Questions C1 (1442 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 2007 January Q8
8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P ( 4,8 )\) lies on \(C\).
  3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3 y = x + 20 .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
  4. Find the length \(P Q\), giving your answer in a simplified surd form.
Edexcel C1 2007 January Q9
9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 □ Row 2 □ 1 Row 3
\includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-11_40_104_566_479} She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.
  1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
  2. Find the total number of sticks Ann uses in making these 10 rows. Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the ( \(k + 1\) )th row,
  3. show that \(k\) satisfies \(( 3 k - 100 ) ( k + 35 ) < 0\).
  4. Find the value of \(k\).
Edexcel C1 2007 January Q10
10. (a) On the same axes sketch the graphs of the curves with equations
  1. \(y = x ^ { 2 } ( x - 2 )\),
  2. \(y = x ( 6 - x )\),
    and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
    (b) Use algebra to find the coordinates of the points where the graphs intersect.
Edexcel C1 2008 January Q1
Find \(\int \left( 3 x ^ { 2 } + 4 x ^ { 5 } - 7 \right) d x\).
Edexcel C1 2008 January Q4
4. The point \(A ( - 6,4 )\) and the point \(B ( 8 , - 3 )\) lie on the line \(L\).
  1. Find an equation for \(L\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
  2. Find the distance \(A B\), giving your answer in the form \(k \sqrt { 5 }\), where \(k\) is an integer.
Edexcel C1 2008 January Q5
5. (a) Write \(\frac { 2 \sqrt { } x + 3 } { x }\) in the form \(2 x ^ { p } + 3 x ^ { q }\) where \(p\) and \(q\) are constants. Given that \(y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.
Edexcel C1 2008 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba0ee180-4c22-49f7-8a8e-a7268828b067-07_693_676_370_632} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 1,0 )\) and \(( 4,0 )\). The maximum point on the curve is \(( 2,5 )\).
In separate diagrams sketch the curves with the following equations.
On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.
  1. \(y = 2 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( - x )\). The maximum point on the curve with equation \(y = \mathrm { f } ( x + a )\) is on the \(y\)-axis.
  3. Write down the value of the constant \(a\).
Edexcel C1 2008 January Q7
  1. A sequence is given by:
$$\begin{aligned} & x _ { 1 } = 1
& x _ { n + 1 } = x _ { n } \left( p + x _ { n } \right) \end{aligned}$$ where \(p\) is a constant ( \(p \neq 0\) ) .
  1. Find \(x _ { 2 }\) in terms of \(p\).
  2. Show that \(x _ { 3 } = 1 + 3 p + 2 p ^ { 2 }\). Given that \(x _ { 3 } = 1\),
  3. find the value of \(p\),
  4. write down the value of \(x _ { 2008 }\).
Edexcel C1 2008 January Q8
8. The equation $$x ^ { 2 } + k x + 8 = k$$ has no real solutions for \(x\).
  1. Show that \(k\) satisfies \(k ^ { 2 } + 4 k - 32 < 0\).
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2008 January Q9
9. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), and \(\mathrm { f } ^ { \prime } ( x ) = 4 x - 6 \sqrt { } x + \frac { 8 } { x ^ { 2 } }\). Given that the point \(P ( 4,1 )\) lies on \(C\),
  1. find \(\mathrm { f } ( x )\) and simplify your answer.
  2. Find an equation of the normal to \(C\) at the point \(P ( 4,1 )\).
Edexcel C1 2008 January Q10
  1. The curve \(C\) has equation
$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$
  1. Sketch \(C\) showing clearly the coordinates of the points where the curve meets the coordinate axes.
  2. Show that the equation of \(C\) can be written in the form $$y = x ^ { 3 } + x ^ { 2 } - 5 x + k ,$$ where \(k\) is a positive integer, and state the value of \(k\). There are two points on \(C\) where the gradient of the tangent to \(C\) is equal to 3 .
  3. Find the \(x\)-coordinates of these two points.
Edexcel C1 2008 January Q11
11. The first term of an arithmetic sequence is 30 and the common difference is - 1.5
  1. Find the value of the 25th term. The \(r\) th term of the sequence is 0 .
  2. Find the value of \(r\). The sum of the first \(n\) terms of the sequence is \(S _ { n }\).
  3. Find the largest positive value of \(S _ { n }\).
Edexcel C1 2009 January Q1
  1. Write down the value of \(125 ^ { \frac { 1 } { 3 } }\).
  2. Find the value of \(125 ^ { - \frac { 2 } { 3 } }\).
Edexcel C1 2009 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{871f5957-180d-4379-88ce-186432f57bad-06_988_1158_285_390} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). There is a maximum at \(( 0,0 )\), a minimum at \(( 2 , - 1 )\) and \(C\) passes through \(( 3,0 )\). On separate diagrams sketch the curve with equation
  1. \(y = \mathrm { f } ( x + 3 )\),
  2. \(y = \mathrm { f } ( - x )\). On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the \(x\)-axis.
Edexcel C1 2009 January Q6
  1. Given that \(\frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }\) can be written in the form \(2 x ^ { p } - x ^ { q }\),
    1. write down the value of \(p\) and the value of \(q\).
    Given that \(y = 5 x ^ { 4 } - 3 + \frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }\),
  2. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.
Edexcel C1 2009 January Q7
7. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 5 k + 4 > 0 .$$
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2009 January Q8
8. The point \(P ( 1 , a )\) lies on the curve with equation \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\).
  1. Find the value of \(a\).
  2. On the axes below sketch the curves with the following equations:
    1. \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\),
    2. \(y = \frac { 2 } { x }\). On your diagram show clearly the coordinates of any points at which the curves meet the axes.
  3. With reference to your diagram in part (b) state the number of real solutions to the equation $$( x + 1 ) ^ { 2 } ( 2 - x ) = \frac { 2 } { x } .$$
    \includegraphics[max width=\textwidth, alt={}]{871f5957-180d-4379-88ce-186432f57bad-10_1347_1344_1245_297}
Edexcel C1 2009 January Q9
9. The first term of an arithmetic series is \(a\) and the common difference is \(d\). The 18th term of the series is 25 and the 21st term of the series is \(32 \frac { 1 } { 2 }\).
  1. Use this information to write down two equations for \(a\) and \(d\).
  2. Show that \(a = - 17.5\) and find the value of \(d\). The sum of the first \(n\) terms of the series is 2750 .
  3. Show that \(n\) is given by $$n ^ { 2 } - 15 n = 55 \times 40 .$$
  4. Hence find the value of \(n\).
Edexcel C1 2009 January Q10
  1. The line \(l _ { 1 }\) passes through the point \(A ( 2,5 )\) and has gradient \(- \frac { 1 } { 2 }\).
    1. Find an equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\).
    The point \(B\) has coordinates (-2, 7).
  2. Show that \(B\) lies on \(l _ { 1 }\).
  3. Find the length of \(A B\), giving your answer in the form \(k \sqrt { } 5\), where \(k\) is an integer. The point \(C\) lies on \(l _ { 1 }\) and has \(x\)-coordinate equal to \(p\).
    The length of \(A C\) is 5 units.
  4. Show that \(p\) satisfies $$p ^ { 2 } - 4 p - 16 = 0 .$$
Edexcel C1 2009 January Q11
  1. The curve \(C\) has equation
$$y = 9 - 4 x - \frac { 8 } { x } , \quad x > 0$$ The point \(P\) on \(C\) has \(x\)-coordinate equal to 2 .
  1. Show that the equation of the tangent to \(C\) at the point \(P\) is \(y = 1 - 2 x\).
  2. Find an equation of the normal to \(C\) at the point \(P\). The tangent at \(P\) meets the \(x\)-axis at \(A\) and the normal at \(P\) meets the \(x\)-axis at \(B\).
  3. Find the area of triangle \(A P B\).
Edexcel C1 2010 January Q1
Given that \(y = x ^ { 4 } + x ^ { \frac { 1 } { 3 } } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
Edexcel C1 2010 January Q4
4. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 35\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.
Edexcel C1 2010 January Q5
5. Solve the simultaneous equations $$\begin{array} { r } y - 3 x + 2 = 0
y ^ { 2 } - x - 6 x ^ { 2 } = 0 \end{array}$$
Edexcel C1 2010 January Q6
6. The curve \(C\) has equation $$y = \frac { ( x + 3 ) ( x - 8 ) } { x } , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
  2. Find an equation of the tangent to \(C\) at the point where \(x = 2\)
Edexcel C1 2010 January Q7
7. Jill gave money to a charity over a 20 -year period, from Year 1 to Year 20 inclusive. She gave \(\pounds 150\) in Year \(1 , \pounds 160\) in Year 2, \(\pounds 170\) in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.
  1. Find the amount of money she gave in Year 10.
  2. Calculate the total amount of money she gave over the 20 -year period. Kevin also gave money to the charity over the same 20 -year period. He gave \(\pounds A\) in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference \(\pounds 30\). The total amount of money that Kevin gave over the 20 -year period was twice the total amount of money that Jill gave.
  3. Calculate the value of \(A\).