Questions — Edexcel (9670 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel C1 2017 June Q1
4 marks Easy -1.3
  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
5 marks Moderate -0.8
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
6 marks Moderate -0.3
  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
7 marks Moderate -0.8
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
8 marks Easy -1.2
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 marks Moderate -0.8
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
9 marks Standard +0.3
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 marks Moderate -0.8
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
11 marks Standard +0.8
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
11 marks Moderate -0.3
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
5 marks Easy -1.3
  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
7 marks Easy -1.3
  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
6 marks Moderate -0.8
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
7 marks Easy -1.3
  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 marks Moderate -0.8
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 \(k\).
Edexcel C1 2018 June Q6
7 marks Moderate -0.8
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = \frac { a _ { n } } { a _ { n } + 1 } , \quad n \geqslant 1 , n \in \mathbb { N } \end{aligned}$$
  1. Find the values of \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\) Write your answers as simplified fractions. Given that $$a _ { n } = \frac { 4 } { p n + q } , \text { where } p \text { and } q \text { are constants }$$
  2. state the value of \(p\) and the value of \(q\).
  3. Hence calculate the value of \(N\) such that \(a _ { N } = \frac { 4 } { 321 }\)
Edexcel C1 2018 June Q7
8 marks Moderate -0.3
  1. The equation \(20 x ^ { 2 } = 4 k x - 13 k x ^ { 2 } + 2\), where \(k\) is a constant, has no real roots.
    1. Show that \(k\) satisfies the inequality
    $$2 k ^ { 2 } + 13 k + 20 < 0$$
  2. Find the set of possible values for \(k\).
Edexcel C1 2018 June Q8
8 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-20_1063_1319_251_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the straight line \(l _ { 1 }\) with equation \(4 y = 5 x + 12\)
  1. State the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the point \(E ( 12,5 )\), as shown in Figure 2.
  2. Find the equation of \(l _ { 2 }\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined. The line \(l _ { 2 }\) cuts the \(x\)-axis at the point \(C\) and the \(y\)-axis at the point \(B\).
  3. Find the coordinates of
    1. the point \(B\),
    2. the point \(C\). The line \(l _ { 1 }\) cuts the \(y\)-axis at the point \(A\).
      The point \(D\) lies on \(l _ { 1 }\) such that \(A B C D\) is a parallelogram, as shown in Figure 2.
  4. Find the area of \(A B C D\).
Edexcel C1 2018 June Q9
12 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where
$$f ^ { \prime } ( x ) = ( x - 3 ) ( 3 x + 5 )$$ Given that the point \(P ( 1,20 )\) lies on \(C\),
  1. find \(\mathrm { f } ( x )\), simplifying each term.
  2. Show that $$f ( x ) = ( x - 3 ) ^ { 2 } ( x + A )$$ where \(A\) is a constant to be found.
  3. Sketch the graph of \(C\). Show clearly the coordinates of the points where \(C\) cuts or meets the \(x\)-axis and where \(C\) cuts the \(y\)-axis.
Edexcel C1 2018 June Q10
10 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-28_643_1171_260_518} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x + \frac { 27 } { x } - 12 , \quad x > 0$$ The point \(A\) lies on \(C\) and has coordinates \(\left( 3 , - \frac { 3 } { 2 } \right)\).
  1. Show that the equation of the normal to \(C\) at \(A\) can be written as \(10 y = 4 x - 27\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.
  2. Use algebra to find the coordinates of \(B\).
Edexcel C1 Q1
3 marks Easy -1.2
  1. Solve the inequality
    \(10 + x ^ { 2 } > x ( x - 2 )\).
    (3)
\begin{center} \begin{tabular}{|l|l|} \hline
Edexcel C1 Q2
4 marks Easy -1.2
2. Find \(\int \left( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } + \sqrt [ 3 ] { x } \right) \mathrm { d } x\) & Leave blank
\hline \end{tabular} \end{center}
Edexcel C1 Q3
4 marks Easy -1.8
Find the value of
  1. \(81 ^ { \frac { 1 } { 2 } }\),
  2. \(81 ^ { \frac { 3 } { 4 } }\),
  3. \(81 ^ { - \frac { 3 } { 4 } }\).
Edexcel C1 Q5
7 marks Moderate -0.8
5. (a) Show that eliminating \(y\) from the equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ produces the equation $$x ^ { 2 } + 8 x - 1 = 0$$ (b) Hence solve the simultaneous equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ giving your answers in the form \(a + b \sqrt { } 17\), where \(a\) and \(b\) are integers.
5. continuedLeave blank
Edexcel C1 Q6
9 marks Easy -1.2
6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { } x } , \quad x > 0$$
  1. Show that \(\mathrm { f } ( x )\) can be written in the form \(P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }\), stating the values of the constants \(P , Q\) and \(R\).
  2. Find f \({ } ^ { \prime } ( x )\).
  3. Show that the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) is parallel to the line with equation \(2 y = 11 x + 3\).
    (3)
    6. continuedLeave blank
    \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}