Questions — Edexcel AEA (167 questions)

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Edexcel AEA 2006 June Q4
14 marks Challenging +1.2
4.The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$
  1. Show that \(m\) satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin \(O\) to \(C _ { 1 }\) touch \(C _ { 1 }\) at the points \(A\) and \(B\) .
  2. Find the coordinates of the points \(A\) and \(B\) .
    (8)
    Another circle \(C _ { 2 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 13\) .The tangents from the point \(( 4 , - 7 )\) to \(C _ { 2 }\) touch it at the points \(P\) and \(Q\) .
  3. Find the coordinates of either the point \(P\) or the point \(Q\) .
    (2)
Edexcel AEA 2006 June Q5
15 marks Challenging +1.8
5.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have vector equations \(L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )\), \(L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )\),
where \(\lambda\) and \(\mu\) are parameters.
  1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
  2. Show that the vector \(( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) . The point \(A\) lies on \(L _ { 1 }\) ,the point \(B\) lies on \(L _ { 2 }\) and \(A B\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) .
  3. Find the position vector of the point \(A\) and the position vector of the point \(B\) .
    (8) \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \sin ( \ln x ) , \quad x \geq 1 .$$ The point \(Q\) ,on \(C\) ,is a maximum.
Edexcel AEA 2006 June Q7
20 marks Hard +2.3
7. \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275} The circle \(C _ { 1 }\) has centre \(O\) and radius \(R\). The tangents \(A P\) and \(B P\) to \(C _ { 1 }\) meet at the point \(P\) and angle \(A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }\). A sequence of circles \(C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots\) is drawn so that each new circle \(C _ { n + 1 }\) touches each of \(C _ { n } , A P\) and \(B P\) for \(n = 1,2,3 , \ldots\) as shown in Figure 2. The centre of each circle lies on the line \(O P\).
  1. Show that the radii of the circles form a geometric sequence with common ratio $$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
  2. Find, in terms of \(R\) and \(\alpha\), the total area enclosed by all the circles, simplifying your answer. The area inside the quadrilateral \(P A O B\), not enclosed by part of \(C _ { 1 }\) or any of the other circles, is \(S\).
  3. Show that $$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
  4. Show that, as \(\alpha\) varies, $$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
  5. Find, in terms of \(R\), the least value of \(S\) for \(\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }\).
Edexcel AEA 2007 June Q1
9 marks Challenging +1.2
1.(a)Write down the binomial expansion of \(\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1\) ,in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) .
(b)Hence,or otherwise,show that $$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$ and state the values of \(\theta\) for which this result is not valid.
(4)
Find
(c) $$\begin{aligned} & 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \\ & 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \end{aligned}$$ (d)
Edexcel AEA 2007 June Q2
10 marks Challenging +1.8
2.(a)On the same diagram,sketch \(y = x\) and \(y = \sqrt { } x\) ,for \(x \geq 0\) ,and mark clearly the coordinates of the points of intersection of the two graphs.
(b)With reference to your sketch,explain why there exists a value \(a\) of \(x ( a > 1 )\) such that $$\int _ { 0 } ^ { a } x \mathrm {~d} x = \int _ { 0 } ^ { a } \sqrt { } x \mathrm {~d} x$$ (c)Find the exact value of \(a\) .
(d)Hence,or otherwise,find a non-constant function \(\mathrm { f } ( x )\) and a constant \(b ( b \neq 0 )\) such that $$\int _ { - b } ^ { b } \mathrm { f } ( x ) \mathrm { d } x = \int _ { - b } ^ { b } \sqrt { } [ \mathrm { f } ( x ) ] \mathrm { d } x$$
Edexcel AEA 2007 June Q3
11 marks Standard +0.8
3.(a)Solve,for \(0 \leq x < 2 \pi\) , $$\cos x + \cos 2 x = 0$$ (b)Find the exact value of \(x , x \geq 0\) ,for which $$\arccos x + \arccos 2 x = \frac { \pi } { 2 }$$ [ \(\arccos x\) is an alternative notation for \(\cos ^ { - 1 } x\) .]
Edexcel AEA 2007 June Q4
11 marks Hard +2.3
4.The function \(\mathrm { h } ( x )\) has domain \(\mathbb { R }\) and range \(\mathrm { h } ( x ) > 0\) ,and satisfies $$\sqrt { \int \mathrm { h } ( x ) \mathrm { d } x } = \int \sqrt { \mathrm { h } ( x ) } \mathrm { d } x$$
  1. By substituting \(\mathrm { h } ( x ) = \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 }\) ,show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( y + c ) ,$$ where \(c\) is constant.
  2. Hence find a general expression for \(y\) in terms of \(x\) .
  3. Given that \(\mathrm { h } ( 0 ) = 1\) ,find \(\mathrm { h } ( x )\) .
Edexcel AEA 2007 June Q5
15 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-4_493_1324_279_367} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of a sequence \(S _ { 1 } , S _ { 2 } , S _ { 3 } , \ldots\), of model snowflakes. The first term \(S _ { 1 }\) consists of a single square of side \(a\). To obtain \(S _ { 2 }\), the middle third of each edge is replaced with a new square, of side \(\frac { a } { 3 }\), as shown in Figure 1 . Subsequent terms are obtained by replacing the middle third of each external edge of a new square formed in the previous snowflake, by a square \(\frac { 1 } { 3 }\) of the size, as illustrated by \(S _ { 3 }\) in Figure 1.
  1. Deduce that to form \(S _ { 4 } , 36\) new squares of side \(\frac { a } { 27 }\) must be added to \(S _ { 3 }\).
  2. Show that the perimeters of \(S _ { 2 }\) and \(S _ { 3 }\) are \(\frac { 20 a } { 3 }\) and \(\frac { 28 a } { 3 }\) respectively.
  3. Find the perimeter of \(S _ { n }\).
  4. Describe what happens to the perimeter of \(S _ { n }\) as \(n\) increases.
  5. Find the areas of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\).
  6. Find the smallest value of the constant \(S\) such that the area of \(S _ { n } < S\), for all values of \(n\). \includegraphics[max width=\textwidth, alt={}, center]{f2290882-b9a4-43ec-a38f-c44d46477242-5_590_1041_283_588} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \tan \frac { t } { 2 } , \quad 0 \leq t \leq \frac { \pi } { 2 }\).
    The point \(P\) on \(C\) has coordinates \(\left( x , \tan \frac { x } { 2 } \right)\).
    The vertices of rectangle \(R\) are at \(( x , 0 ) , \left( \frac { x } { 2 } , 0 \right) , \left( \frac { x } { 2 } , \tan \frac { x } { 2 } \right)\) and \(\left( x , \tan \frac { x } { 2 } \right)\) as shown in Figure 2.
Edexcel AEA 2007 June Q7
20 marks Challenging +1.8
7.The points \(O , P\) and \(Q\) lie on a circle \(C\) with diameter \(O Q\) .The position vectors of \(P\) and \(Q\) , relative to \(O\) ,are \(\mathbf { p }\) and \(\mathbf { q }\) respectively.
  1. Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-6_615_714_412_689} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The point \(R\) also lies on \(C\) and \(O P Q R\) is a kite \(K\) as shown in Figure 3.The point \(S\) has position vector,relative to \(O\) ,of \(\lambda \mathbf { q }\) ,where \(\lambda\) is a constant.Given that \(\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }\) and that \(O Q\) is perpendicular to \(P S\) ,find
  2. the value of \(\lambda\) ,
  3. the position vector of \(R\) ,
  4. the area of \(K\) . Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) .
  5. Find the radius of \(C _ { 1 }\) giving your answer in the form \(( \sqrt { } 2 - 1 ) \sqrt { } n\) ,where \(n\) is an integer. A second kite \(K _ { 1 }\) is similar to \(K\) and is drawn inside \(C _ { 1 }\) .
  6. Find that area of \(K _ { 1 }\) .
Edexcel AEA 2009 June Q1
8 marks Challenging +1.2
  1. (a) On the same diagram, sketch
$$y = ( x + 1 ) ( 2 - x ) \quad \text { and } \quad y = x ^ { 2 } - 2 | x |$$ Mark clearly the coordinates of the points where these curves cross the coordinate axes.
(b) Find the \(x\)-coordinates of the points of intersection of these two curves.
Edexcel AEA 2009 June Q2
9 marks Hard +2.3
2. The curve \(C\) has equation \(y = x ^ { \sin x } , \quad x > 0\).
  1. Find the equation of the tangent to \(C\) at the point where \(x = \frac { \pi } { 2 }\).
  2. Prove that this tangent touches \(C\) at infinitely many points.
Edexcel AEA 2009 June Q3
12 marks Challenging +1.2
3. (a) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin \left( \frac { \pi } { 3 } - \theta \right) = \frac { 1 } { \sqrt { } 3 } \cos \theta$$ (b) Find the value of \(x\) for which $$\begin{aligned} & \arcsin ( 1 - 2 x ) = \frac { \pi } { 3 } - \arcsin x , \quad 0 < x < 0.5 \\ & { \left[ \arcsin x \text { is an alternative notation for } \sin ^ { - 1 } x \right] } \end{aligned}$$
Edexcel AEA 2009 June Q4
14 marks Challenging +1.8
  1. (a) The function \(\mathrm { f } ( x )\) has \(\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { u } ( x ) } { \mathrm { v } ( x ) }\). Given that \(\mathrm { f } ^ { \prime } ( k ) = 0\), show that \(\mathrm { f } ^ { \prime \prime } ( k ) = \frac { \mathrm { u } ^ { \prime } ( k ) } { \mathrm { v } ( k ) }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-3_874_879_486_593} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} (b) The curve \(C\) with equation $$y = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 1 }$$ crosses the \(y\)-axis at the point \(A\). Figure 1 shows a sketch of \(C\) together with its 3 asymptotes.
  1. Find the coordinates of the point \(A\).
  2. Find the equations of the asymptotes of \(C\). The point \(P ( a , b ) , a > 0\) and \(b > 0\), lies on \(C\). The point \(Q\) also lies on \(C\) with \(P Q\) parallel to the \(x\)-axis and \(A P = A Q\).
  3. Show that the area of triangle \(P A Q\) is given by \(\frac { 5 a ^ { 3 } } { a ^ { 2 } - 1 }\).
  4. Find, as \(a\) varies, the minimum area of triangle \(P A Q\), giving your answer in its simplest form.
Edexcel AEA 2009 June Q5
15 marks Challenging +1.8
5.(a)The sides of the triangle \(A B C\) have lengths \(B C = a , A C = b\) and \(A B = c\) ,where \(a < b < c\) .The sizes of the angles \(A , B\) and \(C\) form an arithmetic sequence.
  1. Show that the area of triangle \(A B C\) is \(a c \frac { \sqrt { 3 } } { 4 }\) . Given that \(a = 2\) and \(\sin A = \frac { \sqrt { } 15 } { 5 }\) ,find
  2. the value of \(b\) ,
  3. the value of \(c\) .
    (b)The internal angles of an \(n\)-sided polygon form an arithmetic sequence with first term \(143 ^ { \circ }\) and common difference \(2 ^ { \circ }\) . Given that all of the internal angles are less than \(180 ^ { \circ }\) ,find the value of \(n\) .
Edexcel AEA 2009 June Q6
17 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-5_700_684_246_694} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \sin t , \quad y = \ln ( \sec t ) , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The tangent to \(C\) at the point \(P\) ,where \(t = \frac { \pi } { 3 }\) ,cuts the \(x\)-axis at \(A\) .
  1. Show that the \(x\)-coordinate of \(A\) is \(\frac { \sqrt { } 3 } { 3 } ( 3 - \ln 2 )\) . The shaded region \(R\) lies between \(C\) ,the positive \(x\)-axis and the tangent \(A P\) as shown in Figure 2 .
  2. Show that the area of \(R\) is \(\sqrt { 3 } ( 1 + \ln 2 ) - 2 \ln ( 2 + \sqrt { 3 } ) - \frac { \sqrt { 3 } } { 6 } ( \ln 2 ) ^ { 2 }\) .
Edexcel AEA 2009 June Q7
18 marks Challenging +1.8
7.Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$
  1. Find the cosine of angle \(A B C\) . The quadrilateral \(A B C D\) is a kite \(K\) .
  2. Find the area of \(K\) . A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) .
  3. Find the radius of the circle,giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ,where \(p\) and \(q\) are positive integers.
  4. Find the position vector of the point \(D\) .
    (Total 18 marks)
Edexcel AEA 2010 June Q1
12 marks Standard +0.8
1.(a)Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ (b)Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$
Edexcel AEA 2010 June Q2
11 marks Challenging +1.2
2.The sum of the first \(p\) terms of an arithmetic series is \(q\) and the sum of the first \(q\) terms of the same arithmetic series is \(p\) ,where \(p\) and \(q\) are positive integers and \(p \neq q\) . Giving simplified answers in terms of \(p\) and \(q\) ,find
  1. the common difference of the terms in this series,
  2. the first term of the series,
  3. the sum of the first \(( p + q )\) terms of the series.
Edexcel AEA 2010 June Q3
11 marks Challenging +1.2
3.The curve \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + f x y = g ^ { 2 }$$ where \(f\) and \(g\) are constants and \(g \neq 0\) .
  1. Find an expression in terms of \(\alpha , \beta\) and \(f\) for the gradient of \(C\) at the point \(( \alpha , \beta )\) . Given that \(f < 2\) and \(f \neq - 2\) and that the gradient of \(C\) at the point \(( \alpha , \beta )\) is 1 ,
  2. show that \(\alpha = - \beta = \frac { \pm g } { \sqrt { } ( 2 - f ) }\) . Given that \(f = - 2\) ,
  3. sketch \(C\) .
Edexcel AEA 2010 June Q4
16 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-3_643_332_246_870} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a cuboid \(O A B C D E F G\), where \(O\) is the origin, \(A\) has position vector \(5 \mathbf { i } , C\) has position vector \(10 \mathbf { j }\) and \(D\) has position vector \(20 \mathbf { k }\).
  1. Find the cosine of angle \(C A F\). Given that the point \(X\) lies on \(A C\) and that \(F X\) is perpendicular to \(A C\),
  2. find the position vector of point \(X\) and the distance \(F X\). The line \(l _ { 1 }\) passes through \(O\) and through the midpoint of the face \(A B F E\). The line \(l _ { 2 }\) passes through \(A\) and through the midpoint of the edge \(F G\).
  3. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection.
Edexcel AEA 2010 June Q5
12 marks Challenging +1.8
5. $$I = \int \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x , \quad x > 1$$
  1. Use the substitution \(x = 1 + u ^ { - 1 }\) to show that $$I = - \left( \frac { x + 1 } { x - 1 } \right) ^ { \frac { 1 } { 2 } } + c$$
  2. Hence show that $$\int _ { \sec \alpha } ^ { \sec \beta } \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x = \cot \left( \frac { \alpha } { 2 } \right) - \cot \left( \frac { \beta } { 2 } \right) , \quad 0 < \alpha < \beta < \frac { \pi } { 2 }$$
Edexcel AEA 2010 June Q6
10 marks Challenging +1.8
6.(a)Given that \(x ^ { 4 } + y ^ { 4 } = 1\) ,prove that \(x ^ { 2 } + y ^ { 2 }\) is a maximum when \(x = \pm y\) ,and find the maximum and minimum values of \(x ^ { 2 } + y ^ { 2 }\) .
(b)On the same diagram,sketch the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations \(x ^ { 4 } + y ^ { 4 } = 1\) and \(x ^ { 2 } + y ^ { 2 } = 1\) respectively.
(c)Write down the equation of the circle \(C _ { 3 }\) ,centre the origin,which touches the curve \(C _ { 1 }\) at the points where \(x = \pm y\) .
Edexcel AEA 2010 June Q7
21 marks Challenging +1.2
7. $$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$
  1. Show that \(\mathrm { f } ( x )\) may be written in the form $$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
  2. Find the range of the function \(\mathrm { f } ( x )\). The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
  3. Find the coordinates of all the maximum and minimum points on this curve. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The region \(R\), bounded by \(y = 2\) and \(y = \mathrm { f } ( x )\), is shown shaded in Figure 3.
  4. Find the area of \(R\).
Edexcel AEA 2012 June Q1
8 marks Challenging +1.2
1.The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 6 , \quad x \geqslant 0$$
  1. Find the range of \(f\) . The function \(g\) is given by $$\mathrm { g } ( x ) = 3 + \sqrt { } ( x + 4 ) , \quad x \geqslant 2$$
  2. Find \(\operatorname { gf } ( x )\) .
  3. Find the domain and range of gf.
Edexcel AEA 2012 June Q2
10 marks Challenging +1.8
2.(a)Show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$ Hence find
(b) \(\int \cos x ( 6 \sin x - 2 \sin 3 x ) ^ { \frac { 2 } { 3 } } \mathrm {~d} x\) (c) \(\int ( 3 \sin 2 x - 2 \sin 3 x \cos x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\)