Edexcel FP3 (Further Pure Mathematics 3)

1. An ellipse has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1\).
   1. Sketch the ellipse.
   2. Find the value of the eccentricity \(e\).
   3. State the coordinates of the foci of the ellipse.
   4. Solve the equation
   $$10 \cosh x + 2 \sinh x = 11 .$$ Give each answer in the form \(\ln a\) where \(a\) is a rational number.
   [0pt] [P5 June 2002 Qn 3]

4. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \cos x \mathrm {~d} x , \quad n \geq 0$$

1. Prove that \(I _ { n } = \left( \frac { \pi } { 2 } \right) ^ { n } - n ( n - 1 ) I _ { n - 2 } , n \geq 2\).
2. Find an exact expression for \(I _ { 6 }\).
   [0pt] [P5 June 2002 Qn 4]

5. (a) Given that \(y = \arctan 3 x\), and assuming the derivative of \(\tan x\), prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 1 + 9 x ^ { 2 } }$$ (b) Show that $$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 3 } } 6 x \arctan 3 x \mathrm {~d} x = \frac { 1 } { 9 } ( 4 \pi - 3 \sqrt { } 3 )$$ (6)
[0pt] [P5 June 2002 Qn 6] \section*{6.} \begin{figure}[h] \end{figure} The curve \(C\) shown in Fig. 1 has equation \(y ^ { 2 } = 4 x , 0 \leq x \leq 1\).
The part of the curve in the first quadrant is rotated through \(2 \pi\) radians about the \(x\)-axis.
(a) Show that the surface area of the solid generated is given by $$4 \pi \int _ { 0 } ^ { 1 } \sqrt { ( 1 + x ) } d x$$ (b) Find the exact value of this surface area.
(c) Show also that the length of the curve \(C\), between the points \(( 1 , - 2 )\) and \(( 1,2 )\), is given by $$2 \int _ { 0 } ^ { 1 } \sqrt { \left( \frac { x + 1 } { x } \right) } \mathrm { d } x$$ (d) Use the substitution \(x = \sinh ^ { 2 } \theta\) to show that the exact value of this length is $$2 [ \sqrt { } 2 + \ln ( 1 + \sqrt { } 2 ) ]$$ [P5 June 2002 Qn 8]

7. Prove that \(\sinh ( \mathrm { i } \pi - \theta ) = \sinh \theta\).
[0pt] [P6 June 2002 Qn 1]

8. $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 0 & 4
0 & 5 & 4
4 & 4 & 3 \end{array} \right)$$

1. Verify that \(\left( \begin{array} { r } 2
   - 2
   1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and find the corresponding eigenvalue.
2. Show that 9 is another eigenvalue of \(\mathbf { A }\) and find the corresponding eigenvector.
3. Given that the third eigenvector of \(\mathbf { A }\) is \(\left( \begin{array} { r } 2
   1
   - 2 \end{array} \right)\), write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { A } \mathbf { P } = \mathbf { D } .$$

1. The plane \(\Pi\) passes through the points

$$A ( - 1 , - 1,1 ) , B ( 4,2,1 ) \text { and } C ( 2,1,0 )$$

1. Find a vector equation of the line perpendicular to \(\Pi\) which passes through the point \(D ( 1,2,3 )\).
2. Find the volume of the tetrahedron \(A B C D\).
3. Obtain the equation of \(\Pi\) in the form r.n \(= p\). The perpendicular from \(D\) to the plane \(\Pi\) meets \(\Pi\) at the point \(E\).
4. Find the coordinates of \(E\).
5. Show that \(D E = \frac { 11 \sqrt { 35 } } { 35 }\). The point \(D ^ { \prime }\) is the reflection of \(D\) in \(\Pi\).
6. Find the coordinates of \(D ^ { \prime }\).
   (3)
   [0pt] [P6 June 2002 Qn 7]

10. Find the values of \(x\) for which $$4 \cosh x + \sinh x = 8$$ giving your answer as natural logarithms.
[0pt] [P5 June 2003 Qn 1]

11. (a) Prove that the derivative of \(\operatorname { artanh } x , - 1 < x < 1\), is \(\frac { 1 } { 1 - x ^ { 2 } }\).
(3)
(b) Find \(\int \operatorname { artanh } x \mathrm {~d} x\).
(4)
[0pt] [P5 June 2003 Qn 2]

12. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{6706ed7f-4575-4898-b757-aee8475b2a30-07_588_1518_941_392} Figure 1 shows the cross-section \(R\) of an artificial ski slope. The slope is modelled by the curve with equation $$y = \frac { 10 } { \sqrt { \left( 4 x ^ { 2 } + 9 \right) } } , \quad 0 \leq x \leq 5$$ Given that 1 unit on each axis represents 10 metres, use integration to calculate the area \(R\). Show your method clearly and give your answer to 2 significant figures.
[0pt] [P5 June 2003 Qn 3]

13. \begin{figure}[h] \end{figure} A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation $$y = a \cosh \left( \frac { x } { a } \right) , \quad - k a \leq x \leq k a$$ where \(a\) and \(k\) are positive constants.

1. Prove that the length of the rope is \(2 a \sinh k\). Given that the length of the rope is \(8 a\),
2. find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate.

14. The curve \(C\) has equation $$y = \operatorname { arcsec } \mathrm { e } ^ { x } , \quad x > 0 , \quad 0 < y < \frac { 1 } { 2 } \pi$$

1. Prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { \left( \mathrm { e } ^ { 2 x } - 1 \right) } }\).
2. Sketch the graph of \(C\). The point \(A\) on \(C\) has \(x\)-coordinate \(\ln 2\). The tangent to \(C\) at \(A\) intersects the \(y\)-axis at the point \(B\).
3. Find the exact value of the \(y\)-coordinate of \(B\).

15. $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x \text { and } J _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x , \quad n \geq 0$$

1. Show that, for \(n \geq 1\), $$I _ { n } = \mathrm { e } - n I _ { n - 1 } .$$
2. Find a similar reduction formula for \(J _ { n }\).
3. Show that \(J _ { 2 } = 2 - \frac { 5 } { \mathrm { e } }\).
4. Show that \(\int _ { 0 } ^ { 1 } x ^ { n } \cosh x \mathrm {~d} x = \frac { 1 } { 2 } \left( I _ { n } + J _ { n } \right)\).
5. Hence, or otherwise, evaluate \(\int _ { 0 } ^ { 1 } x ^ { 2 } \cosh x \mathrm {~d} x\), giving your answer in terms of e.
   [0pt] [P5 June 2003 Qn 7]

16. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\).

1. Show that an equation of the normal to \(C\) at the point \(P ( a \sec t , b \tan t )\) is $$a x \sin t + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan t .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and \(S\) is a focus of \(C\). Given that the eccentricity of \(C\) is \(\frac { 3 } { 2 }\), and that \(O A = 3 O S\), where \(O\) is the origin,
2. determine the possible values of \(t\), for \(0 \leq t < 2 \pi\).
   [0pt] [P5 June 2003 Qn 1]

17. Referred to a fixed origin \(O\), the position vectors of three non-collinear points \(A , B\) and \(C\) are \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively. By considering \(\overrightarrow { A B } \times \overrightarrow { A C }\), prove that the area of \(\triangle A B C\) can be expressed in the form \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
[0pt] [P6 June 2003 Qn 1]

18. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5
6 & - 9 \end{array} \right)$$

1. Find the eigenvalues of \(\mathbf { M }\). A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { M }\). There is a line through the origin for which every point on the line is mapped onto itself under \(T\).
2. Find a cartesian equation of this line.
   [0pt] [P6 June 2003 Qn 3]

19. $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 1
1 & 1 & 1
5 & 3 & u \end{array} \right) , \quad u \neq 1$$

1. Show that \(\operatorname { det } \mathbf { A } = 2 ( u - 1 )\).
2. Find the inverse of \(\mathbf { A }\). The image of the vector \(\left( \begin{array} { l } a
   b
   c \end{array} \right)\) when transformed by the matrix \(\left( \begin{array} { r r r } 3 & 1 & - 1
   1 & 1 & 1
   5 & 3 & 6 \end{array} \right)\) is \(\left( \begin{array} { l } 3
   1
   6 \end{array} \right)\).
3. Find the values of \(a , b\) and \(c\).
   (3)
   [0pt] [P6 June 2003 Qn 6]

20. The plane \(\Pi _ { 1 }\) passes through the \(P\), with position vector \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\), and is perpendicular to the line \(L\) with equation $$\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )$$

1. Show that the Cartesian equation of \(\Pi _ { 1 }\) is \(x - 5 y - 3 z = - 6\). The plane \(\Pi _ { 2 }\) contains the line \(L\) and passes through the point \(Q\), with position vector \(\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\).
2. Find the perpendicular distance of \(Q\) from \(\Pi _ { 1 }\).
3. Find the equation of \(\Pi _ { 2 }\) in the form \(\mathbf { r } = \mathbf { a } + s \mathbf { b } + t \mathbf { c }\).
   [0pt] [P6 June 2003 Qn 7]
   21. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of exponentials,
4. prove that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1\),
   (3)
5. solve \(\operatorname { cosech } x - 2 \operatorname { coth } x = 2\),
   giving your answer in the form \(k \ln a\), where \(k\) and \(a\) are integers.
   (4)
   [0pt] [P5 June 2004 Qn 1]
   22. $$4 x ^ { 2 } + 4 x + 17 \equiv ( a x + b ) ^ { 2 } + c , \quad a > 0$$
6. Find the values of \(a , b\) and \(c\).
7. Find the exact value of $$\int _ { - 0.5 } ^ { 1.5 } \frac { 1 } { 4 x ^ { 2 } + 4 x + 17 } \mathrm {~d} x$$ [P5 June 2004 Qn 2]
   23. An ellipse, with equation \(\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1\), has foci \(S\) and \(S ^ { \prime }\).
8. Find the coordinates of the foci of the ellipse.
   (4)
9. Using the focus-directrix property of the ellipse, show that, for any point \(P\) on the ellipse, $$S P + S ^ { \prime } P = 6$$ [P5 June 2004 Qn 3]
   24. Given that \(y = \sinh ^ { n - 1 } x \cosh x\),
10. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( n - 1 ) \sinh ^ { n - 2 } x + n \sinh ^ { n } x\). The integral \(I _ { n }\) is defined by \(I _ { n } = \int _ { 0 } ^ { \text {arsinh } 1 } \sinh ^ { n } x \mathrm {~d} x , \quad n \geq 0\).
11. Using the result in part (a), or otherwise, show that $$n I _ { n } = \sqrt { } 2 - ( n - 1 ) I _ { n - 2 } , \quad n \geq 2$$
12. Hence find the value of \(I _ { 4 }\).
    25. \section*{Figure 1}
    \includegraphics[max width=\textwidth, alt={}]{6706ed7f-4575-4898-b757-aee8475b2a30-13_659_810_1288_575}
    Figure 1 shows the curve with parametric equations $$x = a \cos ^ { 3 } \theta , \quad y = a \sin ^ { 3 } \theta , \quad 0 \leq \theta < 2 \pi$$
13. Find the total length of this curve. The curve is rotated through \(\pi\) radians about the \(x\)-axis.
14. Find the area of the surface generated.
    [0pt] [P5 June 2004 Qn 7]
    26. The points \(A , B\) and \(C\) lie on the plane \(\Pi\) and, relative to a fixed origin \(O\), they have position vectors $$\mathbf { a } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { b } = - \mathbf { i } + 2 \mathbf { j } , \quad \mathbf { c } = 5 \mathbf { i } - 3 \mathbf { j } + 7 \mathbf { k }$$ respectively.
15. Find \(\xrightarrow { A B } \times \overrightarrow { A C }\).
16. Find an equation of \(\Pi\) in the form r.n \(= p\). The point \(D\) has position vector \(5 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\).
17. Calculate the volume of the tetrahedron \(A B C D\).
    [0pt] [P6 June 2004 Qn 3]
    27. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { c c c } 1 & 4 & - 1
    3 & 0 & p
    a & b & c \end{array} \right)$$ where \(p , a , b\) and \(c\) are constants and \(a > 0\).
    Given that \(\mathbf { M } \mathbf { M } ^ { \mathrm { T } } = k \mathbf { I }\) for some constant \(k\), find
18. the value of \(p\),
19. the value of \(k\),
20. the values of \(a , b\) and \(c\),
21. \(| \operatorname { det } \mathbf { M } |\).
    [0pt] [P6 June 2004 Qn 5]
    28. The transformation \(R\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l } 3 & 1
    1 & 3 \end{array} \right)$$
22. Find the eigenvectors of \(\mathbf { A }\).
23. Find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } = \mathbf { P D } \mathbf { P } ^ { - 1 }$$
24. Hence describe the transformation \(R\) as a combination of geometrical transformations, stating clearly their order.
    29. (a) Find \(\int \frac { 1 + x } { \sqrt { } \left( 1 - 4 x ^ { 2 } \right) } \mathrm { d } x\).
25. Find, to 3 decimal places, the value of $$\int _ { 0 } ^ { 0.3 } \frac { 1 + x } { \sqrt { } \left( 1 - 4 x ^ { 2 } \right) } \mathrm { d } x$$ (Total 7 marks)
    [0pt] [FP2/P5 June 2005 Qn 1]
    30. (a) Show that, for \(x = \ln k\), where \(k\) is a positive constant, $$\cosh 2 x = \frac { k ^ { 4 } + 1 } { 2 k ^ { 2 } }$$ Given that \(\mathrm { f } ( x ) = p x - \tanh 2 x\), where \(p\) is a constant,
26. find the value of \(p\) for which \(\mathrm { f } ( x )\) has a stationary value at \(x = \ln 2\), giving your answer as an exact fraction.
    31. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6706ed7f-4575-4898-b757-aee8475b2a30-16_787_821_1110_667}
    \end{figure} Figure 1 shows a sketch of the curve with parametric equations $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leq t \leq \frac { \pi } { 2 }$$ where \(a\) is a positive constant. The curve is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the area of the curved surface generated.
    [0pt] [FP2/P5 June 2005 Qn 3]
    32. $$I _ { n } = \int x ^ { n } \mathrm { e } ^ { 2 x } \mathrm {~d} x , \quad n \geq 0$$
27. Prove that, for \(n \geq 1\), $$I _ { n } = \frac { 1 } { 2 } \left( x ^ { n } \mathrm { e } ^ { 2 x } - n I _ { n - 1 } \right) .$$
28. Find, in terms of e, the exact value of $$\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ 33. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{6706ed7f-4575-4898-b757-aee8475b2a30-17_723_949_1224_577}
    \end{figure} Figure 2 shows a sketch of the curve with equation $$y = x \operatorname { arcosh } x , \quad 1 \leq x \leq 2 .$$ The region \(R\), as shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line \(x = 2\). Show that the area of \(R\) is $$\frac { 7 } { 4 } \ln ( 2 + \sqrt { 3 } ) - \frac { \sqrt { 3 } } { 2 }$$
    1. (a) Show that, for \(0 < x \leq 1\),
    $$\ln \left( \frac { 1 - \sqrt { } \left( 1 - x ^ { 2 } \right) } { x } \right) = - \ln \left( \frac { 1 + \sqrt { } \left( 1 - x ^ { 2 } \right) } { x } \right)$$
29. Using the definition of \(\cosh x\) or sech \(x\) in terms of exponentials, show that, for \(0 < x \leq 1\), $$\operatorname { arsech } x = \ln \left( \frac { 1 + \sqrt { } \left( 1 - x ^ { 2 } \right) } { x } \right)$$
30. Solve the equation $$3 \tanh ^ { 2 } x - 4 \operatorname { sech } x + 1 = 0$$ giving exact answers in terms of natural logarithms.
    (Total 13 marks)
    [0pt] [FP2/P5 June 2005 Qn 8]
    35. (a) (i) Explain why, for any two vectors \(\mathbf { a }\) and \(\mathbf { b } , \mathbf { a } . \mathbf { b } \times \mathbf { a } = 0\).
    (ii) Given vectors \(\mathbf { a }\), \(\mathbf { b }\) and \(\mathbf { c }\) such that \(\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }\), where \(\mathbf { a } \neq \mathbf { 0 }\) and \(\mathbf { b } \neq \mathbf { c }\), show that $$\mathbf { b } - \mathbf { c } = \lambda \mathbf { a } , \quad \text { where } \lambda \text { is a scalar. }$$
31. A, B and \(\mathbf { C }\) are \(2 \times 2\) matrices.
    1. Given that \(\mathbf { A B } = \mathbf { A C }\), and that \(\mathbf { A }\) is not singular, prove that \(\mathbf { B } = \mathbf { C }\).
    2. Given that \(\mathbf { A B } = \mathbf { A C }\), where \(\mathbf { A } = \left( \begin{array} { l l } 3 & 6
       1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & 5
       0 & 1 \end{array} \right)\), find a matrix \(\mathbf { C }\) whose elements are all non-zero.
       36. The line \(l _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 6 \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 3 \mathbf { i } + p \mathbf { j } + \mu ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) , \text { where } p \text { is a constant. }$$ The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and \(l _ { 2 }\).
32. Find a vector which is normal to \(\Pi _ { 1 }\).
33. Show that an equation for \(\Pi _ { 1 }\) is \(6 x + y - 4 z = 16\).
34. Find the value of \(p\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) = 2\).
35. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form $$( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = \mathbf { 0 }$$ [FP3/P6 June 2005 Qn 3]
    37. $$\mathbf { A } = \left( \begin{array} { l l l } 3 & 2 & 4
    2 & 0 & 2
    4 & 2 & k \end{array} \right)$$
36. Show that \(\operatorname { det } \mathbf { A } = 20 - 4 k\).
    (2)
37. Find \(\mathbf { A } ^ { - 1 }\).
    (6) Given that \(k = 3\) and that \(\left( \begin{array} { r } 0
    2
    - 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),
38. find the corresponding eigenvalue.
    (2) Given that the only other distinct eigenvalue of \(\mathbf { A }\) is 8,
39. find a corresponding eigenvector.
    (4)
    [0pt] [FP3/P6 June 2005 Qn 7]
    38. Evaluate \(\int _ { 1 } ^ { 4 } \frac { 1 } { \left. \sqrt { ( } x ^ { 2 } - 2 x + 17 \right) } \mathrm { d } x\), giving your answer as an exact logarithm.
    [0pt] [FP2/P5 January 2006 Qn 1]
    39. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\). Find
40. the value of the eccentricity of \(H\),
41. the distance between the foci of \(H\). The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1\).
42. Sketch \(H\) and \(E\) on the same diagram, showing the coordinates of the points where each curve crosses the axes.
    (3)
    [0pt] [FP2/P5 January 2006 Qn 2]
    40. A curve is defined by $$x = t + \sin t , \quad y = 1 - \cos t$$ where \(t\) is a parameter. Find the length of the curve from \(t = 0\) to \(t = \frac { \pi } { 2 }\), giving your answer in surd form.
    [0pt] [FP2/P5 January 2006 Qn 3]
    41. (a) Using the definition of \(\cosh x\) in terms of exponentials, prove that $$4 \cosh ^ { 3 } x - 3 \cosh x = \cosh 3 x$$
43. Hence, or otherwise, solve the equation $$\cosh 3 x = 5 \cosh x$$ giving your answer as natural logarithms.
    [0pt] [FP2/P5 January 2006 Qn 4]
    42. Given that $$I _ { n } = \int _ { 0 } ^ { 4 } x ^ { n } \sqrt { } ( 4 - x ) \mathrm { d } x , \quad n \geq 0$$
44. show that \(I _ { n } = \frac { 8 n } { 2 n + 3 } I _ { n - 1 } , n \geq 1\).
    (6) Given that \(\int _ { 0 } ^ { 4 } \sqrt { } ( 4 - x ) \mathrm { d } x = \frac { 16 } { 3 }\),
45. use the result in part (a) to find the exact value of \(\int _ { 0 } ^ { 4 } x ^ { 2 } \sqrt { } ( 4 - x ) \mathrm { d } x\).
    (3)
    [0pt] [FP2/P5 January 2006 Qn 7]
    43. (a) Show that \(\operatorname { artanh } \left( \sin \frac { \pi } { 4 } \right) = \ln ( 1 + \sqrt { } 2 )\).
    (3)
46. Given that \(y = \operatorname { artanh } ( \sin x )\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x\).
47. Find the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \operatorname { artanh } ( \sin x ) \mathrm { d } x\).
    [0pt] [FP2/P5 January 2006 Qn 8]
    44. A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix $$\mathbf { A } = \left( \begin{array} { r r } 2 & 2
    2 & - 1 \end{array} \right) , \text { where } k \text { is a constant. }$$ Find
48. the two eigenvalues of \(\mathbf { A }\),
49. a cartesian equation for each of the two lines passing through the origin which are invariant under \(T\).
    (3)
    [0pt] [*FP3/P6 January 2006 Qn 3]
    45. $$\mathbf { A } = \left( \begin{array} { r r r } k & 1 & - 2
    0 & - 1 & k
    9 & 1 & 0 \end{array} \right) , \text { where } k \text { is a real constant. }$$
50. Find values of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
51. find, in terms of \(k , \mathbf { A } ^ { - 1 }\).
    (5)
    [0pt] [FP3/P6 January 2006 Qn 4]
    46. The plane \(\Pi\) passes through the points $$P ( - 1,3 , - 2 ) , Q ( 4 , - 1 , - 1 ) \text { and } R ( 3,0 , c ) \text {, where } c \text { is a constant. }$$
52. Find, in terms of \(c , \overrightarrow { R P } \times \overrightarrow { R Q }\). Given that \(\overrightarrow { R P } \times \overrightarrow { R Q } = 3 \mathbf { i } + d \mathbf { j } + \mathbf { k }\), where \(d\) is a constant,
53. find the value of \(c\) and show that \(d = 4\),
54. find an equation of \(\Pi\) in the form r.n \(= p\), where \(p\) is a constant. The point \(S\) has position vector \(\mathbf { i } + 5 \mathbf { j } + 10 \mathbf { k }\). The point \(S ^ { \prime }\) is the image of \(S\) under reflection in \(\Pi\).
55. Find the position vector of \(S ^ { \prime }\).
    [0pt] [FP3/P6 January 2006 Qn 7]
    47. Find the values of \(x\) for which $$5 \cosh x - 2 \sinh x = 11$$ giving your answers as natural logarithms.
    [0pt] [FP2 June 2006 Qn 1]
    48. The point \(S\), which lies on the positive \(x\)-axis, is a focus of the ellipse with equation $$\frac { x ^ { 2 } } { 4 } + y ^ { 2 } = 1$$ Given that \(S\) is also the focus of a parabola \(P\), with vertex at the origin, find
56. a cartesian equation for \(P\),
57. an equation for the directrix of \(P\).
    49. The curve with equation $$y = - x + \tanh 4 x , \quad x \geq 0$$ has a maximum turning point \(A\).
58. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\).
59. Show that the \(y\)-coordinate of \(A\) is \(\frac { 1 } { 4 } \{ 2 \sqrt { 3 } - \ln ( 2 + \sqrt { 3 } ) \}\).
    50. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6706ed7f-4575-4898-b757-aee8475b2a30-25_663_647_1080_699}
    \end{figure} The curve \(C\), shown in Figure 1, has parametric equations $$\begin{aligned} & x = t - \ln t
    & y = 4 \sqrt { } t , \quad 1 \leq t \leq 4 \end{aligned}$$
60. Show that the length of \(C\) is \(3 + \ln 4\). The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
61. Find the exact area of the curved surface generated. \section*{51.} \section*{Figure 2}
    \includegraphics[max width=\textwidth, alt={}]{6706ed7f-4575-4898-b757-aee8475b2a30-26_666_937_317_703}
    Figure 2 shows a sketch of part of the curve with equation $$y = x ^ { 2 } \operatorname { arsinh } x$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line \(x = 3\). Show that the area of \(R\) is $$9 \ln ( 3 + \sqrt { } 10 ) - \frac { 1 } { 9 } ( 2 + 7 \sqrt { } 10 )$$ 52. $$I _ { n } = \int x ^ { n } \cosh x \quad \mathrm {~d} x , \quad n \geq 0$$
62. Show that, for \(n \geq 2\), $$I _ { n } = x ^ { n } \sinh x - n x ^ { n - 1 } \cosh x + n ( n - 1 ) I _ { n - 2 } .$$ (4)
63. Hence show that $$I _ { 4 } = \mathrm { f } ( x ) \sinh x + \mathrm { g } ( x ) \cosh x + C$$ where \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are functions of \(x\) to be found, and \(C\) is an arbitrary constant.
64. Find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 4 } \cosh x \mathrm {~d} x\), giving your answer in terms of e .
    53. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) and the line \(L\) has equation \(y = m x + c\), where \(m > 0\) and \(c > 0\).
65. Show that, if \(L\) and \(E\) have any points of intersection, the \(x\)-coordinates of these points are the roots of the equation $$\left( b ^ { 2 } + a ^ { 2 } m ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } - b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(E\),
66. show that \(c ^ { 2 } = b ^ { 2 } + a ^ { 2 } m ^ { 2 }\). The tangent \(L\) meets the negative \(x\)-axis at the point \(A\) and the positive \(y\)-axis at the point \(B\), and \(O\) is the origin.
67. Find, in terms of \(m , a\) and \(b\), the area of triangle \(O A B\).
68. Prove that, as \(m\) varies, the minimum area of triangle \(O A B\) is \(a b\).
69. Find, in terms of \(a\), the \(x\)-coordinate of the point of contact of \(L\) and \(E\) when the area of triangle \(O A B\) is a minimum.
    54. $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 1 & 2
    0 & 1 & 1
    0 & 0 & 1 \end{array} \right)$$ Prove by induction, that for all positive integers \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c c } 1 & n & \frac { 1 } { 2 } \left( n ^ { 2 } + 3 n \right)
    0 & 1 & n
    0 & 0 & 1 \end{array} \right)$$ [FP3 June 2006 Qn 1]
    55. The eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r } 4 & - 2
    1 & 1 \end{array} \right)$$ are \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\), where \(\lambda _ { 1 } < \lambda _ { 2 }\).
70. Find the value of \(\lambda _ { 1 }\) and the value of \(\lambda _ { 2 }\).
    (3)
71. Find \(\mathbf { M } ^ { - 1 }\).
    (2)
72. Verify that the eigenvalues of \(\mathbf { M } ^ { - 1 }\) are \(\lambda _ { 1 } { } ^ { - 1 }\) and \(\lambda _ { 2 } { } ^ { - 1 }\).
    (3) A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { M }\). There are two lines, passing through the origin, each of which is mapped onto itself under the transformation \(T\).
73. Find cartesian equations for each of these lines.
    (4)
    [0pt] [FP3 June 2006 Qn 5]
    56. The points \(A , B\) and \(C\) lie on the plane \(\Pi _ { 1 }\) and, relative to a fixed origin \(O\), they have position vectors $$\mathbf { a } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } \quad \text { and } \quad \mathbf { c } = 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k }$$ respectively.
74. Find \(( \mathbf { b } - \mathbf { a } ) \times ( \mathbf { c } - \mathbf { a } )\).
75. Find an equation for \(\Pi _ { 1 }\), giving your answer in the form r.n \(= p\). The plane \(\Pi _ { 2 }\) has cartesian equation \(x + z = 3\) and \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).
76. Find an equation for \(l\), giving your answer in the form \(( \mathbf { r } - \mathbf { p } ) \times \mathbf { q } = \mathbf { 0 }\). The point \(P\) is the point on \(l\) that is the nearest to the origin \(O\).
77. Find the coordinates of \(P\).
    57. Evaluate \(\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { \left( x ^ { 2 } + 4 x - 5 \right) } } \mathrm { d } x\), giving your answer as an exact logarithm.
    [0pt] [FP2 June 2007 Qn 1]
    58. The ellipse \(D\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\) and the ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1\).
78. Sketch \(D\) and \(E\) on the same diagram, showing the coordinates of the points where each curve crosses the axes. The point \(S\) is a focus of \(D\) and the point \(T\) is a focus of \(E\).
79. Find the length of \(S T\).
    59. The curve \(C\) has equation $$y = \frac { 1 } { 4 } \left( 2 x ^ { 2 } - \ln x \right) , x > 0 .$$ Find the length of \(C\) from \(x = 0.5\) to \(x = 2\), giving your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are rational numbers.
    [0pt] [FP2 June 2007 Qn 3]
    60. (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh ( A - B ) = \cosh A \cosh B - \sinh A \sinh B$$
80. Hence, or otherwise, given that \(\cosh ( x - 1 ) = \sinh x\), show that $$\tanh x = \frac { \mathrm { e } ^ { 2 } + 1 } { \mathrm { e } ^ { 2 } + 2 \mathrm { e } - 1 }$$ [FP2 June 2007 Qn 4]
    61. Given that \(I _ { n } = \int _ { 0 } ^ { 8 } x ^ { n } ( 8 - x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x , n \geq 0\),
81. show that \(I _ { n } = \frac { 24 n } { 3 n + 4 } I _ { n - 1 } , \quad n \geq 1\).
82. Hence find the exact value of \(\int _ { 0 } ^ { 8 } x ( x + 5 ) ( 8 - x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).
    [0pt] [FP2 June 2007 Qn 6]
    62.
    \includegraphics[max width=\textwidth, alt={}, center]{6706ed7f-4575-4898-b757-aee8475b2a30-32_503_1412_429_403} \section*{Figure 1} Figure 1 shows part of the curve \(C\) with equation \(y = \operatorname { arsinh } ( \sqrt { } x ) , x \geq 0\).
83. Find the gradient of \(C\) at the point where \(x = 4\).
    (3) The region \(R\), shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line \(x = 4\).
84. Using the substitution \(x = \sinh ^ { 2 } \theta\), or otherwise, show that the area of \(R\) is $$k \ln ( 2 + \sqrt { 5 } ) - \sqrt { 5 }$$ where \(k\) is a constant to be found.
    63. Given that \(\left( \begin{array} { r } 0
    1
    - 1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 4 & p
    - 1 & q & - 4
    1 & 1 & 3 \end{array} \right)$$
85. find the eigenvalue of \(\mathbf { A }\) corresponding to \(\left( \begin{array} { r } 0
    1
    - 1 \end{array} \right)\),
86. find the value of \(p\) and the value of \(q\). The image of the vector \(\left( \begin{array} { r } l
    m
    n \end{array} \right)\) when transformed by \(\mathbf { A }\) is \(\left( \begin{array} { r } 10
    - 4
    3 \end{array} \right)\).
87. Using the values of \(p\) and \(q\) from part (b), find the values of the constants \(l , m\) and \(n\).
    64. The points \(A , B\) and \(C\) have position vectors, relative to a fixed origin \(O\), $$\begin{aligned} & \mathbf { a } = 2 \mathbf { i } - \mathbf { j } ,
    & \mathbf { b } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ,
    & \mathbf { c } = 2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \end{aligned}$$ respectively. The plane \(\Pi\) passes through \(A , B\) and \(C\).
88. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\).
89. Show that a cartesian equation of \(\Pi\) is \(3 x - y + 2 z = 7\). The line \(l\) has equation \(( \mathbf { r } - 5 \mathbf { i } - 5 \mathbf { j } - 3 \mathbf { k } ) \times ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) = \mathbf { 0 }\). The line \(l\) and the plane \(\Pi\) intersect at the point \(T\).
90. Find the coordinates of \(T\).
91. Show that \(A , B\) and \(T\) lie on the same straight line.
    [0pt] [FP3 June 2007 Qn 7]
    65. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } [ \ln ( \tanh x ) ] = 2 \operatorname { cosech } 2 x , \quad x > 0$$ [FP2 June 2008 Qn 1]
    66. Find the values of \(x\) for which $$8 \cosh x - 4 \sinh x = 13$$ giving your answers as natural logarithms.
    [0pt] [FP2 June 2008 Qn 2]
    67. Show that $$\int _ { 5 } ^ { 6 } \frac { 3 + x } { \sqrt { \left( x ^ { 2 } - 9 \right) } } d x = 3 \ln \left( \frac { 2 + \sqrt { 3 } } { 3 } \right) + 3 \sqrt { } 3 - 4$$ [FP2 June 2008 Qn 3]
    68. The curve \(C\) has equation $$y = \operatorname { arsinh } \left( x ^ { 3 } \right) , \quad x \geq 0$$ The point \(P\) on \(C\) has \(x\)-coordinate \(\sqrt { } 2\).
92. Show that an equation of the tangent to \(C\) at \(P\) is $$y = 2 x - 2 \sqrt { } 2 + \ln ( 3 + 2 \sqrt { } 2 )$$ The tangent to \(C\) at the point \(Q\) is parallel to the tangent to \(C\) at \(P\).
93. Find the \(x\)-coordinate of \(Q\), giving your answer to 2 decimal places.
    69. Given that $$I _ { n } = \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin ^ { n } x \quad \mathrm {~d} x , \quad n \geq 0$$
94. show that, for \(n \geq 2\), $$I _ { n } = \frac { n ( n - 1 ) } { n ^ { 2 } + 1 } I _ { n - 2 }$$
95. Find the exact value of \(I _ { 4 }\).
    [0pt] [FP2 June 2008 Qn 5]
    70.
    \includegraphics[max width=\textwidth, alt={}, center]{6706ed7f-4575-4898-b757-aee8475b2a30-36_554_1351_303_356} \section*{Figure 1} Figure 1 shows the curve \(C\) with equation $$y = \frac { 1 } { 10 } \cosh x \arctan ( \sinh x ) , \quad x \geq 0$$ The shaded region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = 2\).
96. Find \(\int \cosh x \arctan ( \sinh x ) \mathrm { d } x\).
97. Hence show that, to 2 significant figures, the area of \(R\) is 0.34 .
    [0pt] [FP2 June 2008 Qn 6]
    71. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$
98. Show that an equation for the normal to \(H\) at a point \(P ( 4 \sec t , 3 \tan t )\) is $$4 x \sin t + 3 y = 25 \tan t$$ The point \(S\), which lies on the positive \(x\)-axis, is a focus of \(H\). Given that \(P S\) is parallel to the \(y\)-axis and that the \(y\)-coordinate of \(P\) is positive,
99. find the values of the coordinates of \(P\). Given that the normal to \(H\) at this point \(P\) intersects the \(x\)-axis at the point \(R\),
100. find the area of triangle \(P R S\).
     [0pt] [FP2 June 2008 Qn 7]
     72. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & p & 2
     0 & 3 & q
     2 & p & 1 \end{array} \right)$$ where \(p\) and \(q\) are constants.
     Given that \(\left( \begin{array} { l } 1
     2
     1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),
101. show that \(q = 4 p\). Given also that \(\lambda = 5\) is an eigenvalue of \(\mathbf { M }\), and \(p < 0\) and \(q < 0\), find
102. the values of \(p\) and \(q\),
103. an eigenvector corresponding to the eigenvalue \(\lambda = 5\).
     73. \begin{figure}[h]
     \includegraphics[alt={},max width=\textwidth]{6706ed7f-4575-4898-b757-aee8475b2a30-38_673_872_310_559} \captionsetup{labelformat=empty} \caption{Figure 1}
     \end{figure} Figure 1 shows a pyramid \(P Q R S T\) with base \(P Q R S\).
     The coordinates of \(P , Q\) and \(R\) are \(P ( 1,0 , - 1 ) , Q ( 2 , - 1,1 )\) and \(R ( 3 , - 3,2 )\).
     Find
104. \(\overrightarrow { P Q } \times \overrightarrow { P R }\)
     (3)
105. a vector equation for the plane containing the face \(P Q R S\), giving your answer in the form \(\mathbf { r } \cdot \mathbf { n } = d\). The plane \(\Pi\) contains the face \(P S T\). The vector equation of \(\Pi\) is \(\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } - 5 \mathbf { k } ) = 6\).
106. Find cartesian equations of the line through \(P\) and \(S\).
107. Hence show that \(P S\) is parallel to \(Q R\). Given that \(P Q R S\) is a parallelogram and that \(T\) has coordinates \(( 5,2 , - 1 )\),
108. find the volume of the pyramid \(P Q R S T\).
     (3)
     [0pt] [FP3 June 2008 Qn 7]