Questions — Edexcel (10514 questions)

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Edexcel Paper 2 Specimen Q9
5 marks Standard +0.3
  1. Given that \(A\) is constant and
$$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 2 A ^ { 2 }$$ show that there are exactly two possible values for \(A\).
Edexcel Paper 2 Specimen Q10
4 marks Standard +0.8
10. In a geometric series the common ratio is \(r\) and sum to \(n\) terms is \(S _ { n }\) Given $$S _ { \infty } = \frac { 8 } { 7 } \times S _ { 6 }$$ show that \(r = \pm \frac { 1 } { \sqrt { k } }\), where \(k\) is an integer to be found.
Edexcel Paper 2 Specimen Q11
6 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a75c9ef7-b648-47be-bad1-fc8b315be3df-14_570_556_205_758} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$
  1. State the range of f
  2. Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has two distinct roots, (c) state the set of possible values for \(k\).
Edexcel Paper 2 Specimen Q12
8 marks Standard +0.3
  1. Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation $$3 \sin ^ { 2 } x + \sin x + 8 = 9 \cos ^ { 2 } x$$ giving your answers to 2 decimal places.
  2. Hence find the smallest positive solution of the equation $$3 \sin ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) + \sin \left( 2 \theta - 30 ^ { \circ } \right) + 8 = 9 \cos ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right)$$ giving your answer to 2 decimal places.
Edexcel Paper 2 Specimen Q13
9 marks Standard +0.3
13.
  1. Express \(10 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the exact value of \(R\) and give the value of \(\alpha\), in degrees, to 2 decimal places. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a75c9ef7-b648-47be-bad1-fc8b315be3df-18_396_1329_388_367} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The height above the ground, \(H\) metres, of a passenger on a Ferris wheel \(t\) minutes after the wheel starts turning, is modelled by the equation $$H = a - 10 \cos ( 80 t ) ^ { \circ } + 3 \sin ( 80 t ) ^ { \circ }$$ where \(a\) is a constant.
    Figure 3 shows the graph of \(H\) against \(t\) for two complete cycles of the wheel.
    Given that the initial height of the passenger above the ground is 1 metre,
    1. find a complete equation for the model,
    2. hence find the maximum height of the passenger above the ground.
  3. Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle.
    (Solutions based entirely on graphical or numerical methods are not acceptable.) It is decided that, to increase profits, the speed of the wheel is to be increased.
  4. How would you adapt the equation of the model to reflect this increase in speed?
Edexcel Paper 2 Specimen Q14
9 marks Standard +0.3
  1. A company decides to manufacture a soft drinks can with a capacity of 500 ml .
The company models the can in the shape of a right circular cylinder with radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). In the model they assume that the can is made from a metal of negligible thickness.
  1. Prove that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the can is given by $$S = 2 \pi r ^ { 2 } + \frac { 1000 } { r }$$ Given that \(r\) can vary,
  2. find the dimensions of a can that has minimum surface area.
  3. With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area.
Edexcel Paper 2 Specimen Q15
10 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a75c9ef7-b648-47be-bad1-fc8b315be3df-22_796_974_244_548} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with equation $$y = 5 x ^ { \frac { 3 } { 2 } } - 9 x + 11 , x \geqslant 0$$ The point \(P\) with coordinates \(( 4,15 )\) lies on \(C\).
The line \(l\) is the tangent to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis. Show that the area of \(R\) is 24 , making your method clear.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 2 Specimen Q16
12 marks Standard +0.8
  1. Express \(\frac { 1 } { P ( 11 - 2 P ) }\) in partial fractions. A population of meerkats is being studied.
    The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 22 } P ( 11 - 2 P ) , \quad t \geqslant 0 , \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began,
  2. determine the time taken, in years, for this population of meerkats to double,
  3. show that $$P = \frac { A } { B + C \mathrm { e } ^ { - \frac { 1 } { 2 } t } }$$ where \(A , B\) and \(C\) are integers to be found.
Edexcel Paper 2 Specimen Q1
4 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-02_364_369_374_849} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a circle with centre \(O\). The points \(A\) and \(B\) lie on the circumference of the circle. The area of the major sector, shown shaded in Figure 1, is \(135 \mathrm {~cm} ^ { 2 }\). The reflex angle \(A O B\) is 4.8 radians. Find the exact length, in cm, of the minor arc \(A B\), giving your answer in the form \(a \pi + b\), where \(a\) and \(b\) are integers to be found.
(4)
Edexcel Paper 2 Specimen Q2
5 marks Moderate -0.8
  1. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$ Adele uses \(\theta = 5 ^ { \circ }\) to test the approximation in part (a).
    Adele's working is shown below. Using my calculator, \(1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962\), to 3 decimal places.
    Using the approximation \(8 - 5 \theta ^ { 2 }\) gives \(8 - 5 ( 5 ) ^ { 2 } = - 117\) Therefore, \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }\) is not true for \(\theta = 5 ^ { \circ }\)
    1. Identify the mistake made by Adele in her working.
    2. Show that \(8 - 5 \theta ^ { 2 }\) can be used to give a good approximation to \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta\) for an angle of size \(5 ^ { \circ }\) (2)
Edexcel Paper 2 Specimen Q3
4 marks Moderate -0.3
  1. A cup of hot tea was placed on a table. At time \(t\) minutes after the cup was placed on the table, the temperature of the tea in the cup, \(\theta ^ { \circ } \mathrm { C }\), is modelled by the equation
$$\theta = 25 + A \mathrm { e } ^ { - 0.03 t }$$ where \(A\) is a constant. The temperature of the tea was \(75 ^ { \circ } \mathrm { C }\) when the cup was placed on the table.
  1. Find a complete equation for the model.
  2. Use the model to find the time taken for the tea to cool from \(75 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\), giving your answer in minutes to one decimal place. Two hours after the cup was placed on the table, the temperature of the tea was measured as \(20.3 ^ { \circ } \mathrm { C }\). Using this information,
  3. evaluate the model, explaining your reasoning.
Edexcel Paper 2 Specimen Q4
6 marks Moderate -0.3
  1. Sketch the graph with equation $$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
  2. Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$
  3. Find the values of \(x\) which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.
Edexcel Paper 2 Specimen Q5
5 marks Standard +0.3
  1. The line \(l\) has equation
$$3 x - 2 y = k$$ where \(k\) is a real constant.
Given that the line \(l\) intersects the curve with equation $$y = 2 x ^ { 2 } - 5$$ at two distinct points, find the range of possible values for \(k\).
Edexcel Paper 2 Specimen Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-12_624_1057_258_504} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 2.
  1. Find the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\).
  2. Show that the \(x\) coordinate of \(Q\) satisfies $$x = \frac { 8 } { 1 + \ln x }$$
  3. Show that the \(x\) coordinate of \(Q\) lies between 3.5 and 3.6
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$ with \(x _ { 1 } = 3.5\) to
    1. find the value of \(x _ { 5 }\) to 4 decimal places,
    2. find the \(x\) coordinate of \(Q\) accurate to 2 decimal places.
Edexcel Paper 2 Specimen Q7
12 marks Moderate -0.8
  1. A bacterial culture has area \(p \mathrm {~mm} ^ { 2 }\) at time \(t\) hours after the culture was placed onto a circular dish.
A scientist states that at time \(t\) hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture.
  1. Show that the scientist's model for \(p\) leads to the equation $$p = a \mathrm { e } ^ { k t }$$ where \(a\) and \(k\) are constants. The scientist measures the values for \(p\) at regular intervals during the first 24 hours after the culture was placed onto the dish. She plots a graph of \(\ln p\) against \(t\) and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95
  2. Estimate, to 2 significant figures, the value of \(a\) and the value of \(k\).
  3. Hence show that the model for \(p\) can be rewritten as $$p = a b ^ { t }$$ stating, to 3 significant figures, the value of the constant \(b\). With reference to this model,
    1. interpret the value of the constant \(a\),
    2. interpret the value of the constant \(b\).
  5. State a long term limitation of the model for \(p\).
Edexcel Paper 2 Specimen Q8
7 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-18_367_709_280_676} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl. When the depth of the water is \(h \mathrm {~cm}\), the volume of water, \(V \mathrm {~cm} ^ { 3 }\), according to the model is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$ The flow of water into the bowl is at a constant rate of \(160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(0 \leqslant h \leqslant 12\)
  1. Find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 10\) Given that the flow of water into the bowl is increased to a constant rate of \(300 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(12 < h \leqslant 24\)
  2. find the rate of change of the depth of the water, in \(\mathrm { cms } ^ { - 1 }\), when \(h = 20\)
Edexcel Paper 2 Specimen Q9
9 marks Standard +0.3
A circle with centre \(A ( 3 , - 1 )\) passes through the point \(P ( - 9,8 )\) and the point \(Q ( 15 , - 10 )\)
  1. Show that \(P Q\) is a diameter of the circle.
  2. Find an equation for the circle. A point \(R\) also lies on the circle. Given that the length of the chord \(P R\) is 20 units,
  3. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
  4. Find the size of angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.
Edexcel Paper 2 Specimen Q10
9 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-22_554_862_260_603} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { t + 1 } , \quad t > - \frac { 2 } { 3 }$$
  1. State the domain of values of \(x\) for the curve \(C\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = \ln 2\), the \(x\)-axis and the line with equation \(x = \ln 4\)
  2. Use calculus to show that the area of \(R\) is \(\ln \left( \frac { 3 } { 2 } \right)\).
Edexcel Paper 2 Specimen Q11
5 marks Standard +0.8
  1. The second, third and fourth terms of an arithmetic sequence are \(2 k , 5 k - 10\) and \(7 k - 14\) respectively, where \(k\) is a constant.
Show that the sum of the first \(n\) terms of the sequence is a square number.
Edexcel Paper 2 Specimen Q12
7 marks Standard +0.8
  1. A curve \(C\) is given by the equation
$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 2 Specimen Q13
10 marks Standard +0.8
13.
  1. Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 2 Specimen Q14
8 marks Standard +0.3
  1. Kayden claims that $$3 ^ { x } \geqslant 2 ^ { x }$$ Determine whether Kayden's claim is always true, sometimes true or never true, justifying your answer.
  2. Prove that \(\sqrt { 3 }\) is an irrational number.
Edexcel Paper 3 2019 June Q1
6 marks Moderate -0.8
  1. \hspace{0pt} [In this question position vectors are given relative to a fixed origin \(O\) ]
At time \(t\) seconds, where \(t \geqslant 0\), a particle, \(P\), moves so that its velocity \(\mathbf { v ~ m ~ s } ^ { - 1 }\) is given by $$\mathbf { v } = 6 t \mathbf { i } - 5 t ^ { \frac { 3 } { 2 } } \mathbf { j }$$ When \(t = 0\), the position vector of \(P\) is \(( - 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m }\).
  1. Find the acceleration of \(P\) when \(t = 4\)
  2. Find the position vector of \(P\) when \(t = 4\)
Edexcel Paper 3 2019 June Q2
8 marks Standard +0.3
  1. A particle, \(P\), moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
At time \(t = 0\), the particle is at the point \(A\) and is moving with velocity ( \(- \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(3 \mathbf { i } - 4 \mathbf { j }\) )
  1. Find the value of \(T\). At time \(t = 4\) seconds, \(P\) is at the point \(B\).
  2. Find the distance \(A B\).
Edexcel Paper 3 2019 June Q3
12 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-06_339_812_242_628} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two blocks, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are attached to the ends of a light string. Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined at angle \(\alpha\) to the horizontal ground, where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley, \(P\), fixed 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. Block \(B\) hangs freely below \(P\), as shown in Figure 1. The coefficient of friction between \(A\) and the plane is \(\frac { 2 } { 3 }\) The blocks are released from rest with the string taut and \(A\) moves up the plane.
The tension in the string immediately after the blocks are released is \(T\).
The blocks are modelled as particles and the string is modelled as being inextensible.
  1. Show that \(T = \frac { 12 m g } { 5 }\) After \(B\) reaches the ground, \(A\) continues to move up the plane until it comes to rest before reaching \(P\).
  2. Determine whether \(A\) will remain at rest, carefully justifying your answer.
  3. Suggest two refinements to the model that would make it more realistic.