Questions — OCR H240/03 (80 questions)

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OCR H240/03 2018 June Q1
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
2 Solve the equation \(| 2 x - 1 | = | x + 3 |\).
OCR H240/03 2018 June Q4
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
    (a) \(\mathrm { fg } ( x )\),
    (b) \(\operatorname { gf } ( x )\).
  2. Hence find the values of \(x\) for which \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24\).
OCR H240/03 2018 June Q5
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
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
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
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 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
    (a) the stone block as a particle,
    (b) the plank as a rigid rod.
OCR H240/03 2018 June Q10
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
    (a) the magnitude of the resultant of the two remaining forces,
    (b) the direction of the resultant of the two remaining forces.
OCR H240/03 2018 June Q11
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
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.
    (a) Find the exact value of \(\mu\) when \(\lambda = \frac { 1 } { 4 }\).
    (b) Show that the value of \(\lambda\) must be less than \(\frac { 1 } { 2 }\).
  2. 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 2019 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-04_239_867_504_255} The diagram shows triangle \(A B C\), with \(A C = 13.5 \mathrm {~cm} , B C = 8.3 \mathrm {~cm}\) and angle \(A B C = 32 ^ { \circ }\).
Find angle \(C A B\).
OCR H240/03 2019 June Q2
2 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 4 = 0\).
  1. Find
    1. the coordinates of \(C\),
    2. the radius of the circle.
  2. Determine the set of values of \(k\) for which the line \(y = k x - 3\) does not intersect or touch the circle.
OCR H240/03 2019 June Q3
3
  1. In this question you must show detailed reasoning.
    Solve the inequality \(| x - 2 | \leqslant | 2 x - 6 |\).
  2. Give full details of a sequence of two transformations needed to transform the graph of \(y = | x - 2 |\) to the graph of \(y = | 2 x - 6 |\).
OCR H240/03 2019 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-05_456_634_260_251} The diagram shows the part of the curve \(y = 3 x \sin 2 x\) for which \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
The maximum point on the curve is denoted by \(P\).
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\tan 2 x + 2 x = 0\).
  2. Use the Newton-Raphson method, with a suitable initial value, to find the \(x\)-coordinate of \(P\), giving your answer correct to 4 decimal places. Show the result of each iteration.
  3. The trapezium rule, with four strips of equal width, is used to find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\). Show that the result can be expressed as \(k \pi ^ { 2 } ( \sqrt { 2 } + 1 )\), where \(k\) is a rational number to be determined.
    1. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\).
    2. Hence determine whether using the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region enclosed by the curve \(y = 3 x \sin 2 x\) and the \(x\)-axis for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
    3. Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region in this case.
OCR H240/03 2019 June Q7
7 A cyclist starting from rest accelerates uniformly at \(1.5 \mathrm {~ms} ^ { - 2 }\) 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. Calculate the distance travelled by the cyclist in the first 10 seconds.
OCR H240/03 2019 June Q8
8 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 \(\frac { 5 } { 3 } d \mathrm {~ms} ^ { - 1 }\).
  1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground.
  2. Find the vertical component of the initial velocity of \(P\).
    \(P\) just clears a vertical wall which is situated at a horizontal distance \(d \mathrm {~m}\) from \(O\).
  3. Find the height of the wall. The speed of \(P\) as it passes over the wall is \(16 \mathrm {~ms} ^ { - 1 }\).
  4. Find the value of \(d\) correct to 3 significant figures.
OCR H240/03 2019 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-08_362_1191_262_438} 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.
  2. Calculate the coefficient of friction between \(B\) and the horizontal surface.
OCR H240/03 2019 June Q10
10 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 \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = \left( p t ^ { 2 } - 3 t \right) \mathbf { i } + ( 8 t + q ) \mathbf { j }\), 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 \(p\). When \(t = 0 , R\) is at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m }\).
  2. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). When \(t = 1 , R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2 \mathbf { i } - 8 \mathbf { j }\).
  3. Find the value of \(q\).
    \includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-09_544_1297_251_255} The diagram shows a ladder \(A B\), of length \(2 a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2 m\) is placed on the ladder at a point \(C\) where \(A C = d\). The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.
  4. Show that the normal contact force between the ladder and the wall is \(\frac { m g ( a + 2 d ) \sqrt { 3 } } { 4 h }\). It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 8 } \sqrt { 3 }\).
  5. Show that \(h = k ( a + 2 d )\), where \(k\) is a constant to be determined.
  6. Hence find, in terms of \(a\), the greatest possible value of \(d\).
  7. State one improvement that could be made to the model.
OCR H240/03 2020 November Q1
1 Triangle \(A B C\) has \(A B = 8.5 \mathrm {~cm} , B C = 6.2 \mathrm {~cm}\) and angle \(B = 35 ^ { \circ }\). Calculate the area of the triangle.
OCR H240/03 2020 November Q2
2 A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
OCR H240/03 2020 November Q3
3 The functions f and g are defined for all real values of \(x\) by
\(f ( x ) = 2 x ^ { 2 } + 6 x\) and \(g ( x ) = 3 x + 2\).
  1. Find the range of f .
  2. Give a reason why f has no inverse.
  3. Given that \(\mathrm { fg } ( - 2 ) = \mathrm { g } ^ { - 1 } ( a )\), where \(a\) is a constant, determine the value of \(a\).
  4. Determine the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\). Give your answer in set notation.
OCR H240/03 2020 November Q4
4 A curve has equation \(y = 2 \ln ( k - 3 x ) + x ^ { 2 } - 3 x\), where \(k\) is a positive constant.
  1. Given that the curve has a point of inflection where \(x = 1\), show that \(k = 6\). 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 .
  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.
  4. By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.
OCR H240/03 2020 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-05_339_869_262_244} The diagram shows the curve \(C\) with parametric equations
\(x = \frac { 3 } { t } , y = t ^ { 3 } \mathrm { e } ^ { - 2 t }\), where \(t > 0\).
The maximum point on \(C\) is denoted by \(P\).
  1. Determine the exact coordinates of \(P\). 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 $$\int _ { a } ^ { b } 3 t \mathrm { e } ^ { - 2 t } \mathrm {~d} t ,$$ where \(a\) and \(b\) are constants to be determined.
  3. Hence determine the exact area of \(R\).
OCR H240/03 2020 November Q7
7 A particle \(P\) moves with constant acceleration \(( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds, \(P\) is moving with velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Determine the speed of \(P\) when \(t = 3\).
  2. Determine the change in displacement of \(P\) between \(t = 0\) and \(t = 3\).