Questions — Edexcel (9670 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 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.
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 \(P ( 4,1 )\).