Questions C2 (1410 questions)

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Edexcel C2 2007 January Q8
  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
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
  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 2008 January Q1
  1. (a) Find the remainder when
$$x ^ { 3 } - 2 x ^ { 2 } - 4 x + 8$$ is divided by
  1. \(x - 3\),
  2. \(x + 2\).
    (b) Hence, or otherwise, find all the solutions to the equation $$x ^ { 3 } - 2 x ^ { 2 } - 4 x + 8 = 0$$
Edexcel C2 2008 January Q2
2. The fourth term of a geometric series is 10 and the seventh term of the series is 80 . For this series, find
  1. the common ratio,
  2. the first term,
  3. the sum of the first 20 terms, giving your answer to the nearest whole number.
Edexcel C2 2008 January Q3
3. (a) Find the first 4 terms of the expansion of \(\left( 1 + \frac { x } { 2 } \right) ^ { 10 }\) in ascending powers of \(x\), giving
each term in its simplest form. each term in its simplest form.
(b) Use your expansion to estimate the value of \(( 1.005 ) ^ { 10 }\), giving your answer to 5 decimal places.
Edexcel C2 2008 January Q4
4. (a) Show that the equation $$3 \sin ^ { 2 } \theta - 2 \cos ^ { 2 } \theta = 1$$ can be written as $$5 \sin ^ { 2 } \theta = 3$$ (b) Hence solve, for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\), the equation $$3 \sin ^ { 2 } \theta - 2 \cos ^ { 2 } \theta = 1$$ giving your answers to 1 decimal place.
Edexcel C2 2008 January Q5
  1. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations
$$\begin{gathered} a = 3 b ,
\log _ { 3 } a + \log _ { 3 } b = 2 . \end{gathered}$$ Give your answers as exact numbers. \section*{6.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-08_687_454_294_703}
\end{figure} Figure 1 shows 3 yachts \(A , B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is \(015 ^ { \circ }\).
  1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. The bearing of yacht \(C\) from yacht \(B\) is \(\theta ^ { \circ }\), as shown in Figure 1.
  2. Calculate the value of \(\theta\).
Edexcel C2 2008 January Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-10_691_995_267_477}
\end{figure} In Figure 2 the curve \(C\) has equation \(y = 6 x - x ^ { 2 }\) and the line \(L\) has equation \(y = 2 x\).
  1. Show that the curve \(C\) intersects the \(x\)-axis at \(x = 0\) and \(x = 6\).
  2. Show that the line \(L\) intersects the curve \(C\) at the points \(( 0,0 )\) and \(( 4,8 )\). The region \(R\), bounded by the curve \(C\) and the line \(L\), is shown shaded in Figure 2.
  3. Use calculus to find the area of \(R\).
Edexcel C2 2008 January Q8
  1. A circle \(C\) has centre \(M ( 6,4 )\) and radius 3 .
    1. Write down the equation of the circle in the form
    $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$ \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-12_833_1276_605_322}
    \end{figure} Figure 3 shows the circle \(C\). The point \(T\) lies on the circle and the tangent at \(T\) passes through the point \(P ( 12,6 )\). The line \(M P\) cuts the circle at \(Q\).
  2. Show that the angle \(T M Q\) is 1.0766 radians to 4 decimal places. The shaded region \(T P Q\) is bounded by the straight lines \(T P , Q P\) and the arc \(T Q\), as shown in Figure 3.
  3. Find the area of the shaded region \(T P Q\). Give your answer to 3 decimal places. \section*{9.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-14_675_844_283_534}
    \end{figure} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle \(x\) metres by \(y\) metres. The height of the tank is \(x\) metres. The capacity of the tank is \(100 \mathrm {~m} ^ { 3 }\).
  4. Show that the area \(A \mathrm {~m} ^ { 2 }\) of the sheet metal used to make the tank is given by $$A = \frac { 300 } { x } + 2 x ^ { 2 }$$
  5. Use calculus to find the value of \(x\) for which \(A\) is stationary.
  6. Prove that this value of \(x\) gives a minimum value of \(A\).
  7. Calculate the minimum area of sheet metal needed to make the tank.
Edexcel C2 2009 January Q1
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
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
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
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
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
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. (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
  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
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
  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
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
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
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
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
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}