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 2011 June Q7
7. $$\mathrm { f } ( x ) = x ^ { 2 } + ( k + 3 ) x + k$$ where \(k\) is a real constant.
  1. Find the discriminant of \(\mathrm { f } ( x )\) in terms of \(k\).
  2. Show that the discriminant of \(\mathrm { f } ( x )\) can be expressed in the form \(( k + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
  3. Show that, for all values of \(k\), the equation \(\mathrm { f } ( x ) = 0\) has real roots.
Edexcel C1 2011 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1bb296f-afb2-43cd-9408-2114d7b66971-09_487_743_210_603} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the origin and through \(( 6,0 )\).
The curve \(C\) has a minimum at the point \(( 3 , - 1 )\). On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( 2 x )\),
  2. \(y = - \mathrm { f } ( x )\),
  3. \(y = \mathrm { f } ( x + p )\), where \(p\) is a constant and \(0 < p < 3\). On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.
Edexcel C1 2011 June Q9
  1. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,
$$2 + 4 + 6 + \ldots \ldots + 100$$ (b) In the arithmetic series $$k + 2 k + 3 k + \ldots \ldots + 100$$ \(k\) is a positive integer and \(k\) is a factor of 100 .
  1. Find, in terms of \(k\), an expression for the number of terms in this series.
  2. Show that the sum of this series is $$50 + \frac { 5000 } { k }$$ (c) Find, in terms of \(k\), the 50th term of the arithmetic sequence $$( 2 k + 1 ) , ( 4 k + 4 ) , ( 6 k + 7 ) , \ldots \ldots ,$$ giving your answer in its simplest form.
Edexcel C1 2011 June Q10
10. The curve \(C\) has equation $$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$
  1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
  3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
  4. Find the \(x\)-coordinate of \(B\).
Edexcel C1 2012 June Q1
  1. Find
$$\int \left( 6 x ^ { 2 } + \frac { 2 } { x ^ { 2 } } + 5 \right) \mathrm { d } x$$ giving each term in its simplest form.
Edexcel C1 2012 June Q2
2. (a) Evaluate \(( 32 ) ^ { \frac { 3 } { 5 } }\), giving your answer as an integer.
(b) Simplify fully \(\left( \frac { 25 x ^ { 4 } } { 4 } \right) ^ { - \frac { 1 } { 2 } }\)
Edexcel C1 2012 June Q3
3. Show that \(\frac { 2 } { \sqrt { } ( 12 ) - \sqrt { } ( 8 ) }\) can be written in the form \(\sqrt { } a + \sqrt { } b\), where \(a\) and \(b\) are integers.
Edexcel C1 2012 June Q4
4. $$y = 5 x ^ { 3 } - 6 x ^ { \frac { 4 } { 3 } } + 2 x - 3$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in its simplest form.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
Edexcel C1 2012 June Q5
5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3
& a _ { n + 1 } = 2 a _ { n } - c \quad ( n \geqslant 1 ) \end{aligned}$$ where \(c\) is a constant.
  1. Write down an expression, in terms of \(c\), for \(a _ { 2 }\)
  2. Show that \(a _ { 3 } = 12 - 3 c\) Given that \(\sum _ { i = 1 } ^ { 4 } a _ { i } \geqslant 23\)
  3. find the range of values of \(c\).
Edexcel C1 2012 June Q6
6. A boy saves some money over a period of 60 weeks. He saves 10 p in week 1 , 15 p in week \(2,20 \mathrm { p }\) in week 3 and so on until week 60 . His weekly savings form an arithmetic sequence.
  1. Find how much he saves in week 15
  2. Calculate the total amount he saves over the 60 week period. The boy's sister also saves some money each week over a period of \(m\) weeks. She saves 10 p in week \(1,20 \mathrm { p }\) in week \(2,30 \mathrm { p }\) in week 3 and so on so that her weekly savings form an arithmetic sequence. She saves a total of \(\pounds 63\) in the \(m\) weeks.
  3. Show that $$m ( m + 1 ) = 35 \times 36$$
  4. Hence write down the value of \(m\).
Edexcel C1 2012 June Q7
7. The point \(P ( 4 , - 1 )\) lies on the curve \(C\) with equation \(y = \mathrm { f } ( x ) , x > 0\), and $$f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - \frac { 6 } { \sqrt { } x } + 3$$
  1. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.
  2. Find \(\mathrm { f } ( x )\).
Edexcel C1 2012 June Q8
8. $$4 x - 5 - x ^ { 2 } = q - ( x + p ) ^ { 2 }$$ where \(p\) and \(q\) are integers.
  1. Find the value of \(p\) and the value of \(q\).
  2. Calculate the discriminant of \(4 x - 5 - x ^ { 2 }\)
  3. On the axes on page 17, sketch the curve with equation \(y = 4 x - 5 - x ^ { 2 }\) showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-11_1143_1143_260_388}
Edexcel C1 2012 June Q9
9. The line \(L _ { 1 }\) has equation \(4 y + 3 = 2 x\) The point \(A ( p , 4 )\) lies on \(L _ { 1 }\)
  1. Find the value of the constant \(p\). The line \(L _ { 2 }\) passes through the point \(C ( 2,4 )\) and is perpendicular to \(L _ { 1 }\)
  2. Find an equation for \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(L _ { 1 }\) and the line \(L _ { 2 }\) intersect at the point \(D\).
  3. Find the coordinates of the point \(D\).
  4. Show that the length of \(C D\) is \(\frac { 3 } { 2 } \sqrt { } 5\) A point \(B\) lies on \(L _ { 1 }\) and the length of \(A B = \sqrt { } ( 80 )\)
    The point \(E\) lies on \(L _ { 2 }\) such that the length of the line \(C D E = 3\) times the length of \(C D\).
  5. Find the area of the quadrilateral \(A C B E\).
Edexcel C1 2012 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-14_515_833_251_552} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } ( 9 - 2 x )$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).
  1. Write down the coordinates of the point \(A\).
  2. On separate diagrams sketch the curve with equation
    1. \(y = \mathrm { f } ( x + 3 )\)
    2. \(y = \mathrm { f } ( 3 x )\) On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
  3. Write down the value of \(k\).
Edexcel C1 2013 June Q1
Given \(y = x ^ { 3 } + 4 x + 1\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 3\)
Edexcel C1 2013 June Q4
4. The line \(L _ { 1 }\) has equation \(4 x + 2 y - 3 = 0\)
  1. Find the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) is perpendicular to \(L _ { 1 }\) and passes through the point \(( 2,5 )\).
  2. Find the equation of \(L _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C1 2013 June Q5
5. Solve
  1. \(2 ^ { y } = 8\)
  2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)
Edexcel C1 2013 June Q6
6. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1
x _ { n + 1 } = \left( x _ { n } \right) ^ { 2 } - k x _ { n } , \quad n \geqslant 1 \end{gathered}$$ where \(k\) is a constant, \(k \neq 0\)
  1. Find an expression for \(x _ { 2 }\) in terms of \(k\).
  2. Show that \(x _ { 3 } = 1 - 3 k + 2 k ^ { 2 }\) Given also that \(x _ { 3 } = 1\),
  3. calculate the value of \(k\).
  4. Hence find the value of \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)
Edexcel C1 2013 June Q7
7. Each year, Abbie pays into a savings scheme. In the first year she pays in \(\pounds 500\). Her payments then increase by \(\pounds 200\) each year so that she pays \(\pounds 700\) in the second year, \(\pounds 900\) in the third year and so on.
  1. Find out how much Abbie pays into the savings scheme in the tenth year. Abbie pays into the scheme for \(n\) years until she has paid in a total of \(\pounds 67200\).
  2. Show that \(n ^ { 2 } + 4 n - 24 \times 28 = 0\)
  3. Hence find the number of years that Abbie pays into the savings scheme.
Edexcel C1 2013 June Q8
  1. A rectangular room has a width of \(x \mathrm {~m}\).
The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m ,
  1. show that \(x > 2.8\) Given also that the area of the room is less than \(21 \mathrm {~m} ^ { 2 }\),
    1. write down an inequality, in terms of \(x\), for the area of the room.
    2. Solve this inequality.
  3. Hence find the range of possible values for \(x\).
Edexcel C1 2013 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfc23548-bf4f-4efa-9ceb-b8d03bb1f019-13_698_1413_118_280} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 2,0 )\).
The curve \(C\) has a maximum at the point ( 0,4 ).
  1. The equation of the curve \(C\) can be written in the form $$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$ where \(a\), \(b\) and \(c\) are integers.
    Calculate the values of \(a , b\) and \(c\).
  2. Sketch the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) in the space provided on page 24 Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.
Edexcel C1 2013 June Q10
10. A curve has equation \(y = \mathrm { f } ( x )\). The point \(P\) with coordinates \(( 9,0 )\) lies on the curve. Given that $$\mathrm { f } ^ { \prime } ( x ) = \frac { x + 9 } { \sqrt { } x } , \quad x > 0$$
  1. find \(\mathrm { f } ( x )\).
  2. Find the \(x\)-coordinates of the two points on \(y = \mathrm { f } ( x )\) where the gradient of the curve is equal to 10
Edexcel C1 2013 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfc23548-bf4f-4efa-9ceb-b8d03bb1f019-16_556_1214_219_370} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The line \(y = x + 2\) meets the curve \(x ^ { 2 } + 4 y ^ { 2 } - 2 x = 35\) at the points \(A\) and \(B\) as shown in Figure 2.
  1. Find the coordinates of \(A\) and the coordinates of \(B\).
  2. Find the distance \(A B\) in the form \(r \sqrt { 2 }\) where \(r\) is a rational number.
Edexcel C1 2013 June Q1
  1. Simplify
$$\frac { 7 + \sqrt { 5 } } { \sqrt { 5 } - 1 }$$ giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
Edexcel C1 2013 June Q2
2. Find $$\int \left( 10 x ^ { 4 } - 4 x - \frac { 3 } { \sqrt { } x } \right) \mathrm { d } x$$ giving each term in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-03_120_51_2599_1900}