Questions — Edexcel (9685 questions)

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Edexcel C1 2011 June Q2
7 marks Easy -1.2
Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form, (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) \(\int y \mathrm {~d} x\).
Edexcel C1 2011 June Q3
5 marks Moderate -0.8
The points \(P\) and \(Q\) have coordinates \(( - 1,6 )\) and \(( 9,0 )\) respectively. The line \(l\) is perpendicular to \(P Q\) and passes through the mid-point of \(P Q\).
Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
Edexcel C1 2013 June Q2
4 marks Easy -1.8
Express \(\frac { 15 } { \sqrt { 3 } } - \sqrt { 27 }\) in the form \(k \sqrt { } 3\), where \(k\) is an integer.
Edexcel C1 2013 June Q3
4 marks Easy -1.3
Find $$\int \left( 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x$$ giving each term in its simplest form.
Edexcel C1 2014 June Q2
4 marks Easy -1.3
  1. Evaluate \(81 ^ { \frac { 3 } { 2 } }\)
  2. Simplify fully \(x ^ { 2 } \left( 4 x ^ { - \frac { 1 } { 2 } } \right) ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-03_83_150_2675_1804}
Edexcel C1 2014 June Q3
5 marks Moderate -0.5
A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{array} { l l } a _ { n + 1 } = 4 a _ { n } - 3 , & n \geqslant 1 \\ a _ { 1 } = k , & \text { where } k \text { is a positive integer. } \end{array}$$
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { r = 1 } ^ { 3 } a _ { r } = 66\)
  2. find the value of \(k\).
Edexcel C1 2015 June Q2
7 marks Moderate -0.3
Solve the simultaneous equations $$\begin{gathered} y - 2 x - 4 = 0 \\ 4 x ^ { 2 } + y ^ { 2 } + 20 x = 0 \end{gathered}$$
Edexcel C1 2015 June Q4
8 marks Moderate -0.8
  1. A sequence \(U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots\) is defined by $$\begin{gathered} U _ { n + 2 } = 2 U _ { n + 1 } - U _ { n } , \quad n \geqslant 1 \\ U _ { 1 } = 4 \text { and } U _ { 2 } = 4 \end{gathered}$$ Find the value of
    (a) \(U _ { 3 }\) (b) \(\sum _ { n = 1 } ^ { 20 } U _ { n }\)
  2. Another sequence \(V _ { 1 } , V _ { 2 } , V _ { 3 } , \ldots\) is defined by
    (a) Find \(V _ { 3 }\) and \(V _ { 4 }\) in terms of \(k\). $$\begin{gathered} V _ { n + 2 } = 2 V _ { n + 1 } - V _ { n } , \quad n \geqslant 1 \\ V _ { 1 } = k \text { and } V _ { 2 } = 2 k , \text { where } k \text { is a constant } \end{gathered}$$ a) Find \(V _ { 3 }\)
Edexcel C1 2015 June Q10
10 marks Standard +0.3
A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 4,9 )\). Given that $$f ^ { \prime } ( x ) = \frac { 3 \sqrt { } x } { 2 } - \frac { 9 } { 4 \sqrt { } x } + 2 , \quad x > 0$$
  1. find \(\mathrm { f } ( x )\), giving each term in its simplest form. Point \(P\) lies on the curve. The normal to the curve at \(P\) is parallel to the line \(2 y + x = 0\)
  2. Find the \(x\) coordinate of \(P\).
Edexcel C1 2016 June Q3
5 marks Easy -1.3
  1. Simplify $$\sqrt { 50 } - \sqrt { 18 }$$ giving your answer in the form \(a \sqrt { 2 }\), where \(a\) is an integer.
  2. Hence, or otherwise, simplify $$\frac { 12 \sqrt { 3 } } { \sqrt { 50 } - \sqrt { 18 } }$$ giving your answer in the form \(b \sqrt { c }\), where \(b\) and \(c\) are integers and \(b \neq 1\)
Edexcel C1 Q4
5 marks Easy -1.2
A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$a _ { 1 } = k , \quad a _ { n + 1 } = 4 a _ { n } - 7 ,$$ where \(k\) is a constant.
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  2. Find \(a _ { 3 }\) in terms of \(k\), simplifying your answer. Given that \(a _ { 3 } = 13\),
  3. find the value of \(k\).
Edexcel P2 2018 Specimen Q5
11 marks Easy -1.2
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 $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$$ A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).
  2. Find the value of \(N\) The company then plans to continue to make 600 mobile phones each week.
  3. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
    \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-16_2673_1948_107_118}
Edexcel C2 2005 January Q2
6 marks Easy -1.2
The points \(A\) and \(B\) have coordinates \(( 5 , - 1 )\) and \(( 13,11 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\). Given that \(A B\) is a diameter of the circle \(C\),
  2. find an equation for \(C\).
Edexcel C2 2009 January Q2
5 marks Moderate -0.5
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-03_870_1027_205_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = ( 1 + x ) ( 4 - x )\).
The curve intersects the \(x\)-axis at \(x = - 1\) and \(x = 4\). The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis. Use calculus to find the exact area of \(R\).
Edexcel C2 2005 June Q2
6 marks Moderate -0.8
Solve
  1. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
  2. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).
Edexcel C2 2005 June Q3
6 marks Moderate -0.8
  1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
  2. Factorise \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\) completely.
Edexcel C2 2005 June Q4
6 marks Standard +0.3
  1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + p x ) ^ { 12 }\), where \(p\) is a non-zero constant. Given that, in the expansion of \(( 1 + p x ) ^ { 12 }\), the coefficient of \(x\) is \(( - q )\) and the coefficient of \(x ^ { 2 }\) is \(11 q\),
  2. find the value of \(p\) and the value of \(q\).
Edexcel C2 2006 June Q2
5 marks Easy -1.2
Use calculus to find the exact value of \(\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
Edexcel C2 2006 June Q3
4 marks Easy -1.2
  1. Write down the value of \(\log _ { 6 } 36\).
  2. Express \(2 \log _ { a } 3 + \log _ { a } 11\) as a single logarithm to base \(a\).
Edexcel C2 2006 June Q4
8 marks Moderate -0.8
$$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2007 June Q2
6 marks Moderate -0.8
$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ). Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 Specimen Q2
4 marks Easy -1.2
The circle \(C\) has centre \(( 3,4 )\) and passes through the point \(( 8 , - 8 )\). Find an equation for C
Edexcel C2 Specimen Q3
6 marks Moderate -0.8
The trapezium rule, with the table below, was used to estimate the area between the curve \(y = \sqrt { x ^ { 3 } + 1 }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis.
\(x\)11.522.53
\(y\)1.4142.0923.000
  1. Calculate, to 3 decimal places, the values of \(y\) for \(x = 2.5\) and \(x = 3\).
  2. Use the values from the table and your answers to part (a) to find an estimate, to 2 decimal places, for this area.
Edexcel C3 2012 January Q2
6 marks Standard +0.3
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{363c5b5d-7ea9-41e5-ae5a-c3efb5626af5-03_716_1122_212_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 3,0 )\) and \(Q ( 2 , - 4 )\) are stationary points on the graph.
Sketch, on separate diagrams, the graphs of
  1. \(y = 3 \mathrm { f } ( x + 2 )\)
  2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of any stationary points.
Edexcel C3 2006 June Q2
6 marks Moderate -0.8
Differentiate, with respect to \(x\),
  1. \(\mathrm { e } ^ { 3 x } + \ln 2 x\),
  2. \(\left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\).