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Edexcel C4 2010 January Q3
9 marks Standard +0.3
  1. The curve \(C\) has the equation
$$\cos 2 x + \cos 3 y = 1 , \quad - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 } , \quad 0 \leqslant y \leqslant \frac { \pi } { 6 }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) lies on \(C\) where \(x = \frac { \pi } { 6 }\).
  2. Find the value of \(y\) at \(P\).
  3. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c \pi = 0\), where \(a , b\) and \(c\) are integers. \section*{LU}
Edexcel C4 2010 January Q4
12 marks Standard +0.3
4. The line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } - 6 \\ 4 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ - 1 \\ 3 \end{array} \right)$$ and the line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } - 6 \\ 4 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\) and the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\).
  1. Write down the coordinates of \(A\).
  2. Find the value of \(\cos \theta\). The point \(X\) lies on \(l _ { 1 }\) where \(\lambda = 4\).
  3. Find the coordinates of \(X\).
  4. Find the vector \(\overrightarrow { A X }\).
  5. Hence, or otherwise, show that \(| \overrightarrow { A X } | = 4 \sqrt { } 26\). The point \(Y\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { Y X }\) is perpendicular to \(l _ { 1 }\),
  6. find the length of \(A Y\), giving your answer to 3 significant figures. \section*{LU}
Edexcel C4 2010 January Q5
8 marks Moderate -0.3
5. (a) Find \(\int \frac { 9 x + 6 } { x } \mathrm {~d} x , x > 0\).
(b) Given that \(y = 8\) at \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 9 x + 6 ) y ^ { \frac { 1 } { 3 } } } { x }$$ giving your answer in the form \(y ^ { 2 } = \mathrm { g } ( x )\). \section*{LU}
Edexcel C4 2010 January Q6
5 marks Standard +0.3
6. The area \(A\) of a circle is increasing at a constant rate of \(1.5 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find, to 3 significant figures, the rate at which the radius \(r\) of the circle is increasing when the area of the circle is \(2 \mathrm {~cm} ^ { 2 }\).
(5)
Edexcel C4 2010 January Q7
9 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef3ae4a-a06d-48c1-8b79-7d7c3f95d120-12_734_1395_210_249} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 5 t ^ { 2 } - 4 , \quad y = t \left( 9 - t ^ { 2 } \right)$$ The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\).
  1. Find the \(x\)-coordinate at the point \(A\) and the \(x\)-coordinate at the point \(B\). The region \(R\), as shown shaded in Figure 2, is enclosed by the loop of the curve.
  2. Use integration to find the area of \(R\).
    \section*{LU}
Edexcel C4 2010 January Q8
10 marks Challenging +1.2
8. (a) Using the substitution \(x = 2 \cos u\), or otherwise, find the exact value of $$\int _ { 1 } ^ { \sqrt { 2 } } \frac { 1 } { x ^ { 2 } \sqrt { } \left( 4 - x ^ { 2 } \right) } d x$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef3ae4a-a06d-48c1-8b79-7d7c3f95d120-14_680_1264_502_338} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \frac { 4 } { x \left( 4 - x ^ { 2 } \right) ^ { \frac { 1 } { 4 } } } , \quad 0 < x < 2\). The shaded region \(S\), shown in Figure 3, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 1\) and \(x = \sqrt { } 2\). The shaded region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(b) Using your answer to part (a), find the exact volume of the solid of revolution formed.
Edexcel C4 2011 January Q1
6 marks Moderate -0.3
  1. Use integration to find the exact value of
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin 2 x \mathrm {~d} x$$
Edexcel C4 2011 January Q2
5 marks Moderate -0.3
2. The current, \(I\) amps, in an electric circuit at time \(t\) seconds is given by $$I = 16 - 16 ( 0.5 ) ^ { t } , \quad t \geqslant 0$$ Use differentiation to find the value of \(\frac { \mathrm { d } I } { \mathrm {~d} t }\) when \(t = 3\).
Give your answer in the form \(\ln a\), where \(a\) is a constant.
Edexcel C4 2011 January Q3
12 marks Standard +0.3
3. (a) Express \(\frac { 5 } { ( x - 1 ) ( 3 x + 2 ) }\) in partial fractions.
(b) Hence find \(\int \frac { 5 } { ( x - 1 ) ( 3 x + 2 ) } \mathrm { d } x\), where \(x > 1\).
(c) Find the particular solution of the differential equation $$( x - 1 ) ( 3 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 5 y , \quad x > 1$$ for which \(y = 8\) at \(x = 2\). Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C4 2011 January Q4
10 marks Moderate -0.3
  1. Relative to a fixed origin \(O\), the point \(A\) has position vector \(\mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\) and the point \(B\) has position vector \(- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The points \(A\) and \(B\) lie on a straight line \(l\).
    1. Find \(\overrightarrow { A B }\).
    2. Find a vector equation of \(l\).
    The point \(C\) has position vector \(2 \mathbf { i } + p \mathbf { j } - 4 \mathbf { k }\) with respect to \(O\), where \(p\) is a constant. Given that \(A C\) is perpendicular to \(l\), find
  2. the value of \(p\),
  3. the distance \(A C\).
Edexcel C4 2011 January Q5
13 marks Standard +0.3
  1. (a) Use the binomial theorem to expand
$$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction. $$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$ In the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), the coefficient of \(x\) is 0 and the coefficient of \(x ^ { 2 }\) is \(\frac { 9 } { 16 }\). Find
(b) the value of \(a\) and the value of \(b\),
(c) the coefficient of \(x ^ { 3 }\), giving your answer as a simplified fraction.
Edexcel C4 2011 January Q6
15 marks Standard +0.8
  1. The curve \(C\) has parametric equations
$$x = \ln t , \quad y = t ^ { 2 } - 2 , \quad t > 0$$ Find
  1. an equation of the normal to \(C\) at the point where \(t = 3\),
  2. a cartesian equation of \(C\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a3ece8a8-8107-4c3a-a6a9-c19b5e35ec5a-10_579_759_740_571} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The finite area \(R\), shown in Figure 1, is bounded by \(C\), the \(x\)-axis, the line \(x = \ln 2\) and the line \(x = \ln 4\). The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  3. Use calculus to find the exact volume of the solid generated.
Edexcel C4 2011 January Q7
14 marks Standard +0.3
7. $$I = \int _ { 2 } ^ { 5 } \frac { 1 } { 4 + \sqrt { } ( x - 1 ) } \mathrm { d } x$$
  1. Given that \(y = \frac { 1 } { 4 + \sqrt { } ( x - 1 ) }\), complete the table below with values of \(y\) corresponding to \(x = 3\) and \(x = 5\). Give your values to 4 decimal places.
    \(x\)2345
    \(y\)0.20.1745
  2. Use the trapezium rule, with all of the values of \(y\) in the completed table, to obtain an estimate of \(I\), giving your answer to 3 decimal places.
  3. Using the substitution \(x = ( u - 4 ) ^ { 2 } + 1\), or otherwise, and integrating, find the exact value of \(I\).
Edexcel C4 2012 January Q1
8 marks Moderate -0.3
  1. The curve \(C\) has the equation \(2 x + 3 y ^ { 2 } + 3 x ^ { 2 } y = 4 x ^ { 2 }\).
The point \(P\) on the curve has coordinates \(( - 1,1 )\).
  1. Find the gradient of the curve at \(P\).
  2. Hence find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C4 2012 January Q2
6 marks Standard +0.8
2. (a) Use integration by parts to find \(\int x \sin 3 x \mathrm {~d} x\).
(b) Using your answer to part (a), find \(\int x ^ { 2 } \cos 3 x \mathrm {~d} x\).
Edexcel C4 2012 January Q3
9 marks Standard +0.3
3. (a) Expand $$\frac { 1 } { ( 2 - 5 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction. Given that the binomial expansion of \(\frac { 2 + k x } { ( 2 - 5 x ) ^ { 2 } } , | x | < \frac { 2 } { 5 }\), is $$\frac { 1 } { 2 } + \frac { 7 } { 4 } x + A x ^ { 2 } + \ldots$$ (b) find the value of the constant \(k\),
(c) find the value of the constant \(A\).
Edexcel C4 2012 January Q4
5 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c963567-d751-4898-b7a7-7095d90514f0-06_606_1185_237_383} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation $$y = \sqrt { } \left( \frac { 2 x } { 3 x ^ { 2 } + 4 } \right) , x \geqslant 0$$ The finite region \(S\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 2\) The region \(S\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
Use integration to find the exact value of the volume of the solid generated, giving your answer in the form \(k \ln a\), where \(k\) and \(a\) are constants.
Edexcel C4 2012 January Q5
8 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c963567-d751-4898-b7a7-7095d90514f0-07_687_1209_214_370} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \sin \left( t + \frac { \pi } { 6 } \right) , \quad y = 3 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the coordinates of all the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
Edexcel C4 2012 January Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c963567-d751-4898-b7a7-7095d90514f0-09_639_1179_246_386} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation \(y = \frac { 2 \sin 2 x } { ( 1 + \cos x ) } , 0 \leqslant x \leqslant \frac { \pi } { 2 }\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 2 \sin 2 x } { ( 1 + \cos x ) }\).
\(x\)0\(\frac { \pi } { 8 }\)\(\frac { \pi } { 4 }\)\(\frac { 3 \pi } { 8 }\)\(\frac { \pi } { 2 }\)
\(y\)01.171571.022800
  1. Complete the table above giving the missing value of \(y\) to 5 decimal places.
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 4 decimal places.
  3. Using the substitution \(u = 1 + \cos x\), or otherwise, show that $$\int \frac { 2 \sin 2 x } { ( 1 + \cos x ) } d x = 4 \ln ( 1 + \cos x ) - 4 \cos x + k$$ where \(k\) is a constant.
  4. Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.
Edexcel C4 2012 January Q7
15 marks Standard +0.3
7. Relative to a fixed origin \(O\), the point \(A\) has position vector ( \(2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }\) ), the point \(B\) has position vector \(( 5 \mathbf { i } + 2 \mathbf { j } + 10 \mathbf { k } )\), and the point \(D\) has position vector \(( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\). The line \(l\) passes through the points \(A\) and \(B\).
  1. Find the vector \(\overrightarrow { A B }\).
  2. Find a vector equation for the line \(l\).
  3. Show that the size of the angle \(B A D\) is \(109 ^ { \circ }\), to the nearest degree. The points \(A , B\) and \(D\), together with a point \(C\), are the vertices of the parallelogram \(A B C D\), where \(\overrightarrow { A B } = \overrightarrow { D C }\).
  4. Find the position vector of \(C\).
  5. Find the area of the parallelogram \(A B C D\), giving your answer to 3 significant figures.
  6. Find the shortest distance from the point \(D\) to the line \(l\), giving your answer to 3 significant figures.
Edexcel C4 2012 January Q8
12 marks Standard +0.3
  1. (a) Express \(\frac { 1 } { P ( 5 - P ) }\) in partial fractions.
A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 15 } P ( 5 - P ) , \quad t \geqslant 0$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that when \(t = 0 , P = 1\),
(b) solve the differential equation, giving your answer in the form, $$P = \frac { a } { b + c \mathrm { e } ^ { - \frac { 1 } { 3 } t } }$$ where \(a\), \(b\) and \(c\) are integers.
(c) Hence show that the population cannot exceed 5000
Edexcel C4 2013 January Q1
5 marks Moderate -0.8
  1. Given
$$f ( x ) = ( 2 + 3 x ) ^ { - 3 } , \quad | x | < \frac { 2 } { 3 }$$ find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
Edexcel C4 2013 January Q2
7 marks Moderate -0.3
2. (a) Use integration to find $$\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$ (b) Hence calculate $$\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$
Edexcel C4 2013 January Q3
4 marks Standard +0.3
3. Express \(\frac { 9 x ^ { 2 } + 20 x - 10 } { ( x + 2 ) ( 3 x - 1 ) }\) in partial fractions.
Edexcel C4 2013 January Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-05_650_1143_223_427} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { x } { 1 + \sqrt { } x }\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the line with equation \(x = 1\) and the line with equation \(x = 4\).
  1. Complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 decimal places.
    (1)
    \(x\)1234
    \(y\)0.50.82841.3333
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate of the area of the region \(R\), giving your answer to 3 decimal places.
  3. Use the substitution \(u = 1 + \sqrt { } x\), to find, by integrating, the exact area of \(R\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-07_743_1568_219_182} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = 1 - \frac { 1 } { 2 } t , \quad y = 2 ^ { t } - 1$$ The curve crosses the \(y\)-axis at the point \(A\) and crosses the \(x\)-axis at the point \(B\).