant. When \(t = 20\) the acceleration of the car is \(1.3 \mathrm {~ms} ^ { - 2 }\). For \(t > 20\) the car continues its journey with constant acceleration \(1.3 \mathrm {~ms} ^ { - 2 }\) until its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).
  1. Find the value of \(k\).
  2. Find the total distance the car has travelled when its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).
OCR H240/03 2018 June Q12
14 marks Standard +0.3
12 One end of a light inextensible string is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a second particle \(B\) of mass \(\lambda m \mathrm {~kg}\), where \(\lambda\) is a constant. Particle \(A\) is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. 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. The particle \(B\) hangs freely below \(P\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-8_405_670_493_685} The coefficient of friction between \(A\) and the plane is \(\mu\).
  1. It is given that \(A\) is on the point of moving down the plane.
    1. Find the exact value of \(\mu\) when \(\lambda = \frac { 1 } { 4 }\).
    2. Show that the value of \(\lambda\) must be less than \(\frac { 1 } { 2 }\).
    3. Given instead that \(\lambda = 2\) and that the acceleration of \(A\) is \(\frac { 1 } { 4 } g \mathrm {~ms} ^ { - 2 }\), find the exact value of \(\mu\). \section*{END OF QUESTION PAPER}
OCR H240/03 2018 June Q3
6 marks Moderate -0.3
3 In this question you must show detailed reasoning. A gardener is planning the design for a rectangular flower bed. The requirements are:
  • the length of the flower bed is to be 3 m longer than the width,
  • the length of the flower bed must be at least 14.5 m ,
  • the area of the flower bed must be less than \(180 \mathrm {~m} ^ { 2 }\).
The width of the flower bed is \(x \mathrm {~m}\).
By writing down and solving appropriate inequalities in \(x\), determine the set of possible values for the width of the flower bed.
OCR H240/03 2018 September Q1
5 marks Easy -1.8
1
  1. Show that \(4 x ^ { 2 } - 12 x + 3 = 4 \left( x - \frac { 3 } { 2 } \right) ^ { 2 } - 6\).
  2. State the coordinates of the minimum point of the curve \(y = 4 x ^ { 2 } - 12 x + 3\).
OCR H240/03 2018 September Q2
6 marks Standard +0.3
2 A curve has equation \(y = a x ^ { 4 } + b x ^ { 3 } - 2 x + 3\).
  1. Given that the curve has a stationary point where \(x = 2\), show that \(16 a + 6 b = 1\).
  2. Given also that this stationary point is a point of inflection, determine the values of \(a\) and \(b\).
OCR H240/03 2018 September Q3
7 marks Standard +0.3
3
  1. Given that \(\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta\), show that \(6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0\).
  2. In this question you must show detailed reasoning. Solve the equation $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\) correct to 1 decimal place.
  3. Explain why not all the solutions from part (ii) are solutions of the equation $$\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta$$
OCR H240/03 2018 September Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-05_787_892_267_568} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = \begin{cases} 4 - 4 x , & x \leqslant a , \\ \ln ( b x - 8 ) - 2 , & x \geqslant a . \end{cases}$$ The range of f is \(\mathrm { f } ( x ) \geqslant - 2\).
  1. Show that \(a = \frac { 3 } { 2 }\).
  2. Find the value of \(b\).
  3. Find the exact value of \(\mathrm { ff } ( - 1 )\).
  4. Explain why the function f does not have an inverse.
OCR H240/03 2018 September Q5
8 marks Challenging +1.2
5 The curve \(C\) has equation $$3 x ^ { 2 } - 5 x y + \mathrm { e } ^ { 2 y - 4 } + 6 = 0$$ The point \(P\) with coordinates \(( 1,2 )\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\) and the normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(B\). Find the exact area of triangle \(A B P\).
OCR H240/03 2018 September Q6
16 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-06_463_702_264_685} The diagram shows the curve \(C\) with parametric equations $$x = \frac { 1 } { 4 } \sin t , \quad y = t \cos t$$ where \(0 \leqslant t \leqslant k\).
  1. Find the value of \(k\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) in terms of \(t\). The maximum point on \(C\) is denoted by \(P\).
  3. Using your answer to part (ii) and the standard small angle approximations, find an approximation for the \(x\)-coordinate of \(P\).
  4. (a) Show that the area of the finite region bounded by \(C\) and the \(x\)-axis is given by $$b \int _ { 0 } ^ { a } t ( 1 + \cos 2 t ) \mathrm { d } t$$ where \(a\) and \(b\) are constants to be determined.
    (b) In this question you must show detailed reasoning. Hence find the exact area of the finite region bounded by \(C\) and the \(x\)-axis.
OCR H240/03 2018 September Q7
5 marks Moderate -0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-07_512_1072_484_502} The diagram shows the velocity-time graph for a train travelling on a straight level track between stations \(A\) and \(B\) that are 2 km apart. The train leaves \(A\), accelerating uniformly from rest for 400 m until reaching a speed of \(32 \mathrm {~ms} ^ { - 1 }\). The train then travels at this steady speed for \(T\) seconds before decelerating uniformly at \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\). Find the total time for the journey.
OCR H240/03 2018 September Q8
8 marks Standard +0.3
8 At time \(t\) seconds a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When \(t = 2 , P\) is moving parallel to the vector \(\mathbf { i }\).
  1. Show that \(k = - 5\).
  2. Find the values of \(t\) when the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\).
OCR H240/03 2018 September Q9
8 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-08_302_992_260_539} The diagram shows a plank of wood \(A B\), of mass 10 kg and length 6 m , resting with its end \(A\) on rough horizontal ground and its end \(B\) in contact with a fixed cylindrical oil drum. The plank is in a vertical plane perpendicular to the axis of the drum, and the line \(A B\) is a tangent to the circular cross-section of the drum, with the point of contact at \(B\). The plank is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The plank is modelled as a uniform rod and the oil drum is modelled as being smooth.
  1. Find, in terms of \(g\), the normal contact force between the drum and the plank.
  2. Given that the plank is in limiting equilibrium, find the coefficient of friction between the plank and the ground.
OCR H240/03 2018 September Q10
14 marks Standard +0.3
10 A small ball \(P\) is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) and moves freely under gravity. The horizontal and vertically upwards displacements of the ball from \(O\) at any subsequent time \(t\) seconds are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The ball is modelled as a particle and the acceleration due to gravity is taken to be \(10 \mathrm {~ms} ^ { - 2 }\).
  1. Show that the equation of the trajectory of \(P\) is $$y = x \tan \theta - \frac { x ^ { 2 } } { 5 } \left( 1 + \tan ^ { 2 } \theta \right)$$ It is given that \(\tan \theta = 3\).
  2. Using part (i), find the maximum height above the level of \(O\) of \(P\) in the subsequent motion.
  3. Find the values of \(t\) when \(P\) is moving at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 1 } { 3 }\).
  4. Give two possible reasons why the values of \(t\) found in part (iii) may not be accurate. \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-09_435_714_267_678} Two particles \(P\) and \(Q\), with masses 2 kg and 8 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. Plane \(\Pi _ { 1 }\) is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and plane \(\Pi _ { 2 }\) is inclined at an angle of \(\theta\) to the horizontal. Particle \(P\) is on \(\Pi _ { 1 }\) and \(Q\) is on \(\Pi _ { 2 }\) with the string taut (see diagram). \(\Pi _ { 1 }\) is rough and the coefficient of friction between \(P\) and \(\Pi _ { 1 }\) is \(\frac { \sqrt { 3 } } { 3 }\). \(\Pi _ { 2 }\) is smooth.
    The particles are released from rest and \(P\) begins to move towards the pulley with an acceleration of \(g \cos \theta \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  5. Show that \(\theta\) satisfies the equation $$4 \sin \theta - 5 \cos \theta = 1 .$$
  6. By expressing \(4 \sin \theta - 5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), find, correct to 3 significant figures, the tension in the string.
OCR H240/03 Q1
7 marks Standard +0.3
1
  1. If \(| x | = 3\), find the possible values of \(| 2 x - 1 |\).
  2. Find the set of values of \(x\) for which \(| 2 x - 1 | > x + 1\). Give your answer in set notation.
OCR H240/03 Q2
4 marks Moderate -0.8
2
  1. Use the trapezium rule, with four strips each of width 0.25 , to find an approximate value for \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } } \mathrm {~d} x\).
  2. Explain how the trapezium rule might be used to give a better approximation to the integral given in part (a).
OCR H240/03 Q3
4 marks Standard +0.8
3 In this question you must show detailed reasoning. Given that \(5 \sin 2 x = 3 \cos x\), where \(0 ^ { \circ } < x < 90 ^ { \circ }\), find the exact value of \(\sin x\).