Questions C2 (1410 questions)

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Edexcel C2 2006 June Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-08_611_682_296_641}
\end{figure} The line \(y = 3 x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(P ( 2,2 )\), as shown in Figure 1. The point \(Q\) is the centre of \(C\).
  1. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
  2. show that the \(x\)-coordinate of \(Q\) is 5,
  3. find an equation for \(C\).
Edexcel C2 2006 June Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-10_620_636_301_660}
\end{figure} Figure 2 shows the cross section \(A B C D\) of a small shed. The straight line \(A B\) is vertical and has length 2.12 m . The straight line \(A D\) is horizontal and has length 1.86 m . The curve \(B C\) is an arc of a circle with centre \(A\), and \(C D\) is a straight line. Given that the size of \(\angle B A C\) is 0.65 radians, find
  1. the length of the arc \(B C\), in m , to 2 decimal places,
  2. the area of the sector \(B A C\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places,
  3. the size of \(\angle C A D\), in radians, to 2 decimal places,
  4. the area of the cross section \(A B C D\) of the shed, in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.
Edexcel C2 2006 June Q9
  1. A geometric series has first term \(a\) and common ratio \(r\). The second term of the series is 4 and the sum to infinity of the series is 25.
    1. Show that \(25 r ^ { 2 } - 25 r + 4 = 0\).
    2. Find the two possible values of \(r\).
    3. Find the corresponding two possible values of \(a\).
    4. Show that the sum, \(S _ { n }\), of the first \(n\) terms of the series is given by
    $$S _ { n } = 25 \left( 1 - r ^ { n } \right) .$$ Given that \(r\) takes the larger of its two possible values,
  2. find the smallest value of \(n\) for which \(S _ { n }\) exceeds 24 .
Edexcel C2 2006 June Q10
10. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-14_636_956_285_513}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).
  1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
    The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
  3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
  4. Hence calculate the exact area of \(R\).
Edexcel C2 2007 June Q1
Evaluate \(\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt { } x } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
Edexcel C2 2007 June Q3
3. (a) Find the first four terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 6 }\), where \(k\) is a non-zero constant. Given that, in this expansion, the coefficients of \(x\) and \(x ^ { 2 }\) are equal, find
(b) the value of \(k\),
(c) the coefficient of \(x ^ { 3 }\).
Edexcel C2 2007 June Q4
4. Figure 1 Figure 1 shows the triangle \(A B C\), with \(A B = 6 \mathrm {~cm} , B C = 4 \mathrm {~cm}\) and \(C A = 5 \mathrm {~cm}\).
  1. Show that \(\cos A = \frac { 3 } { 4 }\).
  2. Hence, or otherwise, find the exact value of \(\sin A\).
Edexcel C2 2007 June Q5
5. The curve \(C\) has equation $$y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , \quad 0 \leqslant x \leqslant 2$$
  1. Complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).
    \(x\)00.511.52
    \(y\)00.5306
  2. Use the trapezium rule, with all the \(y\) values from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } x \sqrt { } \left( x ^ { 3 } + 1 \right) \mathrm { d } x\), giving your answer to 3 significant figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-06_1110_644_1119_648} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the curve \(C\) with equation \(y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , 0 \leqslant x \leqslant 2\), and the straight line segment \(l\), which joins the origin and the point \(( 2,6 )\). The finite region \(R\) is bounded by \(C\) and \(l\).
  3. Use your answer to part (b) to find an approximation for the area of \(R\), giving your answer to 3 significant figures.
    (3) \section*{LU}
Edexcel C2 2007 June Q6
6. (a) Find, to 3 significant figures, the value of \(x\) for which \(8 ^ { x } = 0.8\).
(b) Solve the equation $$2 \log _ { 3 } x - \log _ { 3 } 7 x = 1$$
Edexcel C2 2007 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-09_778_988_223_500} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3.
The point \(A\) has coordinates \(( 1 , - 2 )\) and the mid-point \(M\) of \(A B\) has coordinates \(( 3,1 )\). The line \(l\) passes through the points \(M\) and \(P\).
  1. Find an equation for \(l\). Given that the \(x\)-coordinate of \(P\) is 6 ,
  2. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is - 1 ,
  3. find an equation for the circle.
Edexcel C2 2007 June Q8
8. A trading company made a profit of \(\pounds 50000\) in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r , r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of \(\pounds 50000 r\) will be made.
  1. Write down an expression for the predicted profit in Year \(n\). The model predicts that in Year \(n\), the profit made will exceed \(\pounds 200000\).
  2. Show that \(n > \frac { \log 4 } { \log r } + 1\). Using the model with \(r = 1.09\),
  3. find the year in which the profit made will first exceed \(\pounds 200000\),
  4. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest \(\pounds 10000\).
Edexcel C2 2007 June Q9
9. (a) Sketch, for \(0 \leqslant x \leqslant 2 \pi\), the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.
(c) Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 6 } \right) = 0.65$$ giving your answers in radians to 2 decimal places.
Edexcel C2 2007 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-15_538_529_205_744} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a solid brick in the shape of a cuboid measuring \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The total surface area of the brick is \(600 \mathrm {~cm} ^ { 2 }\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the brick is given by $$V = 200 x - \frac { 4 x ^ { 3 } } { 3 }$$ Given that \(x\) can vary,
  2. use calculus to find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
  3. Justify that the value of \(V\) you have found is a maximum.
Edexcel C2 2008 June Q1
1. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 39 x + 20$$
  1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise f ( \(x\) ) completely.
Edexcel C2 2008 June Q2
2. $$y = \sqrt { } \left( 5 ^ { x } + 2 \right)$$
  1. Complete the table below, giving the values of \(y\) to 3 decimal places.
    \(x\)00.511.52
    \(y\)2.6463.630
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } \sqrt { } \left( 5 ^ { x } + 2 \right) \mathrm { d } x\).
Edexcel C2 2008 June Q3
3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 10 }\), where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 3 }\) is double the coefficient of \(x ^ { 2 }\),
(b) find the value of \(a\).
Edexcel C2 2008 June Q4
4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
(b) Solve the equation \(5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0\).
Edexcel C2 2008 June Q5
5. The circle \(C\) has centre \(( 3,1 )\) and passes through the point \(P ( 8,3 )\).
  1. Find an equation for \(C\).
  2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2008 June Q6
6. A geometric series has first term 5 and common ratio \(\frac { 4 } { 5 }\). Calculate
  1. the 20th term of the series, to 3 decimal places,
  2. the sum to infinity of the series. Given that the sum to \(k\) terms of the series is greater than 24.95,
  3. show that \(k > \frac { \log 0.002 } { \log 0.8 }\),
  4. find the smallest possible value of \(k\).
Edexcel C2 2008 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0454f5f6-b5ee-40b1-bc6a-ff8aeb06a455-09_817_1029_205_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows \(A B C\), a sector of a circle with centre \(A\) and radius 7 cm .
Given that the size of \(\angle B A C\) is exactly 0.8 radians, find
  1. the length of the arc \(B C\),
  2. the area of the sector \(A B C\). The point \(D\) is the mid-point of \(A C\). The region \(R\), shown shaded in Figure 1, is bounded by \(C D , D B\) and the arc \(B C\). Find
  3. the perimeter of \(R\), giving your answer to 3 significant figures,
  4. the area of \(R\), giving your answer to 3 significant figures.
Edexcel C2 2008 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0454f5f6-b5ee-40b1-bc6a-ff8aeb06a455-11_668_1267_292_367} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = 10 + 8 x + x ^ { 2 } - x ^ { 3 }\).
The curve has a maximum turning point \(A\).
  1. Using calculus, show that the \(x\)-coordinate of \(A\) is 2 . The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(y\)-axis and the line from \(O\) to \(A\), where \(O\) is the origin.
  2. Using calculus, find the exact area of \(R\).
Edexcel C2 2008 June Q9
9. Solve, for \(0 \leqslant x < 360 ^ { \circ }\),
  1. \(\quad \sin \left( x - 20 ^ { \circ } \right) = \frac { 1 } { \sqrt { 2 } }\)
  2. \(\cos 3 x = - \frac { 1 } { 2 }\)
Edexcel C2 2009 June Q1
  1. Use calculus to find the value of
$$\int _ { 1 } ^ { 4 } ( 2 x + 3 \sqrt { } x ) d x$$
Edexcel C2 2009 June Q2
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + k x ) ^ { 7 }$$ where \(k\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) is 6 times the coefficient of \(x\),
(b) find the value of \(k\).
Edexcel C2 2009 June Q3
3. $$f ( x ) = ( 3 x - 2 ) ( x - k ) - 8$$ where \(k\) is a constant.
  1. Write down the value of \(\mathrm { f } ( k )\). When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 4
  2. Find the value of \(k\).
  3. Factorise f(x) completely.