Questions C1 (1442 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 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 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 2009 June Q2
2. Given that \(32 \sqrt { } 2 = 2 ^ { a }\), find the value of \(a\).
Edexcel C1 2009 June Q3
3. Given that \(y = 2 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } , x \neq 0\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\), simplifying each term.
Edexcel C1 2009 June Q4
4. Find the set of values of \(x\) for which
  1. \(4 x - 3 > 7 - x\)
  2. \(2 x ^ { 2 } - 5 x - 12 < 0\)
  3. both \(4 x - 3 > 7 - x\) and \(2 x ^ { 2 } - 5 x - 12 < 0\)
Edexcel C1 2009 June Q5
5. A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and finished in 1990 (Year 40). The numbers of houses built each year form an arithmetic sequence with first term \(a\) and common difference \(d\). Given that 2400 new houses were built in 1960 and 600 new houses were built in 1990, find
  1. the value of \(d\),
  2. the value of \(a\),
  3. the total number of houses built in Oldtown over the 40-year period.
Edexcel C1 2009 June Q6
6. The equation \(x ^ { 2 } + 3 p x + p = 0\), where \(p\) is a non-zero constant, has equal roots. Find the value of \(p\).
Edexcel C1 2009 June Q7
7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 2 a _ { n } - 7 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  2. Show that \(a _ { 3 } = 4 k - 21\). Given that \(\sum _ { r = 1 } ^ { 4 } a _ { r } = 43\),
  3. find the value of \(k\).
Edexcel C1 2009 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e72d0d82-af0e-4f36-8446-a67b764fd7f3-09_908_1043_201_495} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(A\) and \(B\) have coordinates \(( 6,7 )\) and \(( 8,2 )\) respectively.
The line \(l\) passes through the point \(A\) and is perpendicular to the line \(A B\), as shown in Figure 1.
  1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. Given that \(l\) intersects the \(y\)-axis at the point \(C\), find
  2. the coordinates of \(C\),
  3. the area of \(\triangle O C B\), where \(O\) is the origin.
Edexcel C1 2009 June Q9
9. $$f ( x ) = \frac { ( 3 - 4 \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , \quad x > 0$$
  1. Show that \(\mathrm { f } ( x ) = 9 x ^ { - \frac { 1 } { 2 } } + A x ^ { \frac { 1 } { 2 } } + B\), where \(A\) and \(B\) are constants to be found.
  2. Find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Evaluate \(\mathrm { f } ^ { \prime } ( 9 )\).
Edexcel C1 2009 June Q10
10. (a) Factorise completely \(x ^ { 3 } - 6 x ^ { 2 } + 9 x\)
(b) Sketch the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. Using your answer to part (b), or otherwise,
(c) sketch, on a separate diagram, the curve with equation $$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$ showing the coordinates of the points at which the curve meets the \(x\)-axis.
Edexcel C1 2009 June Q11
11. The curve \(C\) has equation $$y = x ^ { 3 } - 2 x ^ { 2 } - x + 9 , \quad x > 0$$ The point \(P\) has coordinates (2, 7).
  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. The point \(Q\) also lies on \(C\).
    Given that the tangent to \(C\) at \(Q\) is perpendicular to the tangent to \(C\) at \(P\),
  3. show that the \(x\)-coordinate of \(Q\) is \(\frac { 1 } { 3 } ( 2 + \sqrt { 6 } )\).
Edexcel C1 2010 June Q1
  1. Write
$$\sqrt { } ( 75 ) - \sqrt { } ( 27 )$$ in the form \(k \sqrt { } x\), where \(k\) and \(x\) are integers.
Edexcel C1 2010 June Q2
2. Find $$\int \left( 8 x ^ { 3 } + 6 x ^ { \frac { 1 } { 2 } } - 5 \right) d x$$ giving each term in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-03_40_38_2682_1914}
Edexcel C1 2010 June Q3
3. Find the set of values of \(x\) for which
  1. \(3 ( x - 2 ) < 8 - 2 x\)
  2. \(( 2 x - 7 ) ( 1 + x ) < 0\)
  3. both \(3 ( x - 2 ) < 8 - 2 x\) and \(( 2 x - 7 ) ( 1 + x ) < 0\)
Edexcel C1 2010 June Q4
4. (a) Show that \(x ^ { 2 } + 6 x + 11\) can be written as $$( x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers to be found.
(b) In the space at the top of page 7 , sketch the curve with equation \(y = x ^ { 2 } + 6 x + 11\), showing clearly any intersections with the coordinate axes.
(c) Find the value of the discriminant of \(x ^ { 2 } + 6 x + 11\)
Edexcel C1 2010 June Q5
  1. A sequence of positive numbers is defined by
$$\begin{aligned} a _ { n + 1 } & = \sqrt { } \left( a _ { n } ^ { 2 } + 3 \right) , \quad n \geqslant 1 ,
a _ { 1 } & = 2 \end{aligned}$$
  1. Find \(a _ { 2 }\) and \(a _ { 3 }\), leaving your answers in surd form.
  2. Show that \(a _ { 5 } = 4\)
Edexcel C1 2010 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65d61b2c-2e47-402e-b08f-2d46bb00c188-08_568_942_269_498} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum point \(A\) at \(( - 2,3 )\) and a minimum point \(B\) at \(( 3 , - 5 )\). On separate diagrams sketch the curve with equation
  1. \(y = \mathrm { f } ( x + 3 )\)
  2. \(y = 2 \mathrm { f } ( x )\) On each diagram show clearly the coordinates of the maximum and minimum points.
    The graph of \(y = \mathrm { f } ( x ) + a\) has a minimum at (3, 0), where \(a\) is a constant.
  3. Write down the value of \(a\).
Edexcel C1 2010 June Q7
  1. Given that
$$y = 8 x ^ { 3 } - 4 \sqrt { } x + \frac { 3 x ^ { 2 } + 2 } { x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(6)
Edexcel C1 2010 June Q8
8. (a) Find an equation of the line joining \(A ( 7,4 )\) and \(B ( 2,0 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(b) Find the length of \(A B\), leaving your answer in surd form. The point \(C\) has coordinates ( \(2 , t\) ), where \(t > 0\), and \(A C = A B\).
(c) Find the value of \(t\).
(d) Find the area of triangle \(A B C\).
\(\_\_\_\_\)}
Edexcel C1 2010 June Q9
  1. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays \(\pounds a\) for their first day, \(\pounds ( a + d )\) for their second day, \(\pounds ( a + 2 d )\) for their third day, and so on, thus increasing the daily payment by \(\pounds d\) for each extra day they work.
A picker who works for all 30 days will earn \(\pounds 40.75\) on the final day.
  1. Use this information to form an equation in \(a\) and \(d\). A picker who works for all 30 days will earn a total of \(\pounds 1005\)
  2. Show that \(15 ( a + 40.75 ) = 1005\)
  3. Hence find the value of \(a\) and the value of \(d\).
Edexcel C1 2010 June Q10
10. (a) On the axes below sketch the graphs of
  1. \(y = x ( 4 - x )\)
  2. \(y = x ^ { 2 } ( 7 - x )\)
    showing clearly the coordinates of the points where the curves cross the coordinate axes.
    (b) Show that the \(x\)-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 4 \right) = 0\) The point \(A\) lies on both of the curves and the \(x\) and \(y\) coordinates of \(A\) are both positive.
    (c) Find the exact coordinates of \(A\), leaving your answer in the form ( \(p + q \sqrt { } 3 , r + s \sqrt { } 3\) ), where \(p , q , r\) and \(s\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-14_1178_1203_1407_379}
Edexcel C1 2010 June Q11
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\), where
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x - \frac { 5 } { \sqrt { } x } - 2$$ Given that the point \(P ( 4,5 )\) lies on \(C\), find
  1. \(\mathrm { f } ( x )\),
  2. an equation of the tangent 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.
Edexcel C1 2011 June Q1
Find the value of
  1. \(25 ^ { \frac { 1 } { 2 } }\)
  2. \(25 ^ { - \frac { 3 } { 2 } }\)
Edexcel C1 2011 June Q4
4. Solve the simultaneous equations $$\begin{aligned} x + y & = 2
4 y ^ { 2 } - x ^ { 2 } & = 11 \end{aligned}$$
Edexcel C1 2011 June Q5
5. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 5 a _ { n } + 3 , \quad n \geqslant 1 , \end{aligned}$$ where \(k\) is a positive integer.
  1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  2. Show that \(a _ { 3 } = 25 k + 18\).
    1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\), in its simplest form.
    2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 6 .
Edexcel C1 2011 June Q6
6. Given that \(\frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\) can be written in the form \(6 x ^ { p } + 3 x ^ { q }\),
  1. write down the value of \(p\) and the value of \(q\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\), and that \(y = 90\) when \(x = 4\),
  2. find \(y\) in terms of \(x\), simplifying the coefficient of each term.