Questions — Edexcel C1 (490 questions)

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Edexcel C1 2013 January Q3
  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
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
$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b ,$$ 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
Find the set of values of \(x\) for which $$x ^ { 2 } - 7 x - 18 > 0 .$$
Edexcel C1 2007 June Q2
  1. Find the value of \(8 ^ { \frac { 4 } { 3 } }\).
  2. Simplify \(\frac { 15 x ^ { \frac { 4 } { 3 } } } { 3 x }\).
Edexcel C1 2007 June Q3
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
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
\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
Factorise completely $$x ^ { 3 } - 9 x .$$
Edexcel C1 2008 June Q9
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 \(A\).
Edexcel C1 2011 June Q2
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
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
Express \(\frac { 15 } { \sqrt { 3 } } - \sqrt { 27 }\) in the form \(k \sqrt { } 3\), where \(k\) is an integer.
Edexcel C1 2013 June Q3
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
  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
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
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
  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
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
  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
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 C1 2014 June Q1
  1. Find
$$\int \left( 8 x ^ { 3 } + 4 \right) d x$$ giving each term in its simplest form.
Edexcel C1 2014 June Q2
2．（a）Write down the value of \(32 ^ { \frac { 1 } { 5 } }\)
（b）Simplify fully \(\left( 32 x ^ { 5 } \right) ^ { - \frac { 2 } { 5 } }\)
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Edexcel C1 2014 June Q3
3. Find the set of values of \(x\) for which
  1. \(3 x - 7 > 3 - x\)
  2. \(x ^ { 2 } - 9 x \leqslant 36\)
  3. both \(3 x - 7 > 3 - x\) and \(x ^ { 2 } - 9 x \leqslant 36\)
Edexcel C1 2014 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-05_945_1026_269_466} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$ The curve \(C\) crosses the \(x\)-axis at the point \(A\).
  1. State the \(x\) coordinate of the point \(A\). The curve \(D\) has equation \(y = x ^ { 2 } ( x - 2 )\), for all real values of \(x\).
  2. A copy of Figure 1 is shown on page 7. On this copy, sketch a graph of curve \(D\).
    Show on the sketch the coordinates of each point where the curve \(D\) crosses the coordinate axes.
  3. Using your sketch, state, giving a reason, the number of real solutions to the equation $$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}