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 2010 June Q1
Easy -1.8
  1. Write
$$\sqrt { } ( 75 ) - \sqrt { } ( 27 )$$ in the form \(k \sqrt { } x\), where \(k\) and \(x\) are integers.
Edexcel C1 2010 June Q2
Easy -1.3
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
Moderate -0.8
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
Moderate -0.8
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
Moderate -0.8
  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
Moderate -0.8
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
Moderate -0.8
  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
Moderate -0.3
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
Moderate -0.8
  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
Moderate -0.3
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
Moderate -0.3
  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
Easy -1.5
Find the value of
  1. \(25 ^ { \frac { 1 } { 2 } }\)
  2. \(25 ^ { - \frac { 3 } { 2 } }\)
Edexcel C1 2011 June Q4
Moderate -0.3
4. Solve the simultaneous equations $$\begin{aligned} x + y & = 2 \\ 4 y ^ { 2 } - x ^ { 2 } & = 11 \end{aligned}$$
Edexcel C1 2011 June Q5
Moderate -0.8
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
Easy -1.2
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.
Edexcel C1 2011 June Q7
Moderate -0.3
7. $$\mathrm { f } ( x ) = x ^ { 2 } + ( k + 3 ) x + k$$ where \(k\) is a real constant.
  1. Find the discriminant of \(\mathrm { f } ( x )\) in terms of \(k\).
  2. Show that the discriminant of \(\mathrm { f } ( x )\) can be expressed in the form \(( k + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
  3. Show that, for all values of \(k\), the equation \(\mathrm { f } ( x ) = 0\) has real roots.
Edexcel C1 2011 June Q8
Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1bb296f-afb2-43cd-9408-2114d7b66971-09_487_743_210_603} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the origin and through \(( 6,0 )\).
The curve \(C\) has a minimum at the point \(( 3 , - 1 )\). On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( 2 x )\),
  2. \(y = - \mathrm { f } ( x )\),
  3. \(y = \mathrm { f } ( x + p )\), where \(p\) is a constant and \(0 < p < 3\). On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.
Edexcel C1 2011 June Q9
Moderate -0.8
  1. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,
$$2 + 4 + 6 + \ldots \ldots + 100$$ (b) In the arithmetic series $$k + 2 k + 3 k + \ldots \ldots + 100$$ \(k\) is a positive integer and \(k\) is a factor of 100 .
  1. Find, in terms of \(k\), an expression for the number of terms in this series.
  2. Show that the sum of this series is $$50 + \frac { 5000 } { k }$$ (c) Find, in terms of \(k\), the 50th term of the arithmetic sequence $$( 2 k + 1 ) , ( 4 k + 4 ) , ( 6 k + 7 ) , \ldots \ldots ,$$ giving your answer in its simplest form.
Edexcel C1 2011 June Q10
Moderate -0.3
10. The curve \(C\) has equation $$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$
  1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
  3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
  4. Find the \(x\)-coordinate of \(B\).
Edexcel C1 2012 June Q1
Easy -1.2
  1. Find
$$\int \left( 6 x ^ { 2 } + \frac { 2 } { x ^ { 2 } } + 5 \right) \mathrm { d } x$$ giving each term in its simplest form.
Edexcel C1 2012 June Q2
Easy -1.3
2. (a) Evaluate \(( 32 ) ^ { \frac { 3 } { 5 } }\), giving your answer as an integer.
(b) Simplify fully \(\left( \frac { 25 x ^ { 4 } } { 4 } \right) ^ { - \frac { 1 } { 2 } }\)
Edexcel C1 2012 June Q3
Moderate -0.8
3. Show that \(\frac { 2 } { \sqrt { } ( 12 ) - \sqrt { } ( 8 ) }\) can be written in the form \(\sqrt { } a + \sqrt { } b\), where \(a\) and \(b\) are integers.
Edexcel C1 2012 June Q4
Easy -1.3
4. $$y = 5 x ^ { 3 } - 6 x ^ { \frac { 4 } { 3 } } + 2 x - 3$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in its simplest form.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
Edexcel C1 2012 June Q5
Moderate -0.3
5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3 \\ & a _ { n + 1 } = 2 a _ { n } - c \quad ( n \geqslant 1 ) \end{aligned}$$ where \(c\) is a constant.
  1. Write down an expression, in terms of \(c\), for \(a _ { 2 }\)
  2. Show that \(a _ { 3 } = 12 - 3 c\) Given that \(\sum _ { i = 1 } ^ { 4 } a _ { i } \geqslant 23\)
  3. find the range of values of \(c\).
Edexcel C1 2012 June Q6
Moderate -0.8
6. A boy saves some money over a period of 60 weeks. He saves 10 p in week 1 , 15 p in week \(2,20 \mathrm { p }\) in week 3 and so on until week 60 . His weekly savings form an arithmetic sequence.
  1. Find how much he saves in week 15
  2. Calculate the total amount he saves over the 60 week period. The boy's sister also saves some money each week over a period of \(m\) weeks. She saves 10 p in week \(1,20 \mathrm { p }\) in week \(2,30 \mathrm { p }\) in week 3 and so on so that her weekly savings form an arithmetic sequence. She saves a total of \(\pounds 63\) in the \(m\) weeks.
  3. Show that $$m ( m + 1 ) = 35 \times 36$$
  4. Hence write down the value of \(m\).