Questions — Edexcel Paper 2 (135 questions)

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Edexcel Paper 2 2021 October Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-30_630_630_312_721} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where \(k\) is a positive constant.
  1. Sketch the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = k - | 2 x - 3 k |$$ stating
    • the coordinates of the maximum point
    • the coordinates of any points where the graph cuts the coordinate axes
    • Find, in terms of \(k\), the set of values of \(x\) for which
    $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
  2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 f \left( \frac { 1 } { 2 } x \right)$$
Edexcel Paper 2 2021 October Q12
  1. (a) Use the substitution \(u = 1 + \sqrt { x }\) to show that
$$\int _ { 0 } ^ { 16 } \frac { \mathrm { x } } { 1 + \sqrt { \mathrm { x } } } \mathrm {~d} x = \int _ { p } ^ { q } \frac { 2 ( u - 1 ) ^ { 3 } } { u } \mathrm {~d} u$$ where \(p\) and \(q\) are constants to be found.
(b) Hence show that $$\int _ { 0 } ^ { 16 } \frac { \mathrm { x } } { 1 + \sqrt { \mathrm { x } } } \mathrm {~d} x = A - B \ln 5$$ where \(A\) and \(B\) are constants to be found.
Edexcel Paper 2 2021 October Q13
  1. The curve \(C\) has parametric equations
$$x = \sin 2 \theta \quad y = \operatorname { cosec } ^ { 3 } \theta \quad 0 < \theta < \frac { \pi } { 2 }$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\)
  2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\)
Edexcel Paper 2 2021 October Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-40_513_919_294_548} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Water flows at a constant rate into a large tank.
The tank is a cuboid, with all sides of negligible thickness.
The base of the tank measures 8 m by 3 m and the height of the tank is 5 m .
There is a tap at a point \(T\) at the bottom of the tank, as shown in Figure 5.
At time \(t\) minutes after the tap has been opened
  • the depth of water in the tank is \(h\) metres
  • water is flowing into the tank at a constant rate of \(0.48 \mathrm {~m} ^ { 3 }\) per minute
  • water is modelled as leaving the tank through the tap at a rate of \(0.1 h \mathrm {~m} ^ { 3 }\) per minute
    1. Show that, according to the model,
$$1200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 24 - 5 h$$ Given that when the tap was opened, the depth of water in the tank was 2 m ,
  • show that, according to the model, $$h = A + B \mathrm { e } ^ { - k t }$$ where \(A , B\) and \(k\) are constants to be found. Given that the tap remains open,
  • determine, according to the model, whether the tank will ever become full, giving a reason for your answer.
    •
    Edexcel Paper 2 2021 October Q15
    1. (a) Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
    Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-44_440_1118_463_575} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} Figure 6 shows the cross-section of a water wheel.
    The wheel is free to rotate about a fixed axis through the point \(C\).
    The point \(P\) is at the end of one of the paddles of the wheel, as shown in Figure 6.
    The water level is assumed to be horizontal and of constant height.
    The vertical height, \(H\) metres, of \(P\) above the water level is modelled by the equation $$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$ where \(t\) is the time in seconds after the wheel starts rotating.
    Using the model, find
    (b) (i) the maximum height of \(P\) above the water level,
    (ii) the value of \(t\) when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, \(P\) is below the water level for a total of \(T\) seconds. According to the model,
    (c) find the value of \(T\) giving your answer to 3 significant figures.
    (Solutions based entirely on calculator technology are not acceptable.) In reality, the water level may not be of constant height.
    (d) Explain how the equation of the model should be refined to take this into account.
    Edexcel Paper 2 Specimen Q1
    1. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + a$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).
    Edexcel Paper 2 Specimen Q2
    2. Some A level students were given the following question. Solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation $$\cos \theta = 2 \sin \theta$$ The attempts of two of the students are shown below.
    \(\underline { \text { Student } A }\)
    \(\cos \theta = 2 \sin \theta\)
    \(\tan \theta = 2\)
    \(\theta = 63.4 ^ { \circ }\)
    Student \(B\) $$\begin{aligned} \cos \theta & = 2 \sin \theta
    \cos ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta
    1 - \sin ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta
    \sin ^ { 2 } \theta & = \frac { 1 } { 5 }
    \sin \theta & = \pm \frac { 1 } { \sqrt { 5 } }
    \theta & = \pm 26.6 ^ { \circ } \end{aligned}$$
    1. Identify an error made by student \(A\). Student \(B\) gives \(\theta = - 26.6 ^ { \circ }\) as one of the answers to \(\cos \theta = 2 \sin \theta\).
      1. Explain why this answer is incorrect.
      2. Explain how this incorrect answer arose.
    Edexcel Paper 2 Specimen Q3
    3. Given \(y = x ( 2 x + 1 ) ^ { 4 }\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) ^ { n } ( A x + B )$$ where \(n , A\) and \(B\) are constants to be found.
    Edexcel Paper 2 Specimen Q4
    4. Given $$\begin{aligned} & \mathrm { f } ( x ) = \mathrm { e } ^ { x } , \quad x \in \mathbb { R }
    & \mathrm {~g} ( x ) = 3 \ln x , \quad x > 0 , x \in \mathbb { R } \end{aligned}$$
    1. find an expression for \(\mathrm { gf } ( x )\), simplifying your answer.
    2. Show that there is only one real value of \(x\) for which \(\operatorname { gf } ( x ) = \operatorname { fg } ( x )\)
    Edexcel Paper 2 Specimen Q5
    5. The mass, \(m\) grams, of a radioactive substance, \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { - 0.05 t }$$ According to the model,
    1. find the mass of the radioactive substance six months after it was first observed,
    2. show that \(\frac { \mathrm { d } m } { \mathrm {~d} t } = k m\), where \(k\) is a constant to be found.
    Edexcel Paper 2 Specimen Q6
    6. Complete the table below. The first one has been done for you. For each statement you must state if it is always true, sometimes true or never true, giving a reason in each case.
    StatementAlways TrueSometimes TrueNever TrueReason
    The quadratic equation \(a x ^ { 2 } + b x + c = 0 , \quad ( a \neq 0 )\) has 2 real roots.✓It only has 2 real roots when \(b ^ { 2 } - 4 a c > 0\). When \(b ^ { 2 } - 4 a c = 0\) it has 1 real root and when \(b ^ { 2 } - 4 a c < 0\) it has 0 real roots.
    (i)
    When a real value of \(x\) is substituted into \(x ^ { 2 } - 6 x + 10\) the result is positive.
    (ii)
    If \(a x > b\) then \(x > \frac { b } { a }\)
    (2)
    (iii)
    The difference between consecutive square numbers is odd.
    Edexcel Paper 2 Specimen Q7
    1. (a) Use the binomial expansion, in ascending powers of \(x\), to show that
    $$\sqrt { ( 4 - x ) } = 2 - \frac { 1 } { 4 } x + k x ^ { 2 } + \ldots$$ where \(k\) is a rational constant to be found. A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt { 3 }\).
    (b) State, giving a reason, if the expansion is valid for this value of \(x\).
    Edexcel Paper 2 Specimen Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a75c9ef7-b648-47be-bad1-fc8b315be3df-10_602_999_260_534} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a rectangle \(A B C D\).
    The point \(A\) lies on the \(y\)-axis and the points \(B\) and \(D\) lie on the \(x\)-axis as shown in Figure 1. Given that the straight line through the points \(A\) and \(B\) has equation \(5 y + 2 x = 10\)
    1. show that the straight line through the points \(A\) and \(D\) has equation \(2 y - 5 x = 4\)
    2. find the area of the rectangle \(A B C D\).
    Edexcel Paper 2 Specimen Q9
    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
    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
    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
    1. (a) 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.
    (b) 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
    13. (a) 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,
    (b) (i) find a complete equation for the model,
    (ii) hence find the maximum height of the passenger above the ground.
    (c) 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.
    (d) How would you adapt the equation of the model to reflect this increase in speed?
    Edexcel Paper 2 Specimen Q14
    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
    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
    1. (a) 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,
    (b) determine the time taken, in years, for this population of meerkats to double,
    (c) 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
    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
    1. (a) 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 \(\theta = 5 ^ { \circ }\)
    (b) (i) Identify the mistake made by Adele in her working.
    (ii) 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
    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
    1. (a) Sketch the graph with equation
    $$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
    (b) Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$ (c) Find the values of \(x\) which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.