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 2006 June Q3
3. On separate diagrams, sketch the graphs of
  1. \(y = ( x + 3 ) ^ { 2 }\),
  2. \(y = ( x + 3 ) ^ { 2 } + k\), where \(k\) is a positive constant. Show on each sketch the coordinates of each point at which the graph meets the axes.
Edexcel C1 2006 June Q4
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3
& a _ { n + 1 } = 3 a _ { n } - 5 , \quad n \geqslant 1 . \end{aligned}$$
  1. Find the value of \(a _ { 2 }\) and the value of \(a _ { 3 }\).
  2. Calculate the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\).
Edexcel C1 2006 June Q5
5. Differentiate with respect to \(x\)
  1. \(x ^ { 4 } + 6 \sqrt { } x\),
  2. \(\frac { ( x + 4 ) ^ { 2 } } { x }\).
Edexcel C1 2006 June Q6
6. (a) Expand and simplify \(( 4 + \sqrt { 3 } ) ( 4 - \sqrt { 3 } )\).
(b) Express \(\frac { 26 } { 4 + \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
Edexcel C1 2006 June Q7
7. An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day, he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a \mathrm {~km}\) and common difference \(d \mathrm {~km}\). He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period.
Find the value of \(a\) and the value of \(d\).
Edexcel C1 2006 June Q8
8. The equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\), where \(p\) is a positive constant, has equal roots.
  1. Find the value of \(p\).
  2. For this value of \(p\), solve the equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\).
Edexcel C1 2006 June Q9
9. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } - 6 x \right) ( x - 2 ) + 3 x\),
  1. express \(\mathrm { f } ( x )\) in the form \(x \left( a x ^ { 2 } + b x + c \right)\), where \(a\), \(b\) and \(c\) are constants.
  2. Hence factorise \(\mathrm { f } ( x )\) completely.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of each point at which the graph meets the axes.
Edexcel C1 2006 June Q10
10. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \neq 0\), passes through the point ( \(3,7 \frac { 1 } { 2 }\) ). Given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 3 } { x ^ { 2 } }\),
  1. find \(\mathrm { f } ( x )\).
  2. Verify that \(f ( - 2 ) = 5\).
  3. Find an equation for the tangent to \(C\) at the point ( \(- 2,5\) ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2006 June Q11
  1. The line \(l _ { 1 }\) passes through the points \(P ( - 1,2 )\) and \(Q ( 11,8 )\).
    1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    The line \(l _ { 2 }\) passes through the point \(R ( 10,0 )\) and is perpendicular to \(l _ { 1 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(S\).
  2. Calculate the coordinates of \(S\).
  3. Show that the length of \(R S\) is \(3 \sqrt { 5 }\).
  4. Hence, or otherwise, find the exact area of triangle \(P Q R\).
Edexcel C1 2007 June Q1
Simplify \(( 3 + \sqrt { } 5 ) ( 3 - \sqrt { } 5 )\).
\includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-02_108_93_2614_1786}
Edexcel C1 2007 June Q6
6. (a) By eliminating \(y\) from the equations $$\begin{gathered} y = x - 4
2 x ^ { 2 } - x y = 8 \end{gathered}$$ show that $$x ^ { 2 } + 4 x - 8 = 0$$ (b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} y = x - 4
2 x ^ { 2 } - x y = 8 \end{gathered}$$ giving your answers in the form \(a \pm b \sqrt { } 3\), where \(a\) and \(b\) are integers.
Edexcel C1 2007 June Q7
7. The equation \(x ^ { 2 } + k x + ( k + 3 ) = 0\), where \(k\) is a constant, has different real roots.
  1. Show that \(k ^ { 2 } - 4 k - 12 > 0\).
  2. Find the set of possible values of \(k\).
Edexcel C1 2007 June Q8
8. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 3 a _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a positive integer.
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  2. Show that \(a _ { 3 } = 9 k + 20\).
    1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\).
    2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 10 .
Edexcel C1 2007 June Q9
9. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 5,65 )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12\),
  1. use integration to find \(\mathrm { f } ( x )\).
  2. Hence show that \(\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )\).
  3. In the space provided on page 17, sketch \(C\), showing the coordinates of the points where \(C\) crosses the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-11_76_40_2646_1894}
Edexcel C1 2007 June Q10
10. The curve \(C\) has equation \(y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 2 respectively.
  1. Show that the length of \(P Q\) is \(\sqrt { } 170\).
  2. Show that the tangents to \(C\) at \(P\) and \(Q\) are parallel.
  3. Find an equation for the normal 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 2007 June Q11
  1. The line \(l _ { 1 }\) has equation \(y = 3 x + 2\) and the line \(l _ { 2 }\) has equation \(3 x + 2 y - 8 = 0\).
    1. Find the gradient of the line \(l _ { 2 }\).
    The point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\) is \(P\).
  2. Find the coordinates of \(P\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) cross the line \(y = 1\) at the points \(A\) and \(B\) respectively.
  3. Find the area of triangle \(A B P\).
Edexcel C1 2008 June Q1
Find \(\int \left( 2 + 5 x ^ { 2 } \right) d x\).
Edexcel C1 2008 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9451ec48-d955-44a8-9988-68f7c0fb9821-04_463_703_276_589} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the point ( 0,7 ) and has a minimum point at ( 7,0 ). On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( x ) + 3\),
  2. \(y = \mathrm { f } ( 2 x )\). On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the \(y\)-axis.
Edexcel C1 2008 June Q4
4. $$\mathrm { f } ( x ) = 3 x + x ^ { 3 } , \quad x > 0 .$$
  1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 15\),
  2. find the value of \(x\).
Edexcel C1 2008 June Q5
5. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 ,
x _ { n + 1 } = a x _ { n } - 3 , n \geqslant 1 , \end{gathered}$$ where \(a\) is a constant.
  1. Find an expression for \(x _ { 2 }\) in terms of \(a\).
  2. Show that \(x _ { 3 } = a ^ { 2 } - 3 a - 3\). Given that \(x _ { 3 } = 7\),
  3. find the possible values of \(a\).
Edexcel C1 2008 June Q6
6. The curve \(C\) has equation \(y = \frac { 3 } { x }\) and the line \(l\) has equation \(y = 2 x + 5\).
  1. On the axes below, sketch the graphs of \(C\) and \(l\), indicating clearly the coordinates of any intersections with the axes.
  2. Find the coordinates of the points of intersection of \(C\) and \(l\).
    \includegraphics[max width=\textwidth, alt={}, center]{9451ec48-d955-44a8-9988-68f7c0fb9821-07_1137_1141_1046_397}
Edexcel C1 2008 June Q7
7. Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km .
  1. Show that on the 4th Saturday of training she runs 11 km .
  2. Find an expression, in terms of \(n\), for the length of her training run on the \(n\)th Saturday.
  3. Show that the total distance she runs on Saturdays in \(n\) weeks of training is \(n ( n + 4 ) \mathrm { km }\). On the \(n\)th Saturday Sue runs 43 km .
  4. Find the value of \(n\).
  5. Find the total distance, in km , Sue runs on Saturdays in \(n\) weeks of training.
Edexcel C1 2008 June Q8
Given that the equation \(2 q x ^ { 2 } + q x - 1 = 0\), where \(q\) is a constant, has no real roots,
  1. show that \(q ^ { 2 } + 8 q < 0\).
  2. Hence find the set of possible values of \(q\).
Edexcel C1 2008 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9451ec48-d955-44a8-9988-68f7c0fb9821-14_541_863_287_541} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The points \(Q ( 1,3 )\) and \(R ( 7,0 )\) lie on the line \(l _ { 1 }\), as shown in Figure 2.
The length of \(Q R\) is \(a \sqrt { } 5\).
  1. Find the value of \(a\). The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\), passes through \(Q\) and crosses the \(y\)-axis at the point \(P\), as shown in Figure 2. Find
  2. an equation for \(l _ { 2 }\),
  3. the coordinates of \(P\),
  4. the area of \(\triangle P Q R\).
Edexcel C1 2008 June Q11
  1. The gradient of a curve \(C\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( x ^ { 2 } + 3 \right) ^ { 2 } } { x ^ { 2 } } , x \neq 0\).
    1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } + 6 + 9 x ^ { - 2 }\).
    The point \(( 3,20 )\) lies on \(C\).
  2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).