Questions — Edexcel (10514 questions)

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Edexcel C2 2006 January Q1
8 marks Moderate -0.8
  1. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 5 x + c\), where \(c\) is a constant.
Given that \(\mathrm { f } ( 1 ) = 0\),
  1. find the value of \(c\),
  2. factorise \(\mathrm { f } ( x )\) completely,
  3. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 3\) ).
Edexcel C2 2006 January Q2
6 marks Moderate -0.8
2.
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + p x ) ^ { 9 }$$ where \(p\) is a constant. These first 3 terms are \(1,36 x\) and \(q x ^ { 2 }\), where \(q\) is a constant.
  2. Find the value of \(p\) and the value of \(q\).
Edexcel C2 2006 January Q3
7 marks Easy -1.2
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-04_675_792_287_568}
\end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find
  1. the exact length of \(A B\),
  2. the coordinates of the midpoint \(P\) of \(A B\),
  3. an equation for the circle \(C\).
Edexcel C2 2006 January Q4
11 marks Moderate -0.3
The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
  1. Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\).
  2. Find, to 2 decimal places, the difference between the 5th and 6th term.
  3. Calculate the sum of the first 7 terms. The sum of the first \(n\) terms of the series is greater than 300 .
  4. Calculate the smallest possible value of \(n\).
Edexcel C2 2006 January Q5
8 marks Moderate -0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-07_538_611_301_680}
\end{figure} In Figure \(2 O A B\) is a sector of a circle radius 5 m . The chord \(A B\) is 6 m long.
  1. Show that \(\cos A \hat { O } B = \frac { 7 } { 25 }\).
  2. Hence find the angle \(A \hat { O } B\) in radians, giving your answer to 3 decimal places.
  3. Calculate the area of the sector \(O A B\).
  4. Hence calculate the shaded area.
Edexcel C2 2006 January Q6
6 marks Moderate -0.8
  1. The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a train at time \(t\) seconds is given by
$$v = \sqrt { } \left( 1.2 ^ { t } - 1 \right) , \quad 0 \leqslant t \leqslant 30$$ The following table shows the speed of the train at 5 second intervals.
\(t\)051015202530
\(v\)01.222.286.11
  1. Complete the table, giving the values of \(v\) to 2 decimal places. The distance, \(s\) metres, travelled by the train in 30 seconds is given by $$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
  2. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\).
    (3)
Edexcel C2 2006 January Q7
10 marks Moderate -0.8
7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Using the result from part (a), find the coordinates of the turning points of \(C\).
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  4. Hence, or otherwise, determine the nature of the turning points of \(C\).
Edexcel C2 2006 January Q8
9 marks Moderate -0.8
  1. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3$$
  2. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$\tan ^ { 2 } \theta = 4$$
Edexcel C2 2006 January Q9
10 marks Moderate -0.5
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-14_545_922_312_497}
\end{figure} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = - 2 x ^ { 2 } + 4 x\) and the line \(y = \frac { 3 } { 2 }\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find
  1. the \(x\)-coordinates of the points \(A\) and \(B\),
  2. the exact area of \(R\).
Edexcel C2 2007 January Q1
7 marks Easy -1.8
1. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$ Find
  1. \(\mathrm { f } ^ { \prime \prime } ( x )\),
  2. \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).
Edexcel C2 2007 January Q2
6 marks Easy -1.2
2.
  1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 - 2 x ) ^ { 5 }\). Give each term in its simplest form.
  2. If \(x\) is small, so that \(x ^ { 2 }\) and higher powers can be ignored, show that $$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x$$ DU
Edexcel C2 2007 January Q3
6 marks Moderate -0.8
3. The line joining the points \(( - 1,4 )\) and \(( 3,6 )\) is a diameter of the circle \(C\). Find an equation for \(C\).
Edexcel C2 2007 January Q4
3 marks Easy -1.2
4. Solve the equation $$5 ^ { x } = 17$$ giving your answer to 3 significant figures.
Edexcel C2 2007 January Q5
7 marks Moderate -0.8
5. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
  3. Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0$$
Edexcel C2 2007 January Q6
6 marks Standard +0.3
6. Find all the solutions, in the interval \(0 \leqslant x < 2 \pi\), of the equation $$2 \cos ^ { 2 } x + 1 = 5 \sin x$$ giving each solution in terms of \(\pi\).
Edexcel C2 2007 January Q7
9 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{872356ab-68d3-43ee-8b76-650a2697d80e-08_1052_1116_351_413}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 5 )$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between \(x = 0\) and \(x = 2\) and is bounded by \(C\), the \(x\)-axis and the line \(x = 2\).
(9)
Edexcel C2 2007 January Q8
9 marks Moderate -0.3
  1. A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, \(\pounds C\), is given by
$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$
  1. Find the value of \(v\) for which \(C\) is a minimum.
  2. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
  3. Calculate the minimum total cost of the journey.
Edexcel C2 2007 January Q9
11 marks Standard +0.3
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{872356ab-68d3-43ee-8b76-650a2697d80e-11_627_965_338_502}
\end{figure} Figure 2 shows a plan of a patio. The patio \(P Q R S\) is in the shape of a sector of a circle with centre \(Q\) and radius 6 m . Given that the length of the straight line \(P R\) is \(6 \sqrt { } 3 \mathrm {~m}\),
  1. find the exact size of angle \(P Q R\) in radians.
  2. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
  3. Find the exact area of the triangle \(P Q R\).
  4. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment \(P R S\).
  5. Find, in \(m\) to 1 decimal place, the perimeter of the patio \(P Q R S\).
Edexcel C2 2007 January Q10
11 marks Moderate -0.8
A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
  2. Find $$\sum _ { k = 1 } ^ { 10 } 100 \left( 2 ^ { k } \right)$$
  3. Find the sum to infinity of the geometric series $$\frac { 5 } { 6 } + \frac { 5 } { 18 } + \frac { 5 } { 54 } + \ldots$$
  4. State the condition for an infinite geometric series with common ratio \(r\) to be convergent.
Edexcel C2 2009 January Q1
4 marks Easy -1.2
Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 - 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)
Edexcel C2 2009 January Q3
6 marks Moderate -0.8
3. \(y = \sqrt { } \left( 10 x - x ^ { 2 } \right)\).
  1. Complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)11.41.82.22.63
    \(y\)33.474.39
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximation for the value of \(\int _ { 1 } ^ { 3 } \sqrt { } \left( 10 x - x ^ { 2 } \right) \mathrm { d } x\).
Edexcel C2 2009 January Q4
6 marks Standard +0.3
4. Given that \(0 < x < 4\) and $$\log _ { 5 } ( 4 - x ) - 2 \log _ { 5 } x = 1$$ find the value of \(x\).
(6)
Edexcel C2 2009 January Q5
8 marks Moderate -0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-06_828_956_244_457} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The points \(P ( - 3,2 ) , Q ( 9,10 )\) and \(R ( a , 4 )\) lie on the circle \(C\), as shown in Figure 2. Given that \(P R\) is a diameter of \(C\),
  1. show that \(a = 13\),
  2. find an equation for \(C\).
Edexcel C2 2009 January Q6
8 marks Moderate -0.3
6. $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. The remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).
  1. Find the value of \(a\). Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(b\).
Edexcel C2 2009 January Q7
8 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-09_878_991_233_461} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The shape \(B C D\) shown in Figure 3 is a design for a logo. The straight lines \(D B\) and \(D C\) are equal in length. The curve \(B C\) is an arc of a circle with centre \(A\) and radius 6 cm . The size of \(\angle B A C\) is 2.2 radians and \(A D = 4 \mathrm {~cm}\). Find
  1. the area of the sector \(B A C\), in \(\mathrm { cm } ^ { 2 }\),
  2. the size of \(\angle D A C\), in radians to 3 significant figures,
  3. the complete area of the logo design, to the nearest \(\mathrm { cm } ^ { 2 }\).