Questions — Edexcel C1 (490 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 2014 June Q5
5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$a _ { n + 1 } = 5 a _ { n } - 3 , \quad n \geqslant 1$$ Given that \(a _ { 2 } = 7\),
  1. find the value of \(a _ { 1 }\)
  2. Find the value of \(\sum _ { r = 1 } ^ { 4 } a _ { r }\)
Edexcel C1 2014 June Q6
6
  1. Write \(\sqrt { } 80\) in the form \(c \sqrt { } 5\), where \(c\) is a positive constant. A rectangle \(R\) has a length of ( \(1 + \sqrt { } 5\) ) cm and an area of \(\sqrt { 80 } \mathrm {~cm} ^ { 2 }\).
  2. Calculate the width of \(R\) in cm . Express your answer in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers to be found.
Edexcel C1 2014 June Q7
7. Differentiate with respect to \(x\), giving each answer in its simplest form.
  1. \(( 1 - 2 x ) ^ { 2 }\)
  2. \(\frac { x ^ { 5 } + 6 \sqrt { } x } { 2 x ^ { 2 } }\)
Edexcel C1 2014 June Q8
8. In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 computers in 2002, and so on forming an arithmetic sequence.
  1. Show that the shop sold 220 computers in 2007.
  2. Calculate the total number of computers the shop sold from 2000 to 2013 inclusive. In the year 2000, the selling price of each computer was \(\pounds 900\). The selling price fell by \(\pounds 20\) each year, so that in 2001 the selling price was \(\pounds 880\), in 2002 the selling price was \(\pounds 860\), and so on forming an arithmetic sequence.
  3. In a particular year, the selling price of each computer in \(\pounds s\) was equal to three times the number of computers the shop sold in that year. By forming and solving an equation, find the year in which this occurred.
Edexcel C1 2014 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-12_675_863_267_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The line \(l _ { 1 }\), shown in Figure 2 has equation \(2 x + 3 y = 26\)
The line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\)
  1. Find an equation for the line \(l _ { 2 }\) The line \(l _ { 2 }\) intersects the line \(l _ { 1 }\) at the point \(C\).
    Line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.
  2. Find the area of triangle \(O B C\). Give your answer in the form \(\frac { a } { b }\), where \(a\) and \(b\) are integers to be determined.
Edexcel C1 2014 June Q10
10. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point (4,25). Given that $$f ^ { \prime } ( x ) = \frac { 3 } { 8 } x ^ { 2 } - 10 x ^ { - \frac { 1 } { 2 } } + 1 , \quad x > 0$$
  1. find \(\mathrm { f } ( x )\), simplifying each term.
  2. Find an equation of the normal to the curve at the point ( 4,25 ). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel C1 2014 June Q11
11. Given that $$f ( x ) = 2 x ^ { 2 } + 8 x + 3$$
  1. find the value of the discriminant of \(\mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(p ( x + q ) ^ { 2 } + r\) where \(p , q\) and \(r\) are integers to be found. The line \(y = 4 x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = \mathrm { f } ( x )\).
  3. Calculate the value of \(c\).
Edexcel C1 Specimen Q1
  1. Calculate \(\sum _ { r = 1 } ^ { 20 } 5 + 2 r\)
  2. Find \(\int 5 x + 3 \sqrt { x } d x\)
  3. (a) Express \(\sqrt { } 80\) in the form \(a \sqrt { } 5\), where \(a\) is an integer.
    (b) Express \(( 4 - \sqrt { 5 } ) ^ { 2 }\) in the form \(b + c \sqrt { 5 }\), where \(b\) and \(c\) are integers.
  4. The points \(A\) and \(B\) have coordinates \(( 3,4 )\) and \(( 7 , - 6 )\) respectively. The straight line \(l\) passes through \(A\) and is perpendicular to \(A B\). Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. (5)
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{99113eec-7a88-4e26-9711-89253d0168ec-1_457_736_1316_747}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points \(( 0,1 )\) and \(( 3,0 )\). The maximum point on the curve is \(( 1,2 )\). On separate diagrams, sketch the curve with equation
Edexcel C1 Specimen Q6
6. (a) Solve the simultaneous equations $$\begin{aligned} & y + 2 x = 5
& 2 x ^ { 2 } - 3 x - y = 16 \end{aligned}$$ (b) Hence, or otherwise, find the set of values of \(x\) for which $$2 x ^ { 2 } - 3 x - 16 > 5 - 2 x$$
Edexcel C1 Specimen Q7
  1. Ahmed plans to save \(\pounds 250\) in the year 2001, \(\pounds 300\) in 2002, \(\pounds 350\) in 2003, and so on until the year 2020. His planned savings form an arithmetic sequence with common difference £50.
    1. Find the amount he plans to save in the year 2011.
    2. Calculate his total planned savings over the 20 year period from 2001 to 2020.
    Ben also plans to save money over the same 20 year period. He saves \(\pounds A\) in the year 2001 and his planned yearly savings form an arithmetic sequence with common difference \(\pounds 60\). Given that Ben's total planned savings over the 20 year period are equal to Ahmed's total planned savings over the same period,
  2. calculate the value of \(A\).
Edexcel C1 Specimen Q8
8. Given that $$x ^ { 2 } + 10 x + 36 = ( x + a ) ^ { 2 } + b$$ where \(a\) and \(b\) are constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Hence show that the equation \(x ^ { 2 } + 10 x + 36 = 0\) has no real roots. The equation \(x ^ { 2 } + 10 x + k = 0\) has equal roots.
  3. Find the value of \(k\).
  4. For this value of \(k\), sketch the graph of \(y = x ^ { 2 } + 10 x + k\), showing the coordinates of any points at which the graph meets the coordinate axes.
Edexcel C1 Specimen Q9
9. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) and the point \(P ( 3,5 )\) lies on \(C\). Given that $$f ( x ) = 3 x ^ { 2 } - 8 x + 6$$
  1. find \(\mathrm { f } ( x )\).
  2. Verify that the point \(( 2,0 )\) lies on \(C\). The point \(Q\) also lies on \(C\), and the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  3. Find the \(x\)-coordinate of \(Q\).
Edexcel C1 Specimen Q10
10. The curve \(C\) has equation \(y = x ^ { 3 } - 5 x + \frac { 2 } { x } , x \neq 0\). The points \(A\) and \(B\) both lie on \(C\) and have coordinates \(( 1 , - 2 )\) and \(( - 1,2 )\) respectively.
  1. Show that the gradient of \(C\) at \(A\) is equal to the gradient of \(C\) at \(B\).
  2. Show that an equation for the normal to \(C\) at \(A\) is \(4 y = x - 9\). The normal to \(C\) at \(A\) meets the \(y\)-axis at the point \(P\). The normal to \(C\) at \(B\) meets the \(y\)-axis at the point \(Q\).
  3. Find the length of \(P Q\).
Edexcel C1 Q1
  1. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7 .
    (b) Hence, or otherwise, evaluate \(\sum _ { r = 1 } ^ { 142 } ( 7 r + 2 )\).
  2. Solve the simultaneous equations
$$\begin{gathered} x - 3 y + 1 = 0
x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{gathered}$$
Edexcel C1 Q3
  1. The first three terms of an arithmetic series are \(p , 5 p - 8\), and \(3 p + 8\) respectively.
    1. Show that \(p = 4\).
    2. Find the value of the 40th term of this series.
    3. \(\mathrm { f } ( x ) = x ^ { 2 } - k x + 9\), where \(k\) is a constant.
    4. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions.
    Given that \(k = 4\),
  2. express \(\mathrm { f } ( x )\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found,
Edexcel C1 Q5
5. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } }$$
  1. Use integration to find \(y\) in terms of \(x\).
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
Edexcel C1 Q6
6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$
Edexcel C1 Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{922202a6-3455-433f-ac8f-673daefaa7d2-3_574_574_879_662}
\end{figure} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 1.
  1. Find the coordinates of the centre of the circle.
  2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
  4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.
Edexcel C1 Q1
  1. A sequence is defined by the recurrence relation
$$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where \(a\) is a constant.
  1. Given that \(a = 20\) and \(u _ { 1 } = 3\), find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\), giving your answers to 2 decimal places.
  2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
    1. calculate the value of \(a\),
    2. write down the value of \(u _ { 5 }\).
Edexcel C1 Q2
2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k ( 25 k - 8 ) \geq 0\).
  2. Hence find the set of possible values of \(k\).
  3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
Edexcel C1 Q3
3. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
(b) Solve the simultaneous equations $$\begin{aligned} & x = 2 y - 2
& x ^ { 2 } = y ^ { 2 } + 7 \end{aligned}$$
Edexcel C1 Q4
  1. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0$$
  1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
  2. Using integration, find \(\mathrm { f } ( x )\).
Edexcel C1 Q5
5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
  2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
  3. Find the exact coordinates of the mid-point of \(A C\).
Edexcel C1 Q6
6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
  1. Write down the maximum value of \(\mathrm { f } ( x )\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
  3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
  4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
  5. Find the value of \(k\).
Edexcel C1 Q7
7. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
  2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.