Questions — Edexcel C2 (579 questions)

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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
  1. 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 \(\mathrm { cm } ^ { 2 }\).
Edexcel C2 2009 January Q8
8 marks Moderate -0.3
8. (a) Show that the equation $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ can be written as $$4 \cos ^ { 2 } x - 9 \cos x + 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 720 ^ { \circ }\), $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ giving your answers to 1 decimal place.
Edexcel C2 2009 January Q9
10 marks Moderate -0.3
  1. The first three terms of a geometric series are ( \(k + 4\) ), \(k\) and ( \(2 k - 15\) ) respectively, where \(k\) is a positive constant.
    1. Show that \(k ^ { 2 } - 7 k - 60 = 0\).
    2. Hence show that \(k = 12\).
    3. Find the common ratio of this series.
    4. Find the sum to infinity of this series.
Edexcel C2 2009 January Q10
12 marks Standard +0.3
10. A solid right circular cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The total surface area of the cylinder is \(800 \mathrm {~cm} ^ { 2 }\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 400 r - \pi r ^ { 3 }$$ Given that \(r\) varies,
  2. use calculus to find the maximum value of \(V\), to the nearest \(\mathrm { cm } ^ { 3 }\).
  3. Justify that the value of \(V\) you have found is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-16_103_63_2477_1873}
Edexcel C2 2010 January Q1
4 marks Easy -1.2
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 3 - x ) ^ { 6 }$$ and simplify each term.
Edexcel C2 2010 January Q2
6 marks Moderate -0.3
2. (a) Show that the equation $$5 \sin x = 1 + 2 \cos ^ { 2 } x$$ can be written in the form $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$ (b) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$
Edexcel C2 2010 January Q3
9 marks Moderate -0.3
3. $$f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x - 6$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is - 5 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is no remainder.
  1. Find the value of \(a\) and the value of \(b\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2010 January Q4
7 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-05_556_1189_237_413} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} An emblem, as shown in Figure 1, consists of a triangle \(A B C\) joined to a sector \(C B D\) of a circle with radius 4 cm and centre \(B\). The points \(A , B\) and \(D\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B D = 4 \mathrm {~cm}\). Angle \(B A C = 0.6\) radians and \(A C\) is the longest side of the triangle \(A B C\).
  1. Show that angle \(A B C = 1.76\) radians, correct to 3 significant figures.
  2. Find the area of the emblem.
Edexcel C2 2010 January Q5
8 marks Moderate -0.3
5. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$
Edexcel C2 2010 January Q6
9 marks Moderate -0.8
6. A car was purchased for \(\pounds 18000\) on 1 st January. On 1st January each following year, the value of the car is \(80 \%\) of its value on 1st January in the previous year.
  1. Show that the value of the car exactly 3 years after it was purchased is \(\pounds 9216\). The value of the car falls below \(\pounds 1000\) for the first time \(n\) years after it was purchased.
  2. Find the value of \(n\). An insurance company has a scheme to cover the maintenance of the car. The cost is \(\pounds 200\) for the first year, and for every following year the cost increases by \(12 \%\) so that for the 3rd year the cost of the scheme is \(\pounds 250.88\)
  3. Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
  4. Find the total cost of the insurance scheme for the first 15 years.
    \section*{LU}
Edexcel C2 2010 January Q7
10 marks Moderate -0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-09_696_821_205_516} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve \(C\) has equation \(y = x ^ { 2 } - 5 x + 4\). It cuts the \(x\)-axis at the points \(L\) and \(M\) as shown in Figure 2.
  1. Find the coordinates of the point \(L\) and the point \(M\).
  2. Show that the point \(N ( 5,4 )\) lies on \(C\).
  3. Find \(\int \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\). The finite region \(R\) is bounded by \(L N , L M\) and the curve \(C\) as shown in Figure 2.
  4. Use your answer to part (c) to find the exact value of the area of \(R\).
    \section*{LU}
Edexcel C2 2010 January Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-11_1262_1178_203_386} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the circle \(C\) with centre \(N\) and equation $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = \frac { 169 } { 4 }$$
  1. Write down the coordinates of \(N\).
  2. Find the radius of \(C\). The chord \(A B\) of \(C\) is parallel to the \(x\)-axis, lies below the \(x\)-axis and is of length 12 units as shown in Figure 3.
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
  4. Show that angle \(A N B = 134.8 ^ { \circ }\), to the nearest 0.1 of a degree. The tangents to \(C\) at the points \(A\) and \(B\) meet at the point \(P\).
  5. Find the length \(A P\), giving your answer to 3 significant figures.
Edexcel C2 2010 January Q9
10 marks Moderate -0.8
9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)
  1. Use calculus to find the coordinates of the turning point on \(C\).
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. State the nature of the turning point.
Edexcel C2 2011 January Q1
7 marks Moderate -0.3
1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7 .
  1. Show that \(a + b = 3\). When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is - 8 .
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C2 2011 January Q2
6 marks Moderate -0.8
2. In the triangle \(A B C , A B = 11 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(C A = 8 \mathrm {~cm}\).
  1. Find the size of angle \(C\), giving your answer in radians to 3 significant figures.
  2. Find the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to 3 significant figures.
Edexcel C2 2011 January Q3
7 marks Moderate -0.3
3. The second and fifth terms of a geometric series are 750 and - 6 respectively. Find
  1. the common ratio of the series,
  2. the first term of the series,
  3. the sum to infinity of the series.
Edexcel C2 2011 January Q4
7 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be8f9187-055a-476f-974d-22e8e16e9996-05_547_798_251_575} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = ( x + 1 ) ( x - 5 )$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\).
  1. Write down the \(x\)-coordinates of \(A\) and \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
  2. Use integration to find the area of \(R\).