Questions — Edexcel C1 (490 questions)

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Edexcel C1 2016 June Q1
  1. Find
$$\int \left( 2 x ^ { 4 } - \frac { 4 } { \sqrt { } x } + 3 \right) d x$$ giving each term in its simplest form.
Edexcel C1 2016 June Q2
Express \(9 ^ { 3 x + 1 }\) in the form \(3 ^ { y }\), giving \(y\) in the form \(a x + b\), where \(a\) and \(b\) are constants.
(2)
Edexcel C1 2016 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b0413ecc-b780-4f77-b76a-da7c699c12cb-05_709_744_269_607} \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 has a maximum point \(A\) at \(( - 2,4 )\) and a minimum point \(B\) at \(( 3 , - 8 )\) and passes through the origin \(O\). On separate diagrams, sketch the curve with equation
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( x ) - 4\)
    (3) On each diagram, show clearly the coordinates of the maximum and the minimum points and the coordinates of the point where the curve crosses the \(y\)-axis.
Edexcel C1 2016 June Q5
5. Solve the simultaneous equations $$\begin{gathered} y + 4 x + 1 = 0
y ^ { 2 } + 5 x ^ { 2 } + 2 x = 0 \end{gathered}$$
Edexcel C1 2016 June Q6
6. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4
a _ { n + 1 } & = 5 - k a _ { n } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\). Find
  2. \(\sum _ { r = 1 } ^ { 3 } \left( 1 + a _ { r } \right)\) in terms of \(k\), giving your answer in its simplest form,
  3. \(\sum _ { r = 1 } ^ { 100 } \left( a _ { r + 1 } + k a _ { r } \right)\)
Edexcel C1 2016 June Q7
  1. Given that
$$y = 3 x ^ { 2 } + 6 x ^ { \frac { 1 } { 3 } } + \frac { 2 x ^ { 3 } - 7 } { 3 \sqrt { } x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Give each term in your answer in its simplified form.
Edexcel C1 2016 June Q8
8. The straight line with equation \(y = 3 x - 7\) does not cross or touch the curve with equation \(y = 2 p x ^ { 2 } - 6 p x + 4 p\), where \(p\) is a constant.
  1. Show that \(4 p ^ { 2 } - 20 p + 9 < 0\)
  2. Hence find the set of possible values of \(p\).
    VILM SIHI NITIIIUMI ON OC
    VILV SIHI NI III HM ION OC
    VALV SIHI NI JIIIM ION OO
Edexcel C1 2016 June Q9
9. On John's 10th birthday he received the first of an annual birthday gift of money from his uncle. This first gift was \(\pounds 60\) and on each subsequent birthday the gift was \(\pounds 15\) more than the year before. The amounts of these gifts form an arithmetic sequence.
  1. Show that, immediately after his 12th birthday, the total of these gifts was \(\pounds 225\)
  2. Find the amount that John received from his uncle as a birthday gift on his 18th birthday.
  3. Find the total of these birthday gifts that John had received from his uncle up to and including his 21st birthday. When John had received \(n\) of these birthday gifts, the total money that he had received from these gifts was \(\pounds 3375\)
  4. Show that \(n ^ { 2 } + 7 n = 25 \times 18\)
  5. Find the value of \(n\), when he had received \(\pounds 3375\) in total, and so determine John's age at this time.
Edexcel C1 2016 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b0413ecc-b780-4f77-b76a-da7c699c12cb-12_593_1166_260_397} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The points \(P ( 0,2 )\) and \(Q ( 3,7 )\) lie on the line \(l _ { 1 }\), as shown in Figure 2.
The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\), passes through \(Q\) and crosses the \(x\)-axis at the point \(R\), as shown in Figure 2. Find
  1. an equation for \(l _ { 2 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers,
  2. the exact coordinates of \(R\),
  3. the exact area of the quadrilateral \(O R Q P\), where \(O\) is the origin.
Edexcel C1 2016 June Q11
11. The curve \(C\) has equation \(y = 2 x ^ { 3 } + k x ^ { 2 } + 5 x + 6\), where \(k\) is a constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\), where \(x = - 2\), lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the line with equation \(2 y - 17 x - 1 = 0\)
    Find
  2. the value of \(k\),
  3. the value of the \(y\) coordinate of \(P\),
  4. the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2017 June Q1
  1. Find
$$\int \left( 2 x ^ { 5 } - \frac { 1 } { 4 x ^ { 3 } } - 5 \right) \mathrm { d } x$$ giving each term in its simplest form.
Edexcel C1 2017 June Q2
2. Given $$y = \sqrt { x } + \frac { 4 } { \sqrt { x } } + 4 , \quad x > 0$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\), writing your answer in the form \(a \sqrt { 2 }\), where \(a\) is a rational number.
(5)
Edexcel C1 2017 June Q3
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned} a _ { 1 } & = 1
a _ { n + 1 } & = \frac { k \left( a _ { n } + 1 \right) } { a _ { n } } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a positive constant.
  1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\), giving your answers in their simplest form. Given that \(\sum _ { r = 1 } ^ { 3 } a _ { r } = 10\)
  2. find an exact value for \(k\).
Edexcel C1 2017 June Q4
4. A company, which is making 140 bicycles each week, plans to increase its production. The number of bicycles produced is to be increased by \(d\) each week, starting from 140 in week 1 , to \(140 + d\) in week 2 , to \(140 + 2 d\) in week 3 and so on, until the company is producing 206 in week 12.
  1. Find the value of \(d\). After week 12 the company plans to continue making 206 bicycles each week.
  2. Find the total number of bicycles that would be made in the first 52 weeks starting from and including week 1.
Edexcel C1 2017 June Q5
5. $$f ( x ) = x ^ { 2 } - 8 x + 19$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) crosses the \(y\)-axis at the point \(P\) and has a minimum point at the point \(Q\).
  2. Sketch the graph of \(C\) showing the coordinates of point \(P\) and the coordinates of point \(Q\).
  3. Find the distance \(P Q\), writing your answer as a simplified surd.
Edexcel C1 2017 June Q6
6. (a) Given \(y = 2 ^ { x }\), show that $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$ can be written in the form $$2 y ^ { 2 } - 17 y + 8 = 0$$ (b) Hence solve $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$
Edexcel C1 2017 June Q7
7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$\mathrm { f } ^ { \prime } ( x ) = 30 + \frac { 6 - 5 x ^ { 2 } } { \sqrt { x } }$$ Given that the point \(P ( 4 , - 8 )\) lies on \(C\),
  1. find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
  2. Find \(\mathrm { f } ( x )\), giving each term in its simplest form.
Edexcel C1 2017 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1b0a49d-9def-4289-a4cd-288991f67caf-16_659_1438_267_251} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The straight line \(l _ { 1 }\), shown in Figure 1, has equation \(5 y = 4 x + 10\)
The point \(P\) with \(x\) coordinate 5 lies on \(l _ { 1 }\)
The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(P\).
  1. Find an equation for \(l _ { 2 }\), writing your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers. The lines \(l _ { 1 }\) and \(l _ { 2 }\) cut the \(x\)-axis at the points \(S\) and \(T\) respectively, as shown in Figure 1.
  2. Calculate the area of triangle SPT.
Edexcel C1 2017 June Q9
9. (a) On separate axes sketch the graphs of
  1. \(y = - 3 x + c\), where \(c\) is a positive constant,
  2. \(y = \frac { 1 } { x } + 5\) On each sketch show the coordinates of any point at which the graph crosses the \(y\)-axis and the equation of any horizontal asymptote. Given that \(y = - 3 x + c\), where \(c\) is a positive constant, meets the curve \(y = \frac { 1 } { x } + 5\) at two distinct points,
    (b) show that \(( 5 - c ) ^ { 2 } > 12\)
    (c) Hence find the range of possible values for \(c\).
Edexcel C1 2017 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1b0a49d-9def-4289-a4cd-288991f67caf-24_666_1195_260_370} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\), where $$f ( x ) = ( 2 x - 5 ) ^ { 2 } ( x + 3 )$$
  1. Given that
    1. the curve with equation \(y = \mathrm { f } ( x ) - k , x \in \mathbb { R }\), passes through the origin, find the value of the constant \(k\),
    2. the curve with equation \(y = \mathrm { f } ( x + c ) , x \in \mathbb { R }\), has a minimum point at the origin, find the value of the constant \(c\).
  2. Show that \(\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 16 x - 35\) Points \(A\) and \(B\) are distinct points that lie on the curve \(y = \mathrm { f } ( x )\).
    The gradient of the curve at \(A\) is equal to the gradient of the curve at \(B\).
    Given that point \(A\) has \(x\) coordinate 3
  3. find the \(x\) coordinate of point \(B\).
    \includegraphics[max width=\textwidth, alt={}]{c1b0a49d-9def-4289-a4cd-288991f67caf-28_2630_1826_121_121}
Edexcel C1 2018 June Q1
  1. (i) Simplify
$$\sqrt { 48 } - \frac { 6 } { \sqrt { 3 } }$$ Write your answer in the form \(a \sqrt { 3 }\), where \(a\) is an integer to be found.
(ii) Solve the equation $$3 ^ { 6 x - 3 } = 81$$ Write your answer as a rational number.
Edexcel C1 2018 June Q2
  1. Given
$$y = 3 \sqrt { x } - 6 x + 4 , \quad x > 0$$
  1. find \(\int y \mathrm {~d} x\), simplifying each term.
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. Hence find the value of \(x\) such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
Edexcel C1 2018 June Q3
3. $$f ( x ) = x ^ { 2 } - 10 x + 23$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants to be found.
  2. Hence, or otherwise, find the exact solutions to the equation $$x ^ { 2 } - 10 x + 23 = 0$$
  3. Use your answer to part (b) to find the larger solution to the equation $$y - 10 y ^ { 0.5 } + 23 = 0$$ Write your solution in the form \(p + q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
Edexcel C1 2018 June Q4
  1. Each year, Andy pays into a savings scheme. In year one he pays in \(\pounds 600\). His payments increase by \(\pounds 120\) each year so that he pays \(\pounds 720\) in year two, \(\pounds 840\) in year three and so on, so that his payments form an arithmetic sequence.
    1. Find out how much Andy pays into the savings scheme in year ten.
      (2)
    Kim starts paying money into a different savings scheme at the same time as Andy. In year one she pays in \(\pounds 130\). Her payments increase each year so that she pays \(\pounds 210\) in year two, \(\pounds 290\) in year three and so on, so that her payments form a different arithmetic sequence. At the end of year \(N\), Andy has paid, in total, twice as much money into his savings scheme as Kim has paid, in total, into her savings scheme.
  2. Find the value of \(N\).
Edexcel C1 2018 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-12_963_1239_255_354} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the sketch of a curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve crosses the \(y\)-axis at \(( 0,4 )\) and crosses the \(x\)-axis at \(( 5,0 )\). The curve has a single turning point, a maximum, at (2, 7). The line with equation \(y = 1\) is the only asymptote to the curve.
  1. State the coordinates of the turning point on the curve with equation \(y = \mathrm { f } ( x - 2 )\).
  2. State the solution of the equation f( \(2 x\) ) \(= 0\)
  3. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( - x )\). Given that the line with equation \(y = k\), where \(k\) is a constant, meets the curve \(y = \mathrm { f } ( x )\) at only one point,
  4. state the set of possible values for \(k\).