Questions — Edexcel C1 (574 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
Edexcel C1 2014 June Q5
5 marks Moderate -0.8
5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$a _ { n + 1 } = 5 a _ { n } - 3 , \quad n \geqslant 1$$ Given that \(a _ { 2 } = 7\),
  1. find the value of \(a _ { 1 }\)
  2. Find the value of \(\sum _ { r = 1 } ^ { 4 } a _ { r }\)
Edexcel C1 2014 June Q6
5 marks Easy -1.2
6
  1. Write \(\sqrt { } 80\) in the form \(c \sqrt { } 5\), where \(c\) is a positive constant. A rectangle \(R\) has a length of ( \(1 + \sqrt { } 5\) ) cm and an area of \(\sqrt { 80 } \mathrm {~cm} ^ { 2 }\).
  2. Calculate the width of \(R\) in cm . Express your answer in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers to be found.
Edexcel C1 2014 June Q7
7 marks Easy -1.2
7. Differentiate with respect to \(x\), giving each answer in its simplest form.
  1. \(( 1 - 2 x ) ^ { 2 }\)
  2. \(\frac { x ^ { 5 } + 6 \sqrt { } x } { 2 x ^ { 2 } }\)
Edexcel C1 2014 June Q8
9 marks Moderate -0.3
8. In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 computers in 2002, and so on forming an arithmetic sequence.
  1. Show that the shop sold 220 computers in 2007.
  2. Calculate the total number of computers the shop sold from 2000 to 2013 inclusive. In the year 2000, the selling price of each computer was \(\pounds 900\). The selling price fell by \(\pounds 20\) each year, so that in 2001 the selling price was \(\pounds 880\), in 2002 the selling price was \(\pounds 860\), and so on forming an arithmetic sequence.
  3. In a particular year, the selling price of each computer in \(\pounds s\) was equal to three times the number of computers the shop sold in that year. By forming and solving an equation, find the year in which this occurred.
Edexcel C1 2014 June Q9
10 marks Moderate -0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-12_675_863_267_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The line \(l _ { 1 }\), shown in Figure 2 has equation \(2 x + 3 y = 26\) The line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\)
  1. Find an equation for the line \(l _ { 2 }\) The line \(l _ { 2 }\) intersects the line \(l _ { 1 }\) at the point \(C\).
    Line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.
  2. Find the area of triangle \(O B C\). Give your answer in the form \(\frac { a } { b }\), where \(a\) and \(b\) are integers to be determined.
Edexcel C1 2014 June Q10
10 marks Moderate -0.8
10. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point (4,25). Given that $$f ^ { \prime } ( x ) = \frac { 3 } { 8 } x ^ { 2 } - 10 x ^ { - \frac { 1 } { 2 } } + 1 , \quad x > 0$$
  1. find \(\mathrm { f } ( x )\), simplifying each term.
  2. Find an equation of the normal to the curve at the point ( 4,25 ). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel C1 2014 June Q11
10 marks Moderate -0.5
11. Given that $$f ( x ) = 2 x ^ { 2 } + 8 x + 3$$
  1. find the value of the discriminant of \(\mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(p ( x + q ) ^ { 2 } + r\) where \(p , q\) and \(r\) are integers to be found. The line \(y = 4 x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = \mathrm { f } ( x )\).
  3. Calculate the value of \(c\).
Edexcel C1 Q1
7 marks Standard +0.3
  1. A sequence is defined by the recurrence relation
$$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where \(a\) is a constant.
  1. Given that \(a = 20\) and \(u _ { 1 } = 3\), find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\), giving your answers to 2 decimal places.
  2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
    1. calculate the value of \(a\),
    2. write down the value of \(u _ { 5 }\).
Edexcel C1 Q2
7 marks Moderate -0.8
2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k ( 25 k - 8 ) \geq 0\).
  2. Hence find the set of possible values of \(k\).
  3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
Edexcel C1 Q3
8 marks Moderate -0.3
3. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
(b) Solve the simultaneous equations $$\begin{aligned} & x = 2 y - 2 \\ & x ^ { 2 } = y ^ { 2 } + 7 \end{aligned}$$
Edexcel C1 Q4
9 marks Moderate -0.8
  1. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0$$
  1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
  2. Using integration, find \(\mathrm { f } ( x )\).
Edexcel C1 Q5
10 marks Moderate -0.8
5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
  2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
  3. Find the exact coordinates of the mid-point of \(A C\).
Edexcel C1 Q6
14 marks Easy -1.2
6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
  1. Write down the maximum value of \(\mathrm { f } ( x )\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
  3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
  4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
  5. Find the value of \(k\).
Edexcel C1 Q7
7 marks Moderate -0.8
7. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
  2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 Q8
5 marks Moderate -0.8
8. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
  1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
  2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\). END
Edexcel C1 Q1
7 marks Easy -1.2
1. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
  1. Use integration to find \(y\) in terms of \(x\).
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
Edexcel C1 Q2
6 marks Easy -1.2
2. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r ) .$$
  1. Write down the first two terms of the series.
  2. Find the common difference of the series. Given that \(n = 50\),
  3. find the sum of the series.
Edexcel C1 Q3
6 marks Easy -1.2
3. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\).
  2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).
Edexcel C1 Q4
7 marks Moderate -0.8
4. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c ,$$ where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).
Edexcel C1 Q5
7 marks Moderate -0.3
5. Find the set of values for \(x\) for which
  1. \(6 x - 7 < 2 x + 3\),
  2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
  3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).
Edexcel C1 Q6
5 marks Moderate -0.8
6. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),
  1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
Edexcel C1 Q7
8 marks Moderate -0.5
7. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),
  1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
  2. show that the population of fish dies out during the sixth year.
  3. Find the value of \(d\) which would leave the population each year unchanged.
Edexcel C1 Q8
11 marks Moderate -0.8
8. A curve \(C\) has equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2 . The \(x\)-coordinate of \(P\) is 3 .
  2. Find the \(x\)-coordinate of \(Q\).
  3. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. This tangent intersects the coordinate axes at the points \(R\) and \(S\).
  4. Find the length of \(R S\), giving your answer as a surd.
Edexcel C1 Q9
12 marks Standard +0.3
9. The points \(A ( - 1 , - 2 ) , B ( 7,2 )\) and \(C ( k , 4 )\), where \(k\) is a constant, are the vertices of \(\triangle A B C\). Angle \(A B C\) is a right angle.
  1. Find the gradient of \(A B\).
  2. Calculate the value of \(k\).
  3. Show that the length of \(A B\) may be written in the form \(p \sqrt { 5 }\), where \(p\) is an integer to be found.
  4. Find the exact value of the area of \(\triangle A B C\).
  5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 Q1
5 marks Easy -1.2
  1. Given that \(2 ^ { x } = \frac { 1 } { \sqrt { 2 } }\) and \(2 ^ { y } = 4 \sqrt { } 2\),
    1. find the exact value of \(x\) and the exact value of \(y\),
    2. calculate the exact value of \(2 ^ { y - x }\).
    3. \(f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0\).
    4. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
    5. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
    6. The sum of an arithmetic series is \(\sum _ { r = 1 } ^ { n } ( 80 - 3 r )\).
    7. Write down the first two terms of the series.
    8. Find the common difference of the series.
    Given that \(n = 50\),
  2. find the sum of the series.