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 2012 June Q7
Moderate -0.3
7. The point \(P ( 4 , - 1 )\) lies on the curve \(C\) with equation \(y = \mathrm { f } ( x ) , x > 0\), and $$f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - \frac { 6 } { \sqrt { } x } + 3$$
  1. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.
  2. Find \(\mathrm { f } ( x )\).
Edexcel C1 2012 June Q8
Moderate -0.8
8. $$4 x - 5 - x ^ { 2 } = q - ( x + p ) ^ { 2 }$$ where \(p\) and \(q\) are integers.
  1. Find the value of \(p\) and the value of \(q\).
  2. Calculate the discriminant of \(4 x - 5 - x ^ { 2 }\)
  3. On the axes on page 17, sketch the curve with equation \(y = 4 x - 5 - x ^ { 2 }\) showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-11_1143_1143_260_388}
Edexcel C1 2012 June Q9
Standard +0.3
9. The line \(L _ { 1 }\) has equation \(4 y + 3 = 2 x\) The point \(A ( p , 4 )\) lies on \(L _ { 1 }\)
  1. Find the value of the constant \(p\). The line \(L _ { 2 }\) passes through the point \(C ( 2,4 )\) and is perpendicular to \(L _ { 1 }\)
  2. Find an equation for \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(L _ { 1 }\) and the line \(L _ { 2 }\) intersect at the point \(D\).
  3. Find the coordinates of the point \(D\).
  4. Show that the length of \(C D\) is \(\frac { 3 } { 2 } \sqrt { } 5\) A point \(B\) lies on \(L _ { 1 }\) and the length of \(A B = \sqrt { } ( 80 )\)
    The point \(E\) lies on \(L _ { 2 }\) such that the length of the line \(C D E = 3\) times the length of \(C D\).
  5. Find the area of the quadrilateral \(A C B E\).
Edexcel C1 2012 June Q10
Moderate -0.8
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-14_515_833_251_552} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } ( 9 - 2 x )$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).
  1. Write down the coordinates of the point \(A\).
  2. On separate diagrams sketch the curve with equation
    1. \(y = \mathrm { f } ( x + 3 )\)
    2. \(y = \mathrm { f } ( 3 x )\) On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
  3. Write down the value of \(k\).
Edexcel C1 2013 June Q1
Easy -1.8
Given \(y = x ^ { 3 } + 4 x + 1\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 3\)
Edexcel C1 2013 June Q4
Moderate -0.8
4. The line \(L _ { 1 }\) has equation \(4 x + 2 y - 3 = 0\)
  1. Find the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) is perpendicular to \(L _ { 1 }\) and passes through the point \(( 2,5 )\).
  2. Find the equation of \(L _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C1 2013 June Q5
Easy -1.2
5. Solve
  1. \(2 ^ { y } = 8\)
  2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)
Edexcel C1 2013 June Q6
Moderate -0.5
6. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1 \\ x _ { n + 1 } = \left( x _ { n } \right) ^ { 2 } - k x _ { n } , \quad n \geqslant 1 \end{gathered}$$ where \(k\) is a constant, \(k \neq 0\)
  1. Find an expression for \(x _ { 2 }\) in terms of \(k\).
  2. Show that \(x _ { 3 } = 1 - 3 k + 2 k ^ { 2 }\) Given also that \(x _ { 3 } = 1\),
  3. calculate the value of \(k\).
  4. Hence find the value of \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)
Edexcel C1 2013 June Q7
Moderate -0.8
7. Each year, Abbie pays into a savings scheme. In the first year she pays in \(\pounds 500\). Her payments then increase by \(\pounds 200\) each year so that she pays \(\pounds 700\) in the second year, \(\pounds 900\) in the third year and so on.
  1. Find out how much Abbie pays into the savings scheme in the tenth year. Abbie pays into the scheme for \(n\) years until she has paid in a total of \(\pounds 67200\).
  2. Show that \(n ^ { 2 } + 4 n - 24 \times 28 = 0\)
  3. Hence find the number of years that Abbie pays into the savings scheme.
Edexcel C1 2013 June Q8
Moderate -0.8
  1. A rectangular room has a width of \(x \mathrm {~m}\).
The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m ,
  1. show that \(x > 2.8\) Given also that the area of the room is less than \(21 \mathrm {~m} ^ { 2 }\),
    1. write down an inequality, in terms of \(x\), for the area of the room.
    2. Solve this inequality.
  3. Hence find the range of possible values for \(x\).
Edexcel C1 2013 June Q9
Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfc23548-bf4f-4efa-9ceb-b8d03bb1f019-13_698_1413_118_280} \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 point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 2,0 )\).
The curve \(C\) has a maximum at the point ( 0,4 ).
  1. The equation of the curve \(C\) can be written in the form $$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$ where \(a\), \(b\) and \(c\) are integers.
    Calculate the values of \(a , b\) and \(c\).
  2. Sketch the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) in the space provided on page 24 Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.
Edexcel C1 2013 June Q10
Moderate -0.3
10. A curve has equation \(y = \mathrm { f } ( x )\). The point \(P\) with coordinates \(( 9,0 )\) lies on the curve. Given that $$\mathrm { f } ^ { \prime } ( x ) = \frac { x + 9 } { \sqrt { } x } , \quad x > 0$$
  1. find \(\mathrm { f } ( x )\).
  2. Find the \(x\)-coordinates of the two points on \(y = \mathrm { f } ( x )\) where the gradient of the curve is equal to 10
Edexcel C1 2013 June Q11
Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfc23548-bf4f-4efa-9ceb-b8d03bb1f019-16_556_1214_219_370} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The line \(y = x + 2\) meets the curve \(x ^ { 2 } + 4 y ^ { 2 } - 2 x = 35\) at the points \(A\) and \(B\) as shown in Figure 2.
  1. Find the coordinates of \(A\) and the coordinates of \(B\).
  2. Find the distance \(A B\) in the form \(r \sqrt { 2 }\) where \(r\) is a rational number.
Edexcel C1 2013 June Q1
Easy -1.2
  1. Simplify
$$\frac { 7 + \sqrt { 5 } } { \sqrt { 5 } - 1 }$$ giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
Edexcel C1 2013 June Q2
Easy -1.3
2. Find $$\int \left( 10 x ^ { 4 } - 4 x - \frac { 3 } { \sqrt { } x } \right) \mathrm { d } x$$ giving each term in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-03_120_51_2599_1900}
Edexcel C1 2013 June Q3
Easy -1.3
3. (a) Find the value of \(8 ^ { \frac { 5 } { 3 } }\)
(b) Simplify fully \(\frac { \left( 2 x ^ { \frac { 1 } { 2 } } \right) ^ { 3 } } { 4 x ^ { 2 } }\)
Edexcel C1 2013 June Q4
Standard +0.3
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = k \left( a _ { n } + 2 \right) , \quad \text { for } n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 2\),
  2. find the two possible values of \(k\).
Edexcel C1 2013 June Q5
Easy -1.2
5. Find the set of values of \(x\) for which
  1. \(2 ( 3 x + 4 ) > 1 - x\)
  2. \(3 x ^ { 2 } + 8 x - 3 < 0\)
Edexcel C1 2013 June Q6
Moderate -0.8
6. The straight line \(L _ { 1 }\) passes through the points \(( - 1,3 )\) and \(( 11,12 )\).
  1. Find an equation for \(L _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The line \(L _ { 2 }\) has equation \(3 y + 4 x - 30 = 0\).
  2. Find the coordinates of the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
Edexcel C1 2013 June Q7
Moderate -0.8
7. A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).
  1. Find the value of \(N\). The company then plans to continue to make 600 mobile phones each week.
  2. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
Edexcel C1 2013 June Q8
Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-09_369_956_287_504} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 3 ) ^ { 2 } ( x - 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(( 1,0 )\), touches it at \(( - 3,0 )\) and crosses the \(y\)-axis at \(( 0 , - 9 )\)
  1. In the space below, sketch the curve \(C\) with equation \(y = \mathrm { f } ( x + 2 )\) and state the coordinates of the points where the curve \(C\) meets the \(x\)-axis.
  2. Write down an equation of the curve \(C\).
  3. Use your answer to part (b) to find the coordinates of the point where the curve \(C\) meets the \(y\)-axis.
Edexcel C1 2013 June Q9
Moderate -0.8
9. $$f ^ { \prime } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) ^ { 2 } } { x ^ { 2 } } , \quad x \neq 0$$
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 9 x ^ { - 2 } + A + B x ^ { 2 }\),
    where \(A\) and \(B\) are constants to be found.
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\). Given that the point \(( - 3,10 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\),
  3. find \(\mathrm { f } ( x )\).
Edexcel C1 2013 June Q10
Moderate -0.8
  1. Given the simultaneous equations
$$\begin{aligned} & 2 x + y = 1 \\ & x ^ { 2 } - 4 k y + 5 k = 0 \end{aligned}$$ where \(k\) is a non zero constant,
  1. show that $$x ^ { 2 } + 8 k x + k = 0$$ Given that \(x ^ { 2 } + 8 k x + k = 0\) has equal roots,
  2. find the value of \(k\).
  3. For this value of \(k\), find the solution of the simultaneous equations.
Edexcel C1 2013 June Q11
Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-15_592_1394_274_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(H\) with equation \(y = \frac { 3 } { x } + 4 , x \neq 0\).
  1. Give the coordinates of the point where \(H\) crosses the \(x\)-axis.
  2. Give the equations of the asymptotes to \(H\).
  3. Find an equation for the normal to \(H\) at the point \(P ( - 3,3 )\). This normal crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  4. Find the length of the line segment \(A B\). Give your answer as a surd.
Edexcel C1 2014 June Q1
Easy -1.8
Factorise fully \(25 x - 9 x ^ { 3 }\)
\includegraphics[max width=\textwidth, alt={}, center]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-02_37_42_2700_1909}