Questions — Edexcel (9685 questions)

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Edexcel C12 2015 January Q2
7 marks Moderate -0.3
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-03_473_654_233_603} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the graph of \(y = \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } , x \geqslant 2\) The table below gives values of \(y\) rounded to 3 decimal places.
\(x\)25811
\(y\)8.4852.5021.5241.100
  1. Use the trapezium rule with all the values of \(y\) from the table to find an approximate value, to 2 decimal places, for $$\int _ { 2 } ^ { 11 } \frac { 12 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \mathrm { d } x$$
  2. Use your answer to part (a) to estimate a value for $$\int _ { 2 } ^ { 11 } \left( 1 + \frac { 6 } { \sqrt { \left( x ^ { 2 } - 2 \right) } } \right) d x$$
Edexcel C12 2017 January Q2
7 marks Moderate -0.8
A circle, with centre \(C\) and radius \(r\), has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 4 y - 12 = 0$$ Find
  1. the coordinates of \(C\),
  2. the exact value of \(r\). The circle cuts the \(y\)-axis at the points \(A\) and \(B\).
  3. Find the coordinates of the points \(A\) and \(B\).
Edexcel C12 2019 June Q3
6 marks Moderate -0.3
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de511cb3-35c7-4225-b459-a136b6304b78-06_955_1495_217_226} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points \(( - 6,0 )\) and \(( 0,3 )\), has a stationary point at \(( - 3,9 )\) and has an asymptote with equation \(y = 1\) On separate diagrams, sketch the curve with equation
  1. \(y = - \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } \left( \frac { 3 } { 2 } x \right)\) On each diagram, show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-07_2255_45_316_36}
Edexcel C12 Specimen Q2
4 marks Easy -1.2
Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 - x ) ^ { 6 }$$ and simplify each term.
Edexcel C1 2005 January Q2
8 marks Easy -1.3
  1. Given that \(y = 5 x ^ { 3 } + 7 x + 3\), find
    (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), (b) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find \(\int \left( 1 + 3 \sqrt { } x - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).
Edexcel C1 2008 January Q2
3 marks Easy -1.3
  1. Write down the value of \(16 ^ { \frac { 1 } { 4 } }\).
  2. Simplify \(\left( 16 x ^ { 12 } \right) ^ { \frac { 3 } { 4 } }\).
Edexcel C1 2008 January Q3
4 marks Easy -1.3
Simplify $$\frac { 5 - \sqrt { 3 } } { 2 + \sqrt { 3 } } ,$$ giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are integers.
Edexcel C1 2009 January Q2
4 marks Easy -1.8
Find \(\int \left( 12 x ^ { 5 } - 8 x ^ { 3 } + 3 \right) \mathrm { d } x\), giving each term in its simplest form.
Edexcel C1 2009 January Q3
2 marks Easy -1.8
Expand and simplify \(( \sqrt { } 7 + 2 ) ( \sqrt { } 7 - 2 )\).
Edexcel C1 2009 January Q4
5 marks Moderate -0.8
A curve has equation \(y = \mathrm { f } ( x )\) and passes through the point (4, 22). Given that $$\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x ^ { \frac { 1 } { 2 } } - 7 ,$$ use integration to find \(\mathrm { f } ( x )\), giving each term in its simplest form.
Edexcel C1 2010 January Q2
6 marks Easy -1.2
  1. Expand and simplify \(( 7 + \sqrt { 5 } ) ( 3 - \sqrt { 5 } )\).
  2. Express \(\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
Edexcel C1 2010 January Q3
5 marks Moderate -0.8
The line \(l _ { 1 }\) has equation \(3 x + 5 y - 2 = 0\)
  1. Find the gradient of \(l _ { 1 }\). The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 3,1 )\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C1 2011 January Q2
5 marks Easy -1.3
Find $$\int \left( 12 x ^ { 5 } - 3 x ^ { 2 } + 4 x ^ { \frac { 1 } { 3 } } \right) \mathrm { d } x$$ giving each term in its simplest form.
Edexcel C1 2012 January Q2
6 marks Easy -1.3
  1. Simplify $$\sqrt { } 32 + \sqrt { } 18$$ giving your answer in the form \(a \sqrt { } 2\), where \(a\) is an integer.
  2. Simplify $$\frac { \sqrt { } 32 + \sqrt { } 18 } { 3 + \sqrt { } 2 }$$ giving your answer in the form \(b \sqrt { } 2 + c\), where \(b\) and \(c\) are integers.
Edexcel C1 2013 January Q2
2 marks Easy -1.3
Express \(8 ^ { 2 x + 3 }\) in the form \(2 ^ { y }\), stating \(y\) in terms of \(x\).
Edexcel C1 2013 January Q3
6 marks Easy -1.3
  1. Express $$( 5 - \sqrt { } 8 ) ( 1 + \sqrt { } 2 )$$ in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
  2. Express $$\sqrt { } 80 + \frac { 30 } { \sqrt { } 5 }$$ in the form \(c \sqrt { } 5\), where \(c\) is an integer.
Edexcel C1 2005 June Q2
5 marks Easy -1.2
Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\int y \mathrm {~d} x\).
Edexcel C1 2005 June Q3
6 marks Moderate -0.8
$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b ,$$ where \(a\) and \(b\) are constants.
  1. Find the value of \(a\) and the value of \(b\).
  2. Hence, or otherwise, show that the roots of $$x ^ { 2 } - 8 x - 29 = 0$$ are \(c \pm d \sqrt { } 5\), where \(c\) and \(d\) are integers to be found.
Edexcel C1 2006 June Q2
4 marks Moderate -0.8
Find the set of values of \(x\) for which $$x ^ { 2 } - 7 x - 18 > 0 .$$
Edexcel C1 2007 June Q2
4 marks Easy -1.3
  1. Find the value of \(8 ^ { \frac { 4 } { 3 } }\).
  2. Simplify \(\frac { 15 x ^ { \frac { 4 } { 3 } } } { 3 x }\).
Edexcel C1 2007 June Q3
7 marks Easy -1.3
Given that \(y = 3 x ^ { 2 } + 4 \sqrt { } x , x > 0\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
  3. \(\int y \mathrm {~d} x\).
Edexcel C1 2007 June Q4
6 marks Moderate -0.8
A girl saves money over a period of 200 weeks. She saves 5 p in Week 1,7 p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence.
  1. Find the amount she saves in Week 200.
  2. Calculate her total savings over the complete 200 week period.
Edexcel C1 2007 June Q5
5 marks Easy -1.2
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-06_702_785_242_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 3 } { x } , x \neq 0\).
  1. On a separate diagram, sketch the curve with equation \(y = \frac { 3 } { x + 2 } , x \neq - 2\), showing the coordinates of any point at which the curve crosses a coordinate axis.
  2. Write down the equations of the asymptotes of the curve in part (a).
Edexcel C1 2008 June Q2
3 marks Easy -1.8
Factorise completely $$x ^ { 3 } - 9 x .$$
Edexcel C1 2008 June Q9
8 marks Moderate -0.3
The curve \(C\) has equation \(y = k x ^ { 3 } - x ^ { 2 } + x - 5\), where \(k\) is a constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The point \(A\) with \(x\)-coordinate \(- \frac { 1 } { 2 }\) lies on \(C\). The tangent to \(C\) at \(A\) is parallel to the line with equation \(2 y - 7 x + 1 = 0\). Find
  2. the value of \(k\),
  3. the value of the \(y\)-coordinate of \(A\).