Questions — OCR MEI Paper 1 (118 questions)

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OCR MEI Paper 1 2018 June Q1
1 Show that ( \(x - 2\) ) is a factor of \(3 x ^ { 3 } - 8 x ^ { 2 } + 3 x + 2\).
OCR MEI Paper 1 2018 June Q2
2 By considering a change of sign, show that the equation \(\mathrm { e } ^ { x } - 5 x ^ { 3 } = 0\) has a root between 0 and 1 .
OCR MEI Paper 1 2018 June Q3
3 In this question you must show detailed reasoning.
Solve the equation \(\sec ^ { 2 } \theta + 2 \tan \theta = 4\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
OCR MEI Paper 1 2018 June Q4
4 Rory pushes a box of mass 2.8 kg across a rough horizontal floor against a resistance of 19 N . Rory applies a constant horizontal force. The box accelerates from rest to \(1.2 \mathrm {~ms} ^ { - 1 }\) as it travels 1.8 m .
  1. Calculate the acceleration of the box.
  2. Find the magnitude of the force that Rory applies.
OCR MEI Paper 1 2018 June Q5
5 The position vector \(\mathbf { r }\) metres of a particle at time \(t\) seconds is given by $$\mathbf { r } = \left( 1 + 12 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. Determine whether the particle is ever stationary.
OCR MEI Paper 1 2018 June Q6
6 Aleela and Baraka are saving to buy a car. Aleela saves \(\pounds 50\) in the first month. She increases the amount she saves by \(\pounds 20\) each month.
  1. Calculate how much she saves in two years. Baraka also saves \(\pounds 50\) in the first month. The amount he saves each month is \(12 \%\) more than the amount he saved in the previous month.
  2. Explain why the amounts Baraka saves each month form a geometric sequence.
  3. Determine whether Baraka saves more in two years than Aleela. Answer all the questions
    Section B (77 marks)
OCR MEI Paper 1 2018 June Q7
7 A rod of length 2 m hangs vertically in equilibrium. Parallel horizontal forces of 30 N and 50 N are applied to the top and bottom and the rod is held in place by a horizontal force \(F \mathrm {~N}\) applied \(x \mathrm {~m}\) below the top of the rod as shown in Fig. 7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-05_445_390_609_824} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the value of \(F\).
  2. Find the value of \(x\).
OCR MEI Paper 1 2018 June Q8
8
  1. Show that \(8 \sin ^ { 2 } x \cos ^ { 2 } x\) can be written as \(1 - \cos 4 x\).
  2. Hence find \(\int \sin ^ { 2 } x \cos ^ { 2 } x \mathrm {~d} x\).
OCR MEI Paper 1 2018 June Q9
9 A pebble is thrown horizontally at \(14 \mathrm {~ms} ^ { - 1 }\) from a window which is 5 m above horizontal ground. The pebble goes over a fence 2 m high \(d \mathrm {~m}\) away from the window as shown in Fig. 9. The origin is on the ground directly below the window with the \(x\)-axis horizontal in the direction in which the pebble is thrown and the \(y\)-axis vertically upwards. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-06_538_1082_452_488} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the time the pebble takes to reach the ground.
  2. Find the cartesian equation of the trajectory of the pebble.
  3. Find the range of possible values for \(d\).
OCR MEI Paper 1 2018 June Q10
10 Fig. 10 shows the graph of \(y = ( k - x ) \ln x\) where \(k\) is a constant ( \(k > 1\) ). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-06_454_1266_1564_395} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Find, in terms of \(k\), the area of the finite region between the curve and the \(x\)-axis.
OCR MEI Paper 1 2018 June Q11
11 Fig. 11 shows two blocks at rest, connected by a light inextensible string which passes over a smooth pulley. Block A of mass 4.7 kg rests on a smooth plane inclined at \(60 ^ { \circ }\) to the horizontal. Block B of mass 4 kg rests on a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. On either side of the pulley, the string is parallel to a line of greatest slope of the plane. Block B is on the point of sliding up the plane. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-07_332_931_443_575} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Show that the tension in the string is 39.9 N correct to 3 significant figures.
  2. Find the coefficient of friction between the rough plane and Block B.
OCR MEI Paper 1 2018 June Q12
12 Fig. 12 shows the circle \(( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 25\), the line \(4 y = 3 x - 32\) and the tangent to the circle at the point \(\mathrm { A } ( 5,2 )\). D is the point of intersection of the line \(4 y = 3 x - 32\) and the tangent at A . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-07_750_773_1311_632} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Write down the coordinates of C , the centre of the circle.
  2. (A) Show that the line \(4 y = 3 x - 32\) is a tangent to the circle.
    (B) Find the coordinates of B , the point where the line \(4 y = 3 x - 32\) touches the circle.
  3. Prove that ADBC is a square.
  4. The point E is the lowest point on the circle. Find the area of the sector ECB .
OCR MEI Paper 1 2018 June Q13
13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).
\(x\)\(f ( x )\)\(f ^ { \prime } ( x )\)
030
0.252.9954- 0.056
0.52.9625- 0.228
0.752.8694- 0.547
12.6684- 1.124
1.252.2490- 1.977
1.50ERROR
  1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
  2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
  3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
  4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
    \end{figure}
  5. Explain why the trapezium rule gives an underestimate.
OCR MEI Paper 1 2018 June Q14
14 The velocity of a car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, is being modelled. Initially the car has velocity \(5 \mathrm {~ms} ^ { - 1 }\) and it accelerates to \(11.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 seconds. In model A, the acceleration is assumed to be uniform.
  1. Find an expression for the velocity of the car at time \(t\) using this model.
  2. Explain why this model is not appropriate in the long term. Model A is refined so that the velocity remains constant once the car reaches \(17.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Sketch a velocity-time graph for the motion of the car, making clear the time at which the acceleration changes.
  4. Calculate the displacement of the car in the first 20 seconds according to this refined model. In model B, the velocity of the car is given by $$v = \begin{cases} 5 + 0.6 t ^ { 2 } - 0.05 t ^ { 3 } & \text { for } 0 \leqslant t \leqslant 8
    17.8 & \text { for } 8 < t \leqslant 20 \end{cases}$$
  5. Show that this model gives an appropriate value for \(v\) when \(t = 4\).
  6. Explain why the value of the acceleration immediately before the velocity becomes constant is likely to mean that model B is a better model than model A.
  7. Show that model B gives the same value as model A for the displacement at time 20 s .
OCR MEI Paper 1 2019 June Q2
2 Show that the line which passes through the points \(( 2 , - 4 )\) and \(( - 1,5 )\) does not intersect the line \(3 x + y = 10\).
OCR MEI Paper 1 2019 June Q3
3 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }\), where \(a\) is a non-zero constant. In the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x\) and \(x ^ { 2 }\) are equal.
  1. Find the value of \(a\).
  2. Using this value for \(a\),
    1. state the set of values of \(x\) for which the binomial expansion is valid,
    2. write down the quadratic function which approximates \(\mathrm { f } ( x )\) when \(x\) is small.
OCR MEI Paper 1 2019 June Q4
4 Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q . P is at one end of the beam and Q is 0.3 m from the other end.
Determine whether a person of mass 50 kg can tip the beam by standing on it. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59e924e6-8fa9-4035-9173-705fce487bd9-4_195_977_1676_262} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
OCR MEI Paper 1 2019 June Q5
5 A car of mass 1200 kg travels from rest along a straight horizontal road. The driving force is 4000 N and the total of all resistances to motion is 800 N .
Calculate the velocity of the car after 9 seconds.
OCR MEI Paper 1 2019 June Q6
6
  1. Prove that \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = \cot \theta\).
  2. Hence find the exact roots of the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 3 \tan \theta\) in the interval \(0 \leqslant \theta \leqslant \pi\). Answer all the questions.
    Section B (75 marks)
OCR MEI Paper 1 2019 June Q7
7 The velocity \(v \mathrm {~ms} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by
\(v = 0.5 t ( 7 - t )\). Determine whether the speed of the particle is increasing or decreasing when \(t = 8\).
OCR MEI Paper 1 2019 June Q8
8 An arithmetic series has first term 9300 and 10th term 3900.
  1. Show that the 20th term of the series is negative.
  2. The sum of the first \(n\) terms is denoted by \(S\). Find the greatest value of \(S\) as \(n\) varies.
OCR MEI Paper 1 2019 June Q9
9 A cannonball is fired from a point on horizontal ground at \(100 \mathrm {~ms} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) above the horizontal. Ignoring air resistance, calculate
  1. the greatest height the cannonball reaches,
  2. the range of the cannonball.
OCR MEI Paper 1 2019 June Q10
10
  1. Express \(7 \cos x - 2 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
  2. Give details of a sequence of two transformations which maps the curve \(y = \sec x\) onto the curve \(y = \frac { 1 } { 7 \cos x - 2 \sin x }\).
OCR MEI Paper 1 2019 June Q11
11 In this question, the unit vector \(\mathbf { i }\) is horizontal and the unit vector \(\mathbf { j }\) is vertically upwards. A particle of mass 0.8 kg moves under the action of its weight and two forces given by ( \(k \mathbf { i } + 5 \mathbf { j }\) ) N and \(( 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\). The acceleration of the particle is vertically upwards.
  1. Write down the value of \(k\). Initially the velocity of the particle is \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  2. Find the velocity of the particle 10 seconds later.
OCR MEI Paper 1 2019 June Q12
12 Fig. 12 shows a curve C with parametric equations \(x = 4 t ^ { 2 } , y = 4 t\). The point P , with parameter \(t\), is a general point on the curve. Q is the point on the line \(x + 4 = 0\) such that PQ is parallel to the \(x\)-axis. R is the point \(( 4,0 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59e924e6-8fa9-4035-9173-705fce487bd9-6_766_584_413_255} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Show algebraically that P is equidistant from Q and R .
  2. Find a cartesian equation of C .