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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}
Edexcel AS Paper 1 2018 June Q1
4 marks Easy -1.3
  1. Find
$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - 6 \sqrt { x } + 1 \right) \mathrm { d } x$$ giving your answer in its simplest form.
Edexcel AS Paper 1 2018 June Q2
5 marks Moderate -0.8
  1. Show that \(x ^ { 2 } - 8 x + 17 > 0\) for all real values of \(x\)
  2. "If I add 3 to a number and square the sum, the result is greater than the square of the original number." State, giving a reason, if the above statement is always true, sometimes true or never true.
Edexcel AS Paper 1 2018 June Q3
4 marks Easy -1.3
Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
(b) find \(| \overrightarrow { A B } |\). Give your answer as a simplified surd.
Edexcel AS Paper 1 2018 June Q4
4 marks Moderate -0.8
  1. The line \(l _ { 1 }\) has equation \(4 y - 3 x = 10\)
The line \(l _ { 2 }\) passes through the points \(( 5 , - 1 )\) and \(( - 1,8 )\).
Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.
Edexcel AS Paper 1 2018 June Q5
5 marks Moderate -0.5
  1. A student's attempt to solve the equation \(2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3\) is shown below.
$$\begin{aligned} & 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3 \\ & 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3 \\ & 2 \log _ { 2 } ( \sqrt { x } ) = 3 \\ & \log _ { 2 } x = 3 \\ & x = 3 ^ { 2 } = 9 \end{aligned}$$ using the subtraction law for logs simplifying using the power law for logs using the definition of a log
  1. Identify two errors made by this student, giving a brief explanation of each.
  2. Write out the correct solution.
Edexcel AS Paper 1 2018 June Q6
7 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-12_599_1084_292_486} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A company makes a particular type of children's toy.
The annual profit made by the company is modelled by the equation $$P = 100 - 6.25 ( x - 9 ) ^ { 2 }$$ where \(P\) is the profit measured in thousands of pounds and \(x\) is the selling price of the toy in pounds. A sketch of \(P\) against \(x\) is shown in Figure 1.
Using the model,
  1. explain why \(\pounds 15\) is not a sensible selling price for the toy. Given that the company made an annual profit of more than \(\pounds 80000\)
  2. find, according to the model, the least possible selling price for the toy. The company wishes to maximise its annual profit.
    State, according to the model,
    1. the maximum possible annual profit,
    2. the selling price of the toy that maximises the annual profit.
Edexcel AS Paper 1 2018 June Q7
6 marks Standard +0.3
In a triangle \(A B C\), side \(A B\) has length 10 cm , side \(A C\) has length 5 cm , and angle \(B A C = \theta\) where \(\theta\) is measured in degrees. The area of triangle \(A B C\) is \(15 \mathrm {~cm} ^ { 2 }\)
  1. Find the two possible values of \(\cos \theta\) Given that \(B C\) is the longest side of the triangle,
  2. find the exact length of \(B C\).
Edexcel AS Paper 1 2018 June Q8
9 marks Moderate -0.3
  1. A lorry is driven between London and Newcastle.
In a simple model, the cost of the journey \(\pounds C\) when the lorry is driven at a steady speed of \(v\) kilometres per hour is $$C = \frac { 1500 } { v } + \frac { 2 v } { 11 } + 60$$
  1. Find, according to this model,
    1. the value of \(v\) that minimises the cost of the journey,
    2. the minimum cost of the journey.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)
  2. Prove by using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) that the cost is minimised at the speed found in (a)(i).
  3. State one limitation of this model.
Edexcel AS Paper 1 2018 June Q9
9 marks Standard +0.3
9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\).
  2. Hence show that \(\mathrm { g } ( x )\) can be written in the form \(\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }\), where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { g } ( x )\)
  3. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
    1. \(\mathrm { g } ( x ) \leqslant 0\)
    2. \(\mathrm { g } ( 2 x ) = 0\)
Edexcel AS Paper 1 2018 June Q10
4 marks Moderate -0.5
  1. Prove, from first principles, that the derivative of \(x ^ { 3 }\) is \(3 x ^ { 2 }\)
Edexcel AS Paper 1 2018 June Q11
8 marks Standard +0.3
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { x } { 16 } \right) ^ { 9 }$$ giving each term in its simplest form. $$f ( x ) = ( a + b x ) \left( 2 - \frac { x } { 16 } \right) ^ { 9 } , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm { f } ( x )\) are 128 and \(36 x\),
  2. find the value of \(a\),
  3. find the value of \(b\).
Edexcel AS Paper 1 2018 June Q12
8 marks Standard +0.3
  1. Show that the equation $$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$
  2. Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 2018 June Q13
8 marks Moderate -0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-36_563_1019_244_523} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The value of a rare painting, \(\pounds V\), is modelled by the equation \(V = p q ^ { t }\), where \(p\) and \(q\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1980. The line \(l\) shown in Figure 3 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) since 1st January 1980. The equation of line \(l\) is \(\log _ { 10 } V = 0.05 t + 4.8\)
  1. Find, to 4 significant figures, the value of \(p\) and the value of \(q\).
  2. With reference to the model interpret
    1. the value of the constant \(p\),
    2. the value of the constant \(q\).
  3. Find the value of the painting, as predicted by the model, on 1st January 2010, giving your answer to the nearest hundred thousand pounds.
Edexcel AS Paper 1 2018 June Q14
9 marks Standard +0.3
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\)
    2. the radius of \(C\) The line with equation \(y = k x\), where \(k\) is a constant, cuts \(C\) at two distinct points.
  2. Find the range of values for \(k\).
Edexcel AS Paper 1 2018 June Q15
10 marks Standard +0.8
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-44_595_977_242_536} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point \(P ( 4,6 )\) lies on \(C\).
The line \(l\) is the normal to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 2\) and the \(x\)-axis. Show that the area of \(R\) is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 2019 June Q1
4 marks Easy -1.2
  1. The line \(l _ { 1 }\) has equation \(2 x + 4 y - 3 = 0\)
The line \(l _ { 2 }\) has equation \(y = m x + 7\), where \(m\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,
  1. find the value of \(m\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\).
  2. Find the \(x\) coordinate of \(P\). \includegraphics[max width=\textwidth, alt={}, center]{deba6a2b-1821-4110-bde8-bde18a5f9be9-02_2258_48_313_1980}
Edexcel AS Paper 1 2019 June Q2
8 marks Moderate -0.8
Find, using algebra, all real solutions to the equation
  1. \(16 a ^ { 2 } = 2 \sqrt { a }\)
  2. \(b ^ { 4 } + 7 b ^ { 2 } - 18 = 0\)
Edexcel AS Paper 1 2019 June Q3
6 marks Moderate -0.8
  1. Given that \(k\) is a constant, find $$\int \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x$$ simplifying your answer.
  2. Hence find the value of \(k\) such that $$\int _ { 0.5 } ^ { 2 } \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x = 8$$
Edexcel AS Paper 1 2019 June Q4
5 marks Moderate -0.8
  1. A tree was planted in the ground.
Its height, \(H\) metres, was measured \(t\) years after planting.
Exactly 3 years after planting, the height of the tree was 2.35 metres.
Exactly 6 years after planting, the height of the tree was 3.28 metres.
Using a linear model,
  1. find an equation linking \(H\) with \(t\). The height of the tree was approximately 140 cm when it was planted.
  2. Explain whether or not this fact supports the use of the linear model in part (a).