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Edexcel Paper 2 2021 October Q4
3 marks Standard +0.3
  1. Given that \(\theta\) is small and measured in radians, use the small angle approximations to show that
$$4 \sin \frac { \theta } { 2 } + 3 \cos ^ { 2 } \theta \approx a + b \theta + c \theta ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.
Edexcel Paper 2 2021 October Q5
7 marks Moderate -0.8
  1. The curve \(C\) has equation
$$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$
  1. Find
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
    1. Verify that \(C\) has a stationary point at \(x = 1\)
    2. Show that this stationary point is a point of inflection, giving reasons for your answer.
Edexcel Paper 2 2021 October Q6
5 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-12_487_784_292_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shape \(O A B C D E F O\) shown in Figure 1 is a design for a logo.
In the design
  • \(O A B\) is a sector of a circle centre \(O\) and radius \(r\)
  • sector \(O F E\) is congruent to sector \(O A B\)
  • \(O D C\) is a sector of a circle centre \(O\) and radius \(2 r\)
  • \(A O F\) is a straight line
Given that the size of angle \(C O D\) is \(\theta\) radians,
  1. write down, in terms of \(\theta\), the size of angle \(A O B\)
  2. Show that the area of the logo is $$\frac { 1 } { 2 } r ^ { 2 } ( 3 \theta + \pi )$$
  3. Find the perimeter of the logo, giving your answer in simplest form in terms of \(r , \theta\) and \(\pi\).
Edexcel Paper 2 2021 October Q7
9 marks Standard +0.3
  1. In this question you should show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-16_805_1041_388_511} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point \(P ( 5 , - 13 )\) lies on \(C\) The line \(l\) is the tangent to \(C\) at \(P\)
  1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
  2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).
  3. Use algebraic integration to find the exact area of \(R\).
Edexcel Paper 2 2021 October Q8
9 marks Standard +0.3
  1. The curve \(C\) has equation
$$p x ^ { 3 } + q x y + 3 y ^ { 2 } = 26$$ where \(p\) and \(q\) are constants.
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a p x ^ { 2 } + b q y } { q x + c y }$$ where \(a\), \(b\) and \(c\) are integers to be found. Given that
    • the point \(P ( - 1 , - 4 )\) lies on \(C\)
    • the normal to \(C\) at \(P\) has equation \(19 x + 26 y + 123 = 0\)
    • find the value of \(p\) and the value of \(q\).
Edexcel Paper 2 2021 October Q9
3 marks Standard +0.8
  1. Show that
$$\sum _ { n = 2 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n } \cos ( 180 n ) ^ { \circ } = \frac { 9 } { 28 }$$
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Edexcel Paper 2 2021 October Q10
6 marks Moderate -0.3
  1. The time, \(T\) seconds, that a pendulum takes to complete one swing is modelled by the formula
$$T = a l ^ { b }$$ where \(l\) metres is the length of the pendulum and \(a\) and \(b\) are constants.
  1. Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-26_581_888_749_625} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A student carried out an experiment to find the values of the constants \(a\) and \(b\).
    The student recorded the value of \(T\) for different values of \(l\).
    Figure 3 shows the linear relationship between \(\log _ { 10 } l\) and \(\log _ { 10 } T\) for the student's data.
    The straight line passes through the points \(( - 0.7,0 )\) and \(( 0.21,0.45 )\) Using this information,
  2. find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of \(a\) and the value of \(b\), each to 3 significant figures.
  3. With reference to the model, interpret the value of the constant \(a\).
Edexcel Paper 2 2021 October Q11
10 marks Standard +0.8
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
7 marks Standard +0.8
  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
6 marks Standard +0.3
  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
12 marks Standard +0.3
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
    11 marks Standard +0.3
    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
    3 marks Easy -1.2
    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
    3 marks Easy -1.2
    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
    4 marks Moderate -0.8
    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
    5 marks Moderate -0.5
    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
    4 marks Moderate -0.8
    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 marks Moderate -0.8
    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
    5 marks Moderate -0.3
    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
    7 marks Standard +0.3
    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
    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. (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
    9 marks Standard +0.3
    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?