Questions C1 (1442 questions)

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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
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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 2005 January Q1
  1. Write down the value of \(16 ^ { \frac { 1 } { 2 } }\).
  2. Find the value of \(16 ^ { - \frac { 3 } { 2 } }\).
Edexcel C1 2005 January Q3
3. Given that the equation \(k x ^ { 2 } + 12 x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\).
Edexcel C1 2005 January Q4
4. Solve the simultaneous equations $$\begin{gathered} x + y = 2
x ^ { 2 } + 2 y = 12 \end{gathered}$$
Edexcel C1 2005 January Q5
5. The \(r\) th term of an arithmetic series is ( \(2 r - 5\) ).
  1. Write down the first three terms of this series.
  2. State the value of the common difference.
  3. Show that \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).
Edexcel C1 2005 January Q6
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-10_515_714_292_609}
\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 \(( 2,0 )\) and \(( 4,0 )\). The minimum point on the curve is \(P ( 3 , - 2 )\). In separate diagrams sketch the curve with equation
  1. \(y = - \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\). On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.
Edexcel C1 2005 January Q7
7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .
  1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
  2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
  3. Find the value of \(k\).
Edexcel C1 2005 January Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-14_687_1196_280_388}
\end{figure} The points \(A ( 1,7 ) , B ( 20,7 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 2. The point \(D ( 8,2 )\) is the mid-point of \(A C\).
  1. Find the value of \(p\) and the value of \(q\). The line \(l\), which passes through \(D\) and is perpendicular to \(A C\), intersects \(A B\) at \(E\).
  2. Find an equation for \(l\), in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
  3. Find the exact \(x\)-coordinate of \(E\).
Edexcel C1 2005 January Q9
9. The gradient of the curve \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 } .$$ The point \(P ( 1,4 )\) lies on \(C\).
  1. Find an equation of the normal to \(C\) at \(P\).
  2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
  3. Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).
Edexcel C1 2005 January Q10
10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geqslant 0 ,$$
  1. express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \geqslant 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
  2. In the space provided on page 19, sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). The line \(y = 41\) meets \(C\) at the point \(R\).
  3. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q \sqrt { } 2\), where \(p\) and \(q\) are integers.
Edexcel C1 2006 January Q1
  1. Factorise completely
$$x ^ { 3 } - 4 x ^ { 2 } + 3 x .$$
Edexcel C1 2006 January Q2
2. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1 .$$
  1. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Write down the value of \(u _ { 20 }\).
Edexcel C1 2006 January Q3
3. The line \(L\) has equation \(y = 5 - 2 x\).
  1. Show that the point \(P ( 3 , - 1 )\) lies on \(L\).
  2. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2006 January Q4
4. Given that \(y = 2 x ^ { 2 } - \frac { 6 } { x ^ { 3 } } , x \neq 0\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\int y \mathrm {~d} x\).
Edexcel C1 2006 January Q5
5. (a) Write \(\sqrt { 45 }\) in the form \(a \sqrt { 5 }\), where \(a\) is an integer.
(b) Express \(\frac { 2 ( 3 + \sqrt { 5 } ) } { ( 3 - \sqrt { 5 } ) }\) in the form \(b + c \sqrt { 5 }\), where \(b\) and \(c\) are integers.
\section*{6.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{815e288c-0140-4c12-9e89-b0bb4fb1a8c1-07_607_844_310_555}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the points \(( 0,3 )\) and \(( 4,0 )\) and touches the \(x\)-axis at the point \(( 1,0 )\). On separate diagrams sketch the curve with equation
Edexcel C1 2006 January Q7
  1. On Alice's 11th birthday she started to receive an annual allowance. The first annual allowance was \(\pounds 500\) and on each following birthday the allowance was increased by \(\pounds 200\).
    1. Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was \(\pounds 1200\).
    2. Find the amount of Alice's annual allowance on her 18th birthday.
    3. Find the total of the allowances that Alice had received up to and including her 18th birthday.
    When the total of the allowances that Alice had received reached \(\pounds 32000\) the allowance stopped.
  2. Find how old Alice was when she received her last allowance.
Edexcel C1 2006 January Q8
  1. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 1,6 )\). Given that
$$f ^ { \prime } ( x ) = 3 + \frac { 5 x ^ { 2 } + 2 } { x ^ { \frac { 1 } { 2 } } } , x > 0$$ find \(\mathrm { f } ( x )\) and simplify your answer.
Edexcel C1 2006 January Q9
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{815e288c-0140-4c12-9e89-b0bb4fb1a8c1-12_812_1088_317_427}
\end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the \(x\)-axis at the points \(P , ( 1,0 )\) and \(Q\), as shown in Figure 2.
  1. Write down the \(x\)-coordinate of \(P\), and the \(x\)-coordinate of \(Q\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4\).
  3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point ( \(- 1,6\) ). The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point ( \(- 1,6\) ).
  4. Find the exact coordinates of \(R\).
Edexcel C1 2006 January Q10
10. $$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$
  1. Find the values of the constants \(a\) and \(b\).
  2. In the space provided below, sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
  3. Find the value of the discriminant of \(x ^ { 2 } + 2 x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). The equation \(x ^ { 2 } + k x + 3 = 0\), where \(k\) is a constant, has no real roots.
  4. Find the set of possible values of \(k\), giving your answer in surd form.
Edexcel C1 2007 January Q1
  1. Given that
$$y = 4 x ^ { 3 } - 1 + 2 x ^ { \frac { 1 } { 2 } } , \quad x > 0 ,$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
\includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-02_29_45_2690_1852}
Edexcel C1 2007 January Q2
2. (a) Express \(\sqrt { } 108\) in the form \(a \sqrt { } 3\), where \(a\) is an integer.
(b) Express \(( 2 - \sqrt { 3 } ) ^ { 2 }\) in the form \(b + c \sqrt { 3 }\), where \(b\) and \(c\) are integers to be found.
Edexcel C1 2007 January Q3
3. Given that \(\quad \mathrm { f } ( x ) = \frac { 1 } { x } , \quad x \neq 0\),
  1. sketch the graph of \(y = \mathrm { f } ( x ) + 3\) and state the equations of the asymptotes.
  2. Find the coordinates of the point where \(y = \mathrm { f } ( x ) + 3\) crosses a coordinate axis.
Edexcel C1 2007 January Q4
4. Solve the simultaneous equations $$\begin{aligned} & y = x - 2 ,
& y ^ { 2 } + x ^ { 2 } = 10 . \end{aligned}$$
Edexcel C1 2007 January Q5
5. The equation \(2 x ^ { 2 } - 3 x - ( k + 1 ) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\).
Edexcel C1 2007 January Q6
6. (a) Show that \(( 4 + 3 \sqrt { } x ) ^ { 2 }\) can be written as \(16 + k \sqrt { } x + 9 x\), where \(k\) is a constant to be found.
(b) Find \(\int ( 4 + 3 \sqrt { } x ) ^ { 2 } \mathrm {~d} x\).
Edexcel C1 2007 January Q7
7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \neq 0\), and the point \(P ( 2,1 )\) lies on \(C\). Given that $$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 6 - \frac { 8 } { x ^ { 2 } } ,$$
  1. find \(\mathrm { f } ( x )\).
  2. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.