Questions — Edexcel C1 (490 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 2013 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{099016ad-e742-4679-9669-47dcd1d9cc5f-08_915_1132_214_397} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 2 } { x } , x \neq 0\) The curve \(C\) has equation \(y = \frac { 2 } { x } - 5 , x \neq 0\), and the line \(l\) has equation \(y = 4 x + 2\)
  1. Sketch and clearly label the graphs of \(C\) and \(l\) on a single diagram. On your diagram, show clearly the coordinates of the points where \(C\) and \(l\) cross the coordinate axes.
  2. Write down the equations of the asymptotes of the curve \(C\).
  3. Find the coordinates of the points of intersection of \(y = \frac { 2 } { x } - 5\) and \(y = 4 x + 2\)
Edexcel C1 2013 January Q7
  1. Lewis played a game of space invaders. He scored points for each spaceship that he captured.
Lewis scored 140 points for capturing his first spaceship.
He scored 160 points for capturing his second spaceship, 180 points for capturing his third spaceship, and so on. The number of points scored for capturing each successive spaceship formed an arithmetic sequence.
  1. Find the number of points that Lewis scored for capturing his 20th spaceship.
  2. Find the total number of points Lewis scored for capturing his first 20 spaceships. Sian played an adventure game. She scored points for each dragon that she captured. The number of points that Sian scored for capturing each successive dragon formed an arithmetic sequence. Sian captured \(n\) dragons and the total number of points that she scored for capturing all \(n\) dragons was 8500 . Given that Sian scored 300 points for capturing her first dragon and then 700 points for capturing her \(n\)th dragon,
  3. find the value of \(n\).
Edexcel C1 2013 January Q8
8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 3 } + \frac { 4 x - 5 } { 2 x ^ { 3 } } , \quad x \neq 0$$ Given that \(y = 7\) at \(x = 1\), find \(y\) in terms of \(x\), giving each term in its simplest form.
Edexcel C1 2013 January Q9
9. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2013 January Q10
10. $$4 x ^ { 2 } + 8 x + 3 \equiv a ( x + b ) ^ { 2 } + c$$
  1. Find the values of the constants \(a , b\) and \(c\).
  2. On the axes on page 27, sketch the curve with equation \(y = 4 x ^ { 2 } + 8 x + 3\), showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{099016ad-e742-4679-9669-47dcd1d9cc5f-15_1283_1284_319_322}
Edexcel C1 2013 January Q11
11. The curve \(C\) has equation $$y = 2 x - 8 \sqrt { } x + 5 , \quad x \geqslant 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form. The point \(P\) on \(C\) has \(x\)-coordinate equal to \(\frac { 1 } { 4 }\)
  2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants. The tangent to \(C\) at the point \(Q\) is parallel to the line with equation \(2 x - 3 y + 18 = 0\)
  3. Find the coordinates of \(Q\).
Edexcel C1 2014 January Q1
  1. Simplify fully
    1. \(( 2 \sqrt { } x ) ^ { 2 }\)
    2. \(\frac { 5 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\)
Edexcel C1 2014 January Q2
2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving each term in its simplest form.
    \includegraphics[max width=\textwidth, alt={}, center]{6081d81b-51d2-4140-9834-71ef7fd700b0-05_104_97_2613_1784}
Edexcel C1 2014 January Q3
3. Solve the simultaneous equations $$\begin{gathered} x - 2 y - 1 = 0
x ^ { 2 } + 4 y ^ { 2 } - 10 x + 9 = 0 \end{gathered}$$
Edexcel C1 2014 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6081d81b-51d2-4140-9834-71ef7fd700b0-08_835_777_118_596} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at \(( 0,3 )\) and has a minimum at \(P ( 4,2 )\). On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( x + 4 )\),
  2. \(y = 2 \mathrm { f } ( x )\). On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.
Edexcel C1 2014 January Q5
5. Given that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } a _ { r } = 12 + 4 n ^ { 2 }$$
  1. find the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\)
  2. Find the value of \(a _ { 6 }\)
Edexcel C1 2014 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6081d81b-51d2-4140-9834-71ef7fd700b0-12_650_885_255_603} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 7\)
The line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.
    1. State the gradient of \(l _ { 1 }\)
    2. Write down the coordinates of the point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B ( 1,5 )\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle A B C = 90 ^ { \circ }\),
  2. find an equation of \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The rectangle \(A B C D\), shown shaded in Figure 2, has vertices at the points \(A , B , C\) and \(D\).
  3. Find the exact area of rectangle \(A B C D\).
Edexcel C1 2014 January Q7
  1. Shelim starts his new job on a salary of \(\pounds 14000\). He will receive a rise of \(\pounds 1500\) a year for each full year that he works, so that he will have a salary of \(\pounds 15500\) in year 2 , a salary of \(\pounds 17000\) in year 3 and so on. When Shelim’s salary reaches \(\pounds 26000\), he will receive no more rises. His salary will remain at \(\pounds 26000\).
    1. Show that Shelim will have a salary of \(\pounds 26000\) in year 9 .
    2. Find the total amount that Shelim will earn in his job in the first 9 years.
    Anna starts her new job at the same time as Shelim on a salary of \(\pounds A\). She receives a rise of \(\pounds 1000\) a year for each full year that she works, so that she has a salary of \(\pounds ( A + 1000 )\) in year \(2 , \pounds ( A + 2000 )\) in year 3 and so on. The maximum salary for her job, which is reached in year 10 , is also \(\pounds 26000\).
  2. Find the difference in the total amount earned by Shelim and Anna in the first 10 years.
Edexcel C1 2014 January Q8
  1. The equation \(2 x ^ { 2 } + 2 k x + ( k + 2 ) = 0\), where \(k\) is a constant, has two distinct real roots.
    1. Show that \(k\) satisfies
    $$k ^ { 2 } - 2 k - 4 > 0$$
  2. Find the set of possible values of \(k\).
Edexcel C1 2014 January Q9
9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point ( 3,6 ). Given that $$f ^ { \prime } ( x ) = ( x - 2 ) ( 3 x + 4 )$$
  1. use integration to find \(\mathrm { f } ( x )\). Give your answer as a polynomial in its simplest form.
  2. Show that \(\mathrm { f } ( x ) \equiv ( x - 2 ) ^ { 2 } ( x + p )\), where \(p\) is a positive constant. State the value of \(p\).
  3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the curve touches or crosses the coordinate axes.
Edexcel C1 2014 January Q10
10. The curve \(C\) has equation \(y = x ^ { 3 } - 2 x ^ { 2 } - x + 3\) The point \(P\), which lies on \(C\), has coordinates \(( 2,1 )\).
  1. Show that an equation of the tangent to \(C\) at the point \(P\) is \(y = 3 x - 5\) The point \(Q\) also lies on \(C\).
    Given that the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\),
  2. find the coordinates of the point \(Q\).
Edexcel C1 2005 June Q1
  1. Write down the value of \(8 ^ { \frac { 1 } { 3 } }\).
  2. Find the value of \(8 ^ { - \frac { 2 } { 3 } }\).
Edexcel C1 2005 June Q4
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5a195cf1-37d9-43e9-ab47-c6892a18ba80-05_689_920_292_511}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the origin \(O\) and through the point \(( 6,0 )\). The maximum point on the curve is \(( 3,5 )\). On separate diagrams, sketch the curve with equation
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( x + 2 )\). On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.
Edexcel C1 2005 June Q5
5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 ,
x ^ { 2 } + y ^ { 2 } = 29 . \end{gathered}$$
Edexcel C1 2005 June Q6
6. Find the set of values of \(x\) for which
  1. \(3 ( 2 x + 1 ) > 5 - 2 x\),
  2. \(2 x ^ { 2 } - 7 x + 3 > 0\),
  3. both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\).
Edexcel C1 2005 June Q7
7. (a) Show that \(\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }\) can be written as \(9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0\), and that \(y = \frac { 2 } { 3 }\) at \(x = 1\),
(b) find \(y\) in terms of \(x\).
Edexcel C1 2005 June Q8
8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) passes through the origin \(O\) and has gradient - 2 . The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
  2. Calculate the coordinates of \(P\). Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
  3. calculate the exact area of \(\triangle O C P\).
    \includegraphics[max width=\textwidth, alt={}, center]{5a195cf1-37d9-43e9-ab47-c6892a18ba80-10_187_62_2563_1881}
Edexcel C1 2005 June Q9
9. 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 $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
  2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
  3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0 .$$
  4. Solve the equation in part (c).
  5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.
Edexcel C1 2005 June Q10
10. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3\). The point \(P\) has coordinates \(( 3,0 )\).
  1. Show that \(P\) lies on \(C\).
  2. 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. Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  3. Find the coordinates of \(Q\).
Edexcel C1 2006 June Q1
Find \(\int \left( 6 x ^ { 2 } + 2 + x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving each term in its simplest form.