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 2010 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{280f0f3b-fdb5-4ac9-adc6-150819b03539-10_646_986_246_562} \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 \(( - 2,5 )\) and an asymptote \(y = 1\), as shown in Figure 1. On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( x ) + 2\)
  2. \(y = 4 \mathrm { f } ( x )\)
  3. \(y = \mathrm { f } ( \mathrm { x } + 1 )\) On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.
Edexcel C1 2010 January Q9
  1. (a) Factorise completely \(x ^ { 3 } - 4 x\)
    (b) Sketch the curve \(C\) with equation
$$y = x ^ { 3 } - 4 x ,$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. The point \(A\) with \(x\)-coordinate - 1 and the point \(B\) with \(x\)-coordinate 3 lie on the curve \(C\).
(c) Find an equation of the line which passes through \(A\) and \(B\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
(d) Show that the length of \(A B\) is \(k \sqrt { } 10\), where \(k\) is a constant to be found.
Edexcel C1 2010 January Q10
10. $$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\). Given that the equation \(\mathrm { f } ( x ) = 0\) has no real roots,
  2. find the set of possible values of \(k\). Given that \(k = 1\),
  3. sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any point at which the graph crosses a coordinate axis.
Edexcel C1 2011 January Q1
  1. Find the value of \(16 ^ { - \frac { 1 } { 4 } }\)
  2. Simplify \(x \left( 2 x ^ { - \frac { 1 } { 4 } } \right) ^ { 4 }\)
Edexcel C1 2011 January Q3
3. Simplify $$\frac { 5 - 2 \sqrt { 3 } } { \sqrt { 3 } - 1 }$$ giving your answer in the form \(p + q \sqrt { } 3\), where \(p\) and \(q\) are rational numbers.
Edexcel C1 2011 January Q4
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 2
a _ { n + 1 } & = 3 a _ { n } - c \end{aligned}$$ where \(c\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
  2. find the value of \(c\).
Edexcel C1 2011 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95e11fd7-765c-477d-800b-7574bc1af81f-06_640_1063_322_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { x - 2 } , \quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations \(y = 1\) and \(x = 2\), as shown in Figure 1.
  1. In the space below, sketch the curve with equation \(y = \mathrm { f } ( x - 1 )\) and state the equations of the asymptotes of this curve.
  2. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } ( x - 1 )\) crosses the coordinate axes.
Edexcel C1 2011 January Q6
6. An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 10 terms of the sequence is 162 .
  1. Show that \(10 a + 45 d = 162\) Given also that the sixth term of the sequence is 17 ,
  2. write down a second equation in \(a\) and \(d\),
  3. find the value of \(a\) and the value of \(d\).
Edexcel C1 2011 January Q7
7. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( - 1,0 )\). Given that $$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$ find \(\mathrm { f } ( x )\).
Edexcel C1 2011 January Q8
8. The equation \(x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies $$k ^ { 2 } + 2 k - 3 > 0$$
  2. Find the set of possible values of \(k\).
Edexcel C1 2011 January Q9
9. The line \(L _ { 1 }\) has equation \(2 y - 3 x - k = 0\), where \(k\) is a constant. Given that the point \(A ( 1,4 )\) lies on \(L _ { 1 }\), find
  1. the value of \(k\),
  2. the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) passes through \(A\) and is perpendicular to \(L _ { 1 }\).
  3. Find an equation of \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The line \(L _ { 2 }\) crosses the \(x\)-axis at the point \(B\).
  4. Find the coordinates of \(B\).
  5. Find the exact length of \(A B\).
Edexcel C1 2011 January Q10
10. (a) On the axes below, sketch the graphs of
  1. \(y = x ( x + 2 ) ( 3 - x )\)
  2. \(y = - \frac { 2 } { x }\)
    showing clearly the coordinates of all the points where the curves cross the coordinate axes.
    (b) Using your sketch state, giving a reason, the number of real solutions to the equation $$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$ \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-13_994_997_1270_479}
Edexcel C1 2011 January Q11
11. The curve \(C\) has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
  3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-15_113_129_2405_1816}
Edexcel C1 2012 January Q1
Given that \(y = x ^ { 4 } + 6 x ^ { \frac { 1 } { 2 } }\), find in their simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\)
Edexcel C1 2012 January Q3
3. Find the set of values of \(x\) for which
  1. \(4 x - 5 > 15 - x\)
  2. \(x ( x - 4 ) > 12\)
Edexcel C1 2012 January Q4
4. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{aligned} x _ { 1 } & = 1
x _ { n + 1 } & = a x _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(a\) is a constant.
  1. Write down an expression for \(x _ { 2 }\) in terms of \(a\).
  2. Show that \(x _ { 3 } = a ^ { 2 } + 5 a + 5\) Given that \(x _ { 3 } = 41\)
  3. find the possible values of \(a\).
Edexcel C1 2012 January Q5
5. The curve \(C\) has equation \(y = x ( 5 - x )\) and the line \(L\) has equation \(2 y = 5 x + 4\)
  1. Use algebra to show that \(C\) and \(L\) do not intersect.
  2. In the space on page 11, sketch \(C\) and \(L\) on the same diagram, showing the coordinates of the points at which \(C\) and \(L\) meet the axes.
Edexcel C1 2012 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff1cdb91-0286-4bc8-9e67-451500b2bf74-07_647_927_274_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The line \(l _ { 1 }\) has equation \(2 x - 3 y + 12 = 0\)
  1. Find the gradient of \(l _ { 1 }\). The line \(l _ { 1 }\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 1. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(B\).
  2. Find an equation of \(l _ { 2 }\). The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(C\).
  3. Find the area of triangle \(A B C\).
Edexcel C1 2012 January Q7
  1. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,10 )\). Given that
$$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x + 5$$ find the value of \(\mathrm { f } ( 1 )\).
Edexcel C1 2012 January Q8
8. The curve \(C _ { 1 }\) has equation $$y = x ^ { 2 } ( x + 2 )$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Sketch \(C _ { 1 }\), showing the coordinates of the points where \(C _ { 1 }\) meets the \(x\)-axis.
  3. Find the gradient of \(C _ { 1 }\) at each point where \(C _ { 1 }\) meets the \(x\)-axis. The curve \(C _ { 2 }\) has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where \(k\) is a constant and \(k > 2\)
  4. Sketch \(C _ { 2 }\), showing the coordinates of the points where \(C _ { 2 }\) meets the \(x\) and \(y\) axes.
Edexcel C1 2012 January Q9
  1. A company offers two salary schemes for a 10 -year period, Year 1 to Year 10 inclusive.
Scheme 1: Salary in Year 1 is \(\pounds P\).
Salary increases by \(\pounds ( 2 T )\) each year, forming an arithmetic sequence.
Scheme 2: Salary in Year 1 is \(\pounds ( P + 1800 )\).
Salary increases by \(\pounds T\) each year, forming an arithmetic sequence.
  1. Show that the total earned under Salary Scheme 1 for the 10-year period is $$\pounds ( 10 P + 90 T )$$ For the 10-year period, the total earned is the same for both salary schemes.
  2. Find the value of \(T\). For this value of \(T\), the salary in Year 10 under Salary Scheme 2 is \(\pounds 29850\)
  3. Find the value of \(P\).
Edexcel C1 2012 January Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff1cdb91-0286-4bc8-9e67-451500b2bf74-14_769_935_285_411} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$y = 2 - \frac { 1 } { x } , \quad x \neq 0$$ The curve crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\).
  2. Show that the equation of the normal to \(C\) at \(A\) can be written as $$2 x + 8 y - 1 = 0$$ The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 2 .
  3. Find the coordinates of \(B\).
Edexcel C1 2013 January Q1
Factorise completely \(x - 4 x ^ { 3 }\)
Edexcel C1 2013 January Q4
4. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n + 1 } = 2 u _ { n } - 1 , n \geqslant 1$$ Given that \(u _ { 2 } = 9\),
  1. find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\),
  2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\).
Edexcel C1 2013 January Q5
5. The line \(l _ { 1 }\) has equation \(y = - 2 x + 3\) The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 5,6 )\).
  1. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).
  2. Find the \(x\)-coordinate of \(A\) and the \(y\)-coordinate of \(B\). Given that \(O\) is the origin,
  3. find the area of the triangle \(O A B\).