Questions — Edexcel (10514 questions)

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Edexcel C2 Q7
11 marks Moderate -0.3
  1. Find the coordinates of the points where the curve and line intersect.
  2. Find the area of the shaded region bounded by the curve and line.
Edexcel C4 Q4
11 marks Challenging +1.2
  1. Show that the volume of the solid formed is \(\frac { 1 } { 4 } \pi ( \pi + 2 )\).
  2. Find a cartesian equation for the curve.
Edexcel S1 Q2
9 marks Moderate -0.8
  1. Plot a scatter diagram showing these data. The student wanted to investigate further whether or not her data provided evidence of an increase in temperature in June each year. Using \(Y\) for the number of years since 1993 and \(T\) for the mean temperature, she calculated the following summary statistics. $$\Sigma Y = 28 , \quad \Sigma T = 182.5 , \quad \Sigma Y ^ { 2 } = 140 , \quad \Sigma T ^ { 2 } = 4173.93 , \quad \Sigma Y T = 644.7 .$$
  2. Calculate the product moment correlation coefficient for these data.
  3. Comment on your result in relation to the student's enquiry.
Edexcel M5 2002 June Q6
17 marks Challenging +1.8
  1. Show that the moment of inertia of the rod about the edge of the table is \(\frac { 7 } { 3 } m a ^ { 2 }\). The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle \(\theta\), its angular speed is \(\dot { \theta }\). Assuming that the rod has not started to slip,
  2. show that \(\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }\),
  3. find the angular acceleration of the rod,
  4. find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is \(\mu\).
  5. Show that the rod starts to slip when \(\tan \theta = \frac { 4 } { 13 } \mu\) (6)
Edexcel M5 2005 June Q4
11 marks Challenging +1.8
  1. Show that the moment of inertia of the body about \(L\) is \(\frac { 77 m a ^ { 2 } } { 4 }\). When \(P R\) is vertical, the body has angular speed \(\omega\) and the centre of the disc strikes a stationary particle of mass \(\frac { 1 } { 2 } \mathrm {~m}\). Given that the particle adheres to the centre of the disc,
  2. find, in terms of \(\omega\), the angular speed of the body immediately after the impact.
Edexcel AEA 2005 June Q6
19 marks Challenging +1.8
  1. Find the coordinates of the points \(P , Q\) and \(R\).
  2. Sketch, on separate diagrams, the graphs of
    1. \(y = \mathrm { f } ( 2 x )\),
    2. \(y = \mathrm { f } ( | x | + 1 )\),
      indicating on each sketch the coordinates of any maximum points and the intersections with the \(x\)-axis.
      (6) \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f9d3e02c-cef2-435b-9cda-76c43fcac575-5_1015_1464_232_337}
      \end{figure} Figure 2 shows a sketch of part of the curve \(C\), with equation \(y = \mathrm { f } ( x - v ) + w\), where \(v\) and \(w\) are constants. The \(x\)-axis is a tangent to \(C\) at the minimum point \(T\), and \(C\) intersects the \(y\)-axis at \(S\). The line joining \(S\) to the maximum point \(U\) is parallel to the \(x\)-axis.
  3. Find the value of \(v\) and the value of \(w\) and hence find the roots of the equation $$f ( x - v ) + w = 0$$
Edexcel AEA 2006 June Q6
15 marks Challenging +1.2
  1. Show that the point \(P ( 1,0 )\) lies on \(C\) .
  2. Find the coordinates of the point \(Q\) .
  3. Find the area of the shaded region between \(C\) and the line \(P Q\) .
Edexcel AEA 2007 June Q6
17 marks Hard +2.3
  1. Find an expression, in terms of \(x\), for the area \(A\) of \(R\).
  2. Show that \(\frac { \mathrm { d } A } { \mathrm {~d} x } = \frac { 1 } { 4 } ( \pi - 2 x - 2 \sin x ) \sec ^ { 2 } \frac { x } { 2 }\).
  3. Prove that the maximum value of \(A\) occurs when \(\frac { \pi } { 4 } < x < \frac { \pi } { 3 }\).
  4. Prove that \(\tan \frac { \pi } { 8 } = \sqrt { } 2 - 1\).
  5. Show that the maximum value of \(A > \frac { \pi } { 4 } ( \sqrt { } 2 - 1 )\).
Edexcel CP AS Specimen Q6
15 marks Standard +0.3
  1. Prove by induction that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$
  2. Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r + 6 ) ( r - 6 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n - 8 ) ( n + 9 )$$
  3. Hence find the value of \(n\) that satisfies $$\sum _ { r = 1 } ^ { n } r ( r + 6 ) ( r - 6 ) = 17 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$
Edexcel C1 Q1
Easy -1.3
  1. (a) Write down the value of \(8 ^ { \frac { 1 } { 3 } }\).
    (b) Find the value of \(8 ^ { - \frac { 2 } { 3 } }\).
  2. Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
    (a) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
    (b) find \(\int y \mathrm {~d} x\).
Edexcel C1 Q2
Moderate -0.8
2. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1$$
  1. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Write down the value of \(u _ { 20 }\).
Edexcel C1 Q4
Moderate -0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-006_689_920_292_511}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the origin \(O\) and through the point \(( 6,0 )\). The maximum point on the curve is \(( 3,5 )\). On separate diagrams, sketch the curve with equation
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( x + 2 )\). On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.
Edexcel C1 Q5
Moderate -0.5
5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$
Edexcel C1 Q8
Moderate -0.8
8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) passes through the origin \(O\) and has gradient - 2 . The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
  2. Calculate the coordinates of \(P\). Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
  3. calculate the exact area of \(\triangle O C P\). \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882} \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}
Edexcel C1 Q9
Moderate -0.8
9. An arithmetic series has first term \(a\) and common difference \(d\).
  1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
  2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
  3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
  4. Solve the equation in part (c).
  5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.
Edexcel C3 Q1
8 marks Standard +0.3
  1. Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \tan ^ { 2 } \theta \equiv \sec ^ { 2 } \theta\).
  2. Solve, for \(0 \leq \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta + \sec \theta = 1 ,$$ giving your answers to 1 decimal place.
Edexcel C3 Q2
10 marks Moderate -0.3
2. (a) Differentiate with respect to \(x\)
  1. \(3 \sin ^ { 2 } x + \sec 2 x\),
  2. \(\{ x + \ln ( 2 x ) \} ^ { 3 }\). Given that \(y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , x \neq 1\),
    (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).
Edexcel C3 Q3
12 marks Standard +0.3
3. The function \(f\) is defined by $$f : x \mapsto \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1\).
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } + 5 , \quad x \in \mathbb { R } .$$ (b) Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 4 }\).
Edexcel C3 Q4
10 marks Standard +0.3
4. $$\mathrm { f } ( x ) = 3 \mathrm { e } ^ { x } - \frac { 1 } { 2 } \ln x - 2 , \quad x > 0 .$$
  1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) has a turning point at \(P\). The \(x\)-coordinate of \(P\) is \(\alpha\).
  2. Show that \(\alpha = \frac { 1 } { 6 } \mathrm { e } ^ { - \alpha }\). The iterative formula $$x _ { n + 1 } = \frac { 1 } { 6 } \mathrm { e } ^ { - x _ { n } } , \quad x _ { 0 } = 1$$ is used to find an approximate value for \(\alpha\).
  3. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
  4. By considering the change of sign of \(\mathrm { f } ^ { \prime } ( x )\) in a suitable interval, prove that \(\alpha = 0.1443\) correct to 4 decimal places.
Edexcel C3 Q5
8 marks Standard +0.3
5.
  1. Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$
  2. Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$
  3. Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  4. Hence, for \(0 \leq \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
    Hence, for \(0 \leq \theta < \pi\), solve \includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.
Edexcel C3 Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_371_643_338_1852}
\end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
    Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
  3. the value of \(a\) and the value of \(b\),
  4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).
Edexcel C3 Q7
11 marks Standard +0.8
7. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be $$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,
  1. show that \(a = 0.12\),
  2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850 .
  3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
  4. Hence show that the population cannot exceed 2800.
Edexcel C3 Q8
13 marks Moderate -0.8
8. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + \ln 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$
  1. Prove that the composite function gf is $$\operatorname { gf } : x \mapsto 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
  2. Sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
  3. Write down the range of gf .
  4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.
Edexcel C3 Q9
9 marks Moderate -0.3
9. (i) Find the exact solutions to the equations
  1. \(\ln ( 3 x - 7 ) = 5\),
  2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\).
    (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } , \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , \quad x > 1 . \end{array}$$
    1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
    2. Find fg and state its range.
Edexcel C3 Q6
Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-009_458_876_285_539}
\end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
  3. the value of \(a\) and the value of \(b\),
  4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).