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Edexcel C12 Specimen Q9
6 marks Standard +0.3
9. Given that \(y = 3 x ^ { 2 }\),
  1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
  2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$
Edexcel C12 Specimen Q10
8 marks Moderate -0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-18_508_812_306_644} \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 C12 Specimen Q11
11 marks Moderate -0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-22_337_892_278_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 2.
  2. Use calculus to find the exact area of \(R\).
Edexcel C12 Specimen Q12
11 marks Standard +0.3
12. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\)
  1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
  2. Find an equation for \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
    Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
  3. find the length of \(P Q\), giving your answer in its simplest surd form.
Edexcel C12 Specimen Q13
11 marks Standard +0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-28_374_410_278_776} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),
  1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
  2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
  3. Use calculus to find the minimum value of \(P\).
Edexcel C12 Specimen Q14
10 marks Moderate -0.3
  1. In this question you must show all stages of your working. (Solutions based entirely on graphical or numerical methods are not acceptable.)
    1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place,
    $$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$
  2. Find, for \(0 \leqslant x < 2 \pi\), all the solutions of $$2 \sin ^ { 2 } x + 2 = 7 \cos x$$ giving your answers in radians. \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-35_108_95_2572_1804}
Edexcel C12 Specimen Q15
12 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-36_394_608_287_676} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The triangle \(X Y Z\) in Figure 4 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 4 is a major arc of the circle with centre \(X\) and radius 4 cm .
  1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
  2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 4. Calculate
  3. the area of this shaded region,
  4. the perimeter \(Z W Y Z\) of this shaded region. \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-39_90_54_2576_1868}
Edexcel C12 Specimen Q16
13 marks Moderate -0.8
16. Maria trains for a triathlon, which involves swimming, cycling and running. On the first day of training she swims 1.5 km and then she swims 1.5 km on each of the following days.
  1. Find the total distance that Maria swims in the first 17 days of training. Maria also runs 1.5 km on the first day of training and on each of the following days she runs 0.25 km further than on the previous day. So she runs 1.75 km on the second day and 2 km on the third day and so on.
  2. Find how far Maria runs on the 17th day of training. Maria also cycles 1.5 km on the first day of training and on each of the following days she cycles \(5 \%\) further than on the previous day.
  3. Find the total distance that Maria cycles in the first 17 days of training.
  4. Find the total distance Maria travels by swimming, running and cycling in the first 17 days of training. Maria needs to cycle 40 km in the triathlon.
  5. On which day of training does Maria first cycle more than 40 km ?
Edexcel C1 2005 January Q1
3 marks Easy -1.8
  1. Write down the value of \(16 ^ { \frac { 1 } { 2 } }\).
  2. Find the value of \(16 ^ { - \frac { 3 } { 2 } }\).
Edexcel C1 2005 January Q3
4 marks Moderate -0.8
3. Given that the equation \(k x ^ { 2 } + 12 x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\).
Edexcel C1 2005 January Q4
6 marks Moderate -0.8
4. Solve the simultaneous equations $$\begin{gathered} x + y = 2 \\ x ^ { 2 } + 2 y = 12 \end{gathered}$$
Edexcel C1 2005 January Q5
6 marks Moderate -0.8
5. The \(r\) th term of an arithmetic series is ( \(2 r - 5\) ).
  1. Write down the first three terms of this series.
  2. State the value of the common difference.
  3. Show that \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).
Edexcel C1 2005 January Q6
6 marks Moderate -0.8
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-10_515_714_292_609}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 2,0 )\) and \(( 4,0 )\). The minimum point on the curve is \(P ( 3 , - 2 )\). In separate diagrams sketch the curve with equation
  1. \(y = - \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\). On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.
Edexcel C1 2005 January Q7
10 marks Moderate -0.8
7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .
  1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
  2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
  3. Find the value of \(k\).
Edexcel C1 2005 January Q8
9 marks Moderate -0.8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-14_687_1196_280_388}
\end{figure} The points \(A ( 1,7 ) , B ( 20,7 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 2. The point \(D ( 8,2 )\) is the mid-point of \(A C\).
  1. Find the value of \(p\) and the value of \(q\). The line \(l\), which passes through \(D\) and is perpendicular to \(A C\), intersects \(A B\) at \(E\).
  2. Find an equation for \(l\), in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
  3. Find the exact \(x\)-coordinate of \(E\).
Edexcel C1 2005 January Q9
11 marks Moderate -0.3
9. The gradient of the curve \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 } .$$ The point \(P ( 1,4 )\) lies on \(C\).
  1. Find an equation of the normal to \(C\) at \(P\).
  2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
  3. Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).
Edexcel C1 2005 January Q10
12 marks Easy -1.2
10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geqslant 0 ,$$
  1. express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \geqslant 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
  2. In the space provided on page 19, sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). The line \(y = 41\) meets \(C\) at the point \(R\).
  3. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q \sqrt { } 2\), where \(p\) and \(q\) are integers.
Edexcel C1 2006 January Q1
3 marks Easy -1.2
  1. Factorise completely
$$x ^ { 3 } - 4 x ^ { 2 } + 3 x .$$
Edexcel C1 2006 January Q2
4 marks Moderate -0.8
2. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1 .$$
  1. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Write down the value of \(u _ { 20 }\).
Edexcel C1 2006 January Q3
5 marks Moderate -0.8
3. The line \(L\) has equation \(y = 5 - 2 x\).
  1. Show that the point \(P ( 3 , - 1 )\) lies on \(L\).
  2. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2006 January Q4
5 marks Easy -1.2
4. Given that \(y = 2 x ^ { 2 } - \frac { 6 } { x ^ { 3 } } , x \neq 0\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\int y \mathrm {~d} x\).
Edexcel C1 2006 January Q5
6 marks Easy -1.8
5. (a) Write \(\sqrt { 45 }\) in the form \(a \sqrt { 5 }\), where \(a\) is an integer.
(b) Express \(\frac { 2 ( 3 + \sqrt { 5 } ) } { ( 3 - \sqrt { 5 } ) }\) in the form \(b + c \sqrt { 5 }\), where \(b\) and \(c\) are integers.
\section*{6.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{815e288c-0140-4c12-9e89-b0bb4fb1a8c1-07_607_844_310_555}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the points \(( 0,3 )\) and \(( 4,0 )\) and touches the \(x\)-axis at the point \(( 1,0 )\). On separate diagrams sketch the curve with equation
Edexcel C1 2006 January Q7
13 marks Easy -1.3
  1. On Alice's 11th birthday she started to receive an annual allowance. The first annual allowance was \(\pounds 500\) and on each following birthday the allowance was increased by \(\pounds 200\).
    1. Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was \(\pounds 1200\).
    2. Find the amount of Alice's annual allowance on her 18th birthday.
    3. Find the total of the allowances that Alice had received up to and including her 18th birthday.
    When the total of the allowances that Alice had received reached \(\pounds 32000\) the allowance stopped.
  2. Find how old Alice was when she received her last allowance.
Edexcel C1 2006 January Q8
7 marks Moderate -0.5
  1. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 1,6 )\). Given that
$$f ^ { \prime } ( x ) = 3 + \frac { 5 x ^ { 2 } + 2 } { x ^ { \frac { 1 } { 2 } } } , x > 0$$ find \(\mathrm { f } ( x )\) and simplify your answer.
Edexcel C1 2006 January Q9
12 marks Easy -1.2
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{815e288c-0140-4c12-9e89-b0bb4fb1a8c1-12_812_1088_317_427}
\end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the \(x\)-axis at the points \(P , ( 1,0 )\) and \(Q\), as shown in Figure 2.
  1. Write down the \(x\)-coordinate of \(P\), and the \(x\)-coordinate of \(Q\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4\).
  3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point ( \(- 1,6\) ). The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point ( \(- 1,6\) ).
  4. Find the exact coordinates of \(R\).