Questions — Edexcel (10514 questions)

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Edexcel C1 2013 June Q9
8 marks 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
10 marks 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
9 marks 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
4 marks 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
4 marks 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
5 marks 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
7 marks 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
6 marks 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
7 marks 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
7 marks 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
6 marks 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
10 marks 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
8 marks 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
11 marks 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
3 marks 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}
Edexcel C1 2014 June Q4
6 marks Easy -1.2
4. Given that \(y = 2 x ^ { 5 } + \frac { 6 } { \sqrt { } x } , x > 0\), find in their simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\)
Edexcel C1 2014 June Q5
4 marks Moderate -0.8
5. Solve the equation $$10 + x \sqrt { 8 } = \frac { 6 x } { \sqrt { 2 } }$$ Give your answer in the form \(a \sqrt { } b\) where \(a\) and \(b\) are integers.
Edexcel C1 2014 June Q6
9 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-08_917_1322_239_303} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres.
Given that the perimeter of the garden is greater than 40 m ,
  1. show that \(x > 1.7\) Given that the area of the garden is less than \(120 \mathrm {~m} ^ { 2 }\),
  2. form and solve a quadratic inequality in \(x\).
  3. Hence state the range of the possible values of \(x\).
Edexcel C1 2014 June Q7
9 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-10_869_1073_267_440} \captionsetup{labelformat=empty} \caption{Diagram NOT to scale}
\end{figure} Figure 2 Figure 2 shows a right angled triangle \(L M N\). The points \(L\) and \(M\) have coordinates ( \(- 1,2\) ) and ( \(7 , - 4\) ) respectively.
  1. Find an equation for the straight line passing through the points \(L\) and \(M\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. Given that the coordinates of point \(N\) are ( \(16 , p\) ), where \(p\) is a constant, and angle \(L M N = 90 ^ { \circ }\),
  2. find the value of \(p\). Given that there is a point \(K\) such that the points \(L , M , N\), and \(K\) form a rectangle,
  3. find the \(y\) coordinate of \(K\).
Edexcel C1 2014 June Q8
7 marks Moderate -0.8
8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 37\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.
Edexcel C1 2014 June Q9
9 marks Moderate -0.3
9. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 2 } + 8\) The line \(L\) has equation \(y = 3 x + k\), where \(k\) is a positive constant.
  1. Sketch \(C\) and \(L\) on separate diagrams, showing the coordinates of the points at which \(C\) and \(L\) cut the axes. Given that line \(L\) is a tangent to \(C\),
  2. find the value of \(k\).
Edexcel C1 2014 June Q10
8 marks Moderate -0.8
  1. Xin has been given a 14 day training schedule by her coach.
Xin will run for \(A\) minutes on day 1 , where \(A\) is a constant.
She will then increase her running time by ( \(d + 1\) ) minutes each day, where \(d\) is a constant.
  1. Show that on day 14 , Xin will run for $$( A + 13 d + 13 ) \text { minutes. }$$ Yi has also been given a 14 day training schedule by her coach.
    Yi will run for \(( A - 13 )\) minutes on day 1 .
    She will then increase her running time by ( \(2 d - 1\) ) minutes each day.
    Given that Yi and Xin will run for the same length of time on day 14,
  2. find the value of \(d\). Given that Xin runs for a total time of 784 minutes over the 14 days,
  3. find the value of \(A\).
Edexcel C1 2014 June Q11
11 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-17_700_1556_276_201} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A sketch of part of the curve \(C\) with equation $$y = 20 - 4 x - \frac { 18 } { x } , \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has an \(x\) coordinate equal to 2
  1. Show that the equation of the normal to \(C\) at \(A\) is \(y = - 2 x + 7\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3 .
  2. Use algebra to find the coordinates of \(B\).
Edexcel C1 2015 June Q1
5 marks Easy -1.3
Simplify
  1. \(( 2 \sqrt { } 5 ) ^ { 2 }\)
  2. \(\frac { \sqrt { } 2 } { 2 \sqrt { } 5 - 3 \sqrt { } 2 }\) giving your answer in the form \(a + \sqrt { } b\), where \(a\) and \(b\) are integers.
Edexcel C1 2015 June Q3
6 marks Easy -1.2
Given that \(y = 4 x ^ { 3 } - \frac { 5 } { x ^ { 2 } } , x \neq 0\), find in their simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\)