Questions — Edexcel C2 (476 questions)

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Edexcel C2 2005 January Q2
The points \(A\) and \(B\) have coordinates \(( 5 , - 1 )\) and \(( 13,11 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\). Given that \(A B\) is a diameter of the circle \(C\),
  2. find an equation for \(C\).
Edexcel C2 2009 January Q2
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-03_870_1027_205_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = ( 1 + x ) ( 4 - x )\).
The curve intersects the \(x\)-axis at \(x = - 1\) and \(x = 4\). The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis. Use calculus to find the exact area of \(R\).
Edexcel C2 2005 June Q2
Solve
  1. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
  2. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).
Edexcel C2 2005 June Q3
  1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
  2. Factorise \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\) completely.
Edexcel C2 2005 June Q4
  1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + p x ) ^ { 12 }\), where \(p\) is a non-zero constant. Given that, in the expansion of \(( 1 + p x ) ^ { 12 }\), the coefficient of \(x\) is \(( - q )\) and the coefficient of \(x ^ { 2 }\) is \(11 q\),
  2. find the value of \(p\) and the value of \(q\).
Edexcel C2 2006 June Q2
Use calculus to find the exact value of \(\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
Edexcel C2 2006 June Q3
  1. Write down the value of \(\log _ { 6 } 36\).
  2. Express \(2 \log _ { a } 3 + \log _ { a } 11\) as a single logarithm to base \(a\).
Edexcel C2 2006 June Q4
$$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2007 June Q2
$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ). Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 Specimen Q2
The circle \(C\) has centre \(( 3,4 )\) and passes through the point \(( 8 , - 8 )\). Find an equation for C
Edexcel C2 Specimen Q3
The trapezium rule, with the table below, was used to estimate the area between the curve \(y = \sqrt { x ^ { 3 } + 1 }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis.
\(x\)11.522.53
\(y\)1.4142.0923.000
  1. Calculate, to 3 decimal places, the values of \(y\) for \(x = 2.5\) and \(x = 3\).
  2. Use the values from the table and your answers to part (a) to find an estimate, to 2 decimal places, for this area.
Edexcel C2 Q1
  1. Given that \(p = \log _ { q } 16\), express in terms of \(p\),
    1. \(\log _ { q } 2\),
    2. \(\log _ { q } ( 8 q )\).
    3. The expansion of \(( 2 - p x ) ^ { 6 }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\), is
    $$64 + A x + 135 x ^ { 2 }$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\).
    (7)
Edexcel C2 Q3
3. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y - 75 = 0$$
  1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). A second circle has centre at the point \(( 15,12 )\) and radius 10 .
  2. Sketch both circles on a single diagram and find the coordinates of the point where they touch.
    (4)
Edexcel C2 Q4
4. (a) Sketch, for \(0 \leq x \leq 360 ^ { \circ }\), the graph of \(y = \sin \left( x + 30 ^ { \circ } \right)\).
(b) Write down the coordinates of the points at which the graph meets the axes.
(c) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation $$\sin \left( x + 30 ^ { \circ } \right) = - \frac { 1 } { 2 }$$
Edexcel C2 Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c85316fe-5c59-4cb3-8cb8-d95a4e97af70-3_549_620_276_689}
\end{figure} The shape of a badge is a sector \(A B C\) of a circle with centre \(A\) and radius \(A B\), as shown in Fig 1. The triangle \(A B C\) is equilateral and has a perpendicular height 3 cm .
  1. Find, in surd form, the length \(A B\).
  2. Find, in terms of \(\pi\), the area of the badge.
  3. Prove that the perimeter of the badge is \(\frac { 2 \sqrt { 3 } } { 3 } ( \pi + 6 ) \mathrm { cm }\).
Edexcel C2 Q6
6. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + p x ^ { 2 } + q x + 8\), where \(p\) and \(q\) are constants. Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(2 x - 1\) ), and also that when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) the remainder is - 7 ,
  1. find the value of \(p\) and the value of \(q\).
  2. Hence factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 Q7
7. A geometric series has first term 1200. Its sum to infinity is 960 .
  1. Show that the common ratio of the series is \(- \frac { 1 } { 4 }\).
  2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series.
  3. Write down an expression for the sum of the first \(n\) terms of the series. Given that \(n\) is odd,
  4. prove that the sum of the first \(n\) terms of the series is $$960 \left( 1 + 0.25 ^ { n } \right)$$
Edexcel C2 Q8
  1. A circle \(C\) has centre \(( 3,4 )\) and radius \(3 \sqrt { } 2\). A straight line \(l\) has equation \(y = x + 3\).
    1. Write down an equation of the circle \(C\).
    2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds.
    3. Find the distance between these two points.
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c85316fe-5c59-4cb3-8cb8-d95a4e97af70-5_730_983_278_404}
    \end{figure} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac { x ^ { 2 } } { 25 } , x \geq 0 .$$ The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates 5 and 10 respectively.
  2. Write down the \(y\)-coordinates of \(A\) and \(B\).
  3. Find an equation of the tangent to \(C\) at \(A\). The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
  4. For points \(( x , y )\) on \(C\), express \(x\) in terms of \(y\).
  5. Use integration to find the area of \(R\). END
Edexcel C2 Q1
1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ),
  1. find the value of \(p\). Given also that \(q = 3\), and \(p\) has the value found in part (a),
  2. find the value of the remainder.
Edexcel C2 Q2
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-2_615_833_941_598}
\end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.
  1. Write down the value of \(a\) and the value of \(b\).
  2. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
  3. Find, to 3 significant figures, the length of \(P T\).
Edexcel C2 Q4
4
4
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Edexcel C2 Q5
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Edexcel C2 Q6
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Edexcel C2 Q7
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\hline \end{tabular} \end{center} 1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ),
  1. find the value of \(p\). Given also that \(q = 3\), and \(p\) has the value found in part (a),
  2. find the value of the remainder.
    2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-2_615_833_941_598}
    \end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.
  3. Write down the value of \(a\) and the value of \(b\).
  4. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
  5. Find, to 3 significant figures, the length of \(P T\).
    3. (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
  6. Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
    4. The first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + p x ) ^ { n }\), are \(1 - 18 x + 36 p ^ { 2 } x ^ { 2 }\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\).
    (7 marks)
    5. Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which
  7. \(\cos ( \theta + 75 ) ^ { \circ } = 0\).
  8. \(\sin 2 \theta ^ { \circ } = 0.7\), giving your answers to one decima1 place.
    6. Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
  9. \(\log _ { 2 } ( 16 x )\),
  10. \(\log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\).
  11. Hence, or otherwise, solve $$\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }$$ giving your answer in its simplest surd form.
    7. The curve \(C\) has equation \(y = \cos \left( x + \frac { \pi } { 4 } \right) , 0 \leq x \leq 2 \pi\).
  12. Sketch \(C\).
  13. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
  14. Solve, for \(x\) in the interval \(0 \leq x \leq 2 \pi\), $$\cos \left( x + \frac { \pi } { 4 } \right) = 0.5$$ giving your answers in terms of \(\pi\).
Edexcel C2 Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-4_769_1150_269_379}
\end{figure} Figure 2 shows part of the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x .$$ The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).
  1. Show that the equation of the curve may be written as $$y = x ( x - 3 ) ^ { 2 } ,$$ and hence write down the coordinates of \(A\).
  2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  3. Find the area of \(R\).