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Edexcel FP3 2010 June Q4
8 marks Challenging +1.8
4. \(\quad I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a > 0 , \quad n \geqslant 0\)
  1. Show that, for \(n \geqslant 2\), $$I _ { n } = n \tilde { a } ^ { - 1 } - n ( n - 1 ) I _ { n - 2 }$$
  2. Hence evaluate \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x \mathrm {~d} x\).
Edexcel FP3 2010 June Q5
9 marks Standard +0.8
  1. Given that \(y = ( \operatorname { arcosh } 3 x ) ^ { 2 }\), where \(3 x > 1\), show that
    1. \(\left( 9 x ^ { 2 } - 1 \right) \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 36 y\),
    2. \(\left( 9 x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 18\).
Edexcel FP3 2010 June Q6
13 marks Standard +0.3
6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant. Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),
  1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\),
  2. show that \(k = 3\),
  3. show that \(\mathbf { M }\) has exactly two eigenvalues. A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\).
    The transformation \(T\) maps the line \(l _ { 1 }\), with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\), onto the line \(l _ { 2 }\).
  4. Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).
Edexcel FP3 2010 June Q7
14 marks Challenging +1.2
7. The plane \(\Pi\) has vector equation $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$
  1. Find an equation of \(\Pi\) in the form \(\mathbf { r } \cdot \mathbf { n } = p\), where \(\mathbf { n }\) is a vector perpendicular to \(\Pi\) and \(p\) is a constant. The point \(P\) has coordinates \(( 6,13,5 )\). The line \(l\) passes through \(P\) and is perpendicular to \(\Pi\). The line \(l\) intersects \(\Pi\) at the point \(N\).
  2. Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\). The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
  3. Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.
Edexcel FP3 2010 June Q8
13 marks Challenging +1.2
8. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\). The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).
  1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
  2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$
Edexcel FP3 2011 June Q1
5 marks Challenging +1.2
  1. The curve \(C\) has equation \(y = 2 x ^ { 3 } , 0 \leqslant x \leqslant 2\).
The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Using calculus, find the area of the surface generated, giving your answer to 3 significant figures.
Edexcel FP3 2011 June Q2
8 marks Standard +0.8
2. (a) Given that \(y = x \arcsin x , 0 \leqslant x \leqslant 1\), find
  1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 2 }\).
    (b) Given that \(y = \arctan \left( 3 \mathrm { e } ^ { 2 x } \right)\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 5 \cosh 2 x + 4 \sinh 2 x }$$
Edexcel FP3 2011 June Q3
9 marks Standard +0.8
  1. Show that
    1. \(\int _ { 5 } ^ { 8 } \frac { 1 } { x ^ { 2 } - 10 x + 34 } \mathrm {~d} x = k \pi\), giving the value of the fraction \(k\),
    2. \(\int _ { 5 } ^ { 8 } \frac { 1 } { \sqrt { } \left( x ^ { 2 } - 10 x + 34 \right) } \mathrm { d } x = \ln ( A + \sqrt { } n )\), giving the values of the integers \(A\) and \(n\).
    $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x , \quad n \geqslant 0$$
  2. Prove that, for \(n \geqslant 1\), $$I _ { n } = \frac { \mathrm { e } ^ { 3 } } { 3 } - \frac { n } { 3 } I _ { n - 1 }$$
  3. Find the exact value of \(I _ { 3 }\).
Edexcel FP3 2011 June Q5
9 marks Standard +0.3
  1. The curve \(C _ { 1 }\) has equation \(y = 3 \sinh 2 x\), and the curve \(C _ { 2 }\) has equation \(y = 13 - 3 \mathrm { e } ^ { 2 x }\).
    1. Sketch the graph of the curves \(C _ { 1 }\) and \(C _ { 2 }\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes.
    2. Solve the equation \(3 \sinh 2 x = 13 - 3 \mathrm { e } ^ { 2 x }\), giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is an integer.
Edexcel FP3 2011 June Q6
10 marks Standard +0.3
  1. The plane \(P\) has equation
$$\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 2 \\ 2 \end{array} \right)$$
  1. Find a vector perpendicular to the plane \(P\). The line \(l\) passes through the point \(A ( 1,3,3 )\) and meets \(P\) at \(( 3,1,2 )\). The acute angle between the plane \(P\) and the line \(l\) is \(\alpha\).
  2. Find \(\alpha\) to the nearest degree.
  3. Find the perpendicular distance from \(A\) to the plane \(P\).
Edexcel FP3 2011 June Q7
12 marks Challenging +1.2
  1. The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r } k & - 1 & 1 \\ 1 & 0 & - 1 \\ 3 & - 2 & 1 \end{array} \right) , \quad k \neq 1$$
  1. Show that \(\operatorname { det } \mathbf { M } = 2 - 2 k\).
  2. Find \(\mathbf { M } ^ { - 1 }\), in terms of \(k\). The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\left( \begin{array} { r r r } 2 & - 1 & 1 \\ 1 & 0 & - 1 \\ 3 & - 2 & 1 \end{array} \right)\). The equation of \(l _ { 2 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 4 \mathbf { i } + \mathbf { j } + 7 \mathbf { k }\) and \(\mathbf { b } = 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
  3. Find a vector equation for the line \(l _ { 1 }\).
Edexcel FP3 2011 June Q8
14 marks Challenging +1.2
  1. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
  1. Use calculus to show that the equation of the tangent to \(H\) at the point \(( a \cosh \theta , b \sinh \theta )\) may be written in the form $$x b \cosh \theta - y a \sinh \theta = a b$$ The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(( a \cosh \theta , b \sinh \theta ) , \theta \neq 0\).
    Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(P\),
  2. find, in terms of \(a\) and \(\theta\), the coordinates of \(P\). The line \(l _ { 2 }\) is the tangent to \(H\) at the point ( \(a , 0\) ).
    Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(Q\),
  3. find, in terms of \(a , b\) and \(\theta\), the coordinates of \(Q\).
  4. Show that, as \(\theta\) varies, the locus of the mid-point of \(P Q\) has equation $$x \left( 4 y ^ { 2 } + b ^ { 2 } \right) = a b ^ { 2 }$$
Edexcel FP3 2012 June Q1
5 marks Standard +0.3
  1. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ Find
  1. the coordinates of the foci of \(H\),
  2. the equations of the directrices of \(H\).
Edexcel FP3 2012 June Q2
6 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb604886-6671-441a-b03d-427b5176df6e-03_606_1271_212_335} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has equation $$y = \frac { 1 } { 3 } \cosh 3 x , \quad 0 \leqslant x \leqslant \ln a$$ where \(a\) is a constant and \(a > 1\) Using calculus, show that the length of curve \(C\) is $$k \left( a ^ { 3 } - \frac { 1 } { a ^ { 3 } } \right)$$ and state the value of the constant \(k\).
Edexcel FP3 2012 June Q3
8 marks Standard +0.3
3. The position vectors of the points \(A , B\) and \(C\) relative to an origin \(O\) are \(\mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , 7 \mathbf { i } - 3 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j }\) respectively. Find
  1. \(\overrightarrow { A C } \times \overrightarrow { B C }\),
  2. the area of triangle \(A B C\),
  3. an equation of the plane \(A B C\) in the form \(\mathbf { r } . \mathbf { n } = p\)
Edexcel FP3 2012 June Q4
11 marks Challenging +1.2
4. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } x ^ { n } \sin 2 x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Prove that, for \(n \geqslant 2\), $$I _ { n } = \frac { 1 } { 4 } n \left( \frac { \pi } { 4 } \right) ^ { n - 1 } - \frac { 1 } { 4 } n ( n - 1 ) I _ { n - 2 }$$
  2. Find the exact value of \(I _ { 2 }\)
  3. Show that \(I _ { 4 } = \frac { 1 } { 64 } \left( \pi ^ { 3 } - 24 \pi + 48 \right)\)
Edexcel FP3 2012 June Q5
10 marks Standard +0.8
  1. (a) Differentiate \(x \operatorname { arsinh } 2 x\) with respect to \(x\).
    (b) Hence, or otherwise, find the exact value of
$$\int _ { 0 } ^ { \sqrt { 2 } } \operatorname { arsinh } 2 x \mathrm {~d} x$$ giving your answer in the form \(A \ln B + C\), where \(A , B\) and \(C\) are real.
Edexcel FP3 2012 June Q6
11 marks Challenging +1.2
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l _ { 1 }\) is a tangent to \(E\) at the point \(P ( a \cos \theta , b \sin \theta )\).
  1. Using calculus, show that an equation for \(l _ { 1 }\) is $$\frac { x \cos \theta } { a } + \frac { y \sin \theta } { b } = 1$$ The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$$ The line \(l _ { 2 }\) is a tangent to \(C\) at the point \(Q ( a \cos \theta , a \sin \theta )\).
  2. Find an equation for the line \(l _ { 2 }\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\),
  3. find, in terms of \(a , b\) and \(\theta\), the coordinates of \(R\).
  4. Find the locus of \(R\), as \(\theta\) varies.
Edexcel FP3 2012 June Q7
11 marks Standard +0.3
7. $$\mathrm { f } ( x ) = 5 \cosh x - 4 \sinh x , \quad x \in \mathbb { R }$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + 9 \mathrm { e } ^ { - x } \right)\) Hence
  2. solve \(\mathrm { f } ( x ) = 5\)
  3. show that \(\int _ { \frac { 1 } { 2 } \ln 3 } ^ { \ln 3 } \frac { 1 } { 5 \cosh x - 4 \sinh x } \mathrm {~d} x = \frac { \pi } { 18 }\)
Edexcel FP3 2012 June Q8
13 marks
  1. The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$
  1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
  2. For the eigenvalue 4, find a corresponding eigenvector. The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { M }\). The equation of \(l _ { 1 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\).
  3. Find a vector equation for the line \(l _ { 2 }\).
Edexcel FP3 2013 June Q1
7 marks Standard +0.8
  1. The hyperbola \(H\) has foci at \(( 5,0 )\) and \(( - 5,0 )\) and directrices with equations \(x = \frac { 9 } { 5 }\) and \(x = - \frac { 9 } { 5 }\).
Find a cartesian equation for \(H\).
Edexcel FP3 2013 June Q2
7 marks Standard +0.8
2. Two skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k } ) + \mu ( - 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ respectively, where \(\lambda\) and \(\mu\) are real parameters.
  1. Find a vector in the direction of the common perpendicular to \(l _ { 1 }\) and \(l _ { 2 }\)
  2. Find the shortest distance between these two lines.
Edexcel FP3 2013 June Q3
8 marks Challenging +1.2
  1. The point \(P\) lies on the ellipse \(E\) with equation
$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$$ \(N\) is the foot of the perpendicular from point \(P\) to the line \(x = 8\) \(M\) is the midpoint of \(P N\).
  1. Sketch the graph of the ellipse \(E\), showing also the line \(x = 8\) and a possible position for the line \(P N\).
  2. Find an equation of the locus of \(M\) as \(P\) moves around the ellipse.
  3. Show that this locus is a circle and state its centre and radius.
Edexcel FP3 2013 June Q4
9 marks Challenging +1.2
  1. The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 2 \\ - 2 \end{array} \right) ,$$ where \(s\) and \(t\) are real parameters. The plane \(\Pi _ { 1 }\) is transformed to the plane \(\Pi _ { 2 }\) by the transformation represented by the matrix \(\mathbf { T }\), where $$\mathbf { T } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & 2 & - 1 \\ 0 & 1 & 2 \end{array} \right)$$ Find an equation of the plane \(\Pi _ { 2 }\) in the form r.n=p
Edexcel FP3 2013 June Q5
10 marks Challenging +1.8
5. $$I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } ( 2 x - 1 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} x , \quad n \geqslant 0$$
  1. Prove that, for \(n \geqslant 1\), $$( 2 n + 1 ) I _ { n } = n I _ { n - 1 } + 3 \times 5 ^ { n } - 1$$
  2. Using the reduction formula given in part (a), find the exact value of \(I _ { 2 }\)