Questions - Edexcel FP3 - A-Level Maths

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 $$\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )$$

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 $\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$.
Find the perpendicular distance of $Q$ from $\Pi _ { 1 }$.
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,
prove that $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$,
(3)
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. $$4 x ^ { 2 } + 4 x + 17 \equiv ( a x + b ) ^ { 2 } + c , \quad a > 0$$
Find the values of $a , b$ and $c$.
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 $S ^ { \prime }$.
Find the coordinates of the foci of the ellipse.
(4)
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 $y = \sinh ^ { n - 1 } x \cosh x$,
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 $I _ { n } = \int _ { 0 } ^ { \text {arsinh } 1 } \sinh ^ { n } x \mathrm {~d} x , \quad n \geq 0$.
Using the result in part (a), or otherwise, show that $$n I _ { n } = \sqrt { } 2 - ( n - 1 ) I _ { n - 2 } , \quad n \geq 2$$
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 $$x = a \cos ^ { 3 } \theta , \quad y = a \sin ^ { 3 } \theta , \quad 0 \leq \theta < 2 \pi$$
Find the total length of this curve. The curve is rotated through $\pi$ radians about the $x$-axis.
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.
Find $\xrightarrow { A B } \times \overrightarrow { A C }$.
Find an equation of $\Pi$ in the form r.n $= p$. The point $D$ has position vector $5 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$.
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
the value of $p$,
the value of $k$,
the values of $a , b$ and $c$,
$| \operatorname { det } \mathbf { M } |$.
[0pt] [P6 June 2004 Qn 5]
28. The transformation $R$ is represented by the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { l l } 3 & 1
1 & 3 \end{array} \right)$$
Find the eigenvectors of $\mathbf { A }$.
Find an orthogonal matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\mathbf { A } = \mathbf { P D } \mathbf { P } ^ { - 1 }$$
Hence describe the transformation $R$ as a combination of geometrical transformations, stating clearly their order.
29. (a) Find $\int \frac { 1 + x } { \sqrt { } \left( 1 - 4 x ^ { 2 } \right) } \mathrm { d } x$.
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,
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. $$I _ { n } = \int x ^ { n } \mathrm { e } ^ { 2 x } \mathrm {~d} x , \quad n \geq 0$$
Prove that, for $n \geq 1$, $$I _ { n } = \frac { 1 } { 2 } \left( x ^ { n } \mathrm { e } ^ { 2 x } - n I _ { n - 1 } \right) .$$
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 $0 < x \leq 1$,
$$\ln \left( \frac { 1 - \sqrt { } \left( 1 - x ^ { 2 } \right) } { x } \right) = - \ln \left( \frac { 1 + \sqrt { } \left( 1 - x ^ { 2 } \right) } { x } \right)$$
Using the definition of $\cosh x$ or sech $x$ in terms of exponentials, show that, for $0 < x \leq 1$, $$\operatorname { arsech } x = \ln \left( \frac { 1 + \sqrt { } \left( 1 - x ^ { 2 } \right) } { x } \right)$$
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 $$\mathbf { b } - \mathbf { c } = \lambda \mathbf { a } , \quad \text { where } \lambda \text { is a scalar. }$$
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 $l _ { 2 }$.
Find a vector which is normal to $\Pi _ { 1 }$.
Show that an equation for $\Pi _ { 1 }$ is $6 x + y - 4 z = 16$.
Find the value of $p$. The plane $\Pi _ { 2 }$ has equation $\mathbf { r } . ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) = 2$.
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. $$\mathbf { A } = \left( \begin{array} { l l l } 3 & 2 & 4
2 & 0 & 2
4 & 2 & k \end{array} \right)$$
Show that $\operatorname { det } \mathbf { A } = 20 - 4 k$.
(2)
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 $\mathbf { A }$,
find the corresponding eigenvalue.
(2) Given that the only other distinct eigenvalue of $\mathbf { A }$ is 8,
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
the value of the eccentricity of $H$,
the distance between the foci of $H$. The ellipse $E$ has equation $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$.
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 $$4 \cosh ^ { 3 } x - 3 \cosh x = \cosh 3 x$$
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 $$I _ { n } = \int _ { 0 } ^ { 4 } x ^ { n } \sqrt { } ( 4 - x ) \mathrm { d } x , \quad n \geq 0$$
show that $I _ { n } = \frac { 8 n } { 2 n + 3 } I _ { n - 1 } , n \geq 1$.
(6) Given that $\int _ { 0 } ^ { 4 } \sqrt { } ( 4 - x ) \mathrm { d } x = \frac { 16 } { 3 }$,
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)
Given that $y = \operatorname { artanh } ( \sin x )$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x$.
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
the two eigenvalues of $\mathbf { A }$,
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. $$\mathbf { A } = \left( \begin{array} { r r r } k & 1 & - 2
0 & - 1 & k
9 & 1 & 0 \end{array} \right) , \text { where } k \text { is a real constant. }$$
Find values of $k$ for which $\mathbf { A }$ is singular. Given that $\mathbf { A }$ is non-singular,
find, in terms of $k , \mathbf { A } ^ { - 1 }$.
(5)
[0pt] [FP3/P6 January 2006 Qn 4]
46. The plane $\Pi$ passes through the points $$P ( - 1,3 , - 2 ) , Q ( 4 , - 1 , - 1 ) \text { and } R ( 3,0 , c ) \text {, where } c \text { is a constant. }$$
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,
find the value of $c$ and show that $d = 4$,
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 $\Pi$.
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
a cartesian equation for $P$,
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 $A$.
Find, in exact logarithmic form, the $x$-coordinate of $A$.
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 $$\begin{aligned} & x = t - \ln t
& y = 4 \sqrt { } t , \quad 1 \leq t \leq 4 \end{aligned}$$
Show that the length of $C$ is $3 + \ln 4$. The curve is rotated through $2 \pi$ radians about the $x$-axis.
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. $$I _ { n } = \int x ^ { n } \cosh x \quad \mathrm {~d} x , \quad n \geq 0$$
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)
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.
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 $c > 0$.
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 $E$,
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.
Find, in terms of $m , a$ and $b$, the area of triangle $O A B$.
Prove that, as $m$ varies, the minimum area of triangle $O A B$ is $a b$.
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 $\lambda _ { 1 } < \lambda _ { 2 }$.
Find the value of $\lambda _ { 1 }$ and the value of $\lambda _ { 2 }$.
(3)
Find $\mathbf { M } ^ { - 1 }$.
(2)
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 $T$.
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.
Find $( \mathbf { b } - \mathbf { a } ) \times ( \mathbf { c } - \mathbf { a } )$.
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 $l$.
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 $O$.
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 $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$.
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 $E$.
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 $$\cosh ( A - B ) = \cosh A \cosh B - \sinh A \sinh B$$
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 $I _ { n } = \int _ { 0 } ^ { 8 } x ^ { n } ( 8 - x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x , n \geq 0$,
show that $I _ { n } = \frac { 24 n } { 3 n + 4 } I _ { n - 1 } , \quad n \geq 1$.
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 $y = \operatorname { arsinh } ( \sqrt { } x ) , x \geq 0$.
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 $x = 4$.
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 $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 4 & p
- 1 & q & - 4
1 & 1 & 3 \end{array} \right)$$
find the eigenvalue of $\mathbf { A }$ corresponding to $\left( \begin{array} { r } 0
1
- 1 \end{array} \right)$,
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 $\left( \begin{array} { r } 10
- 4
3 \end{array} \right)$.
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 $C$.
Find $\overrightarrow { A B } \times \overrightarrow { A C }$.
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 $T$.
Find the coordinates of $T$.
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 $\sqrt { } 2$.
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 $P$.
Find the $x$-coordinate of $Q$, giving your answer to 2 decimal places.
69. Given that $$I _ { n } = \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin ^ { n } x \quad \mathrm {~d} x , \quad n \geq 0$$
show that, for $n \geq 2$, $$I _ { n } = \frac { n ( n - 1 ) } { n ^ { 2 } + 1 } I _ { n - 2 }$$
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 $x = 2$.
Find $\int \cosh x \arctan ( \sinh x ) \mathrm { d } x$.
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 $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$
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,
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 $R$,
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 $\mathbf { M }$,
show that $q = 4 p$. Given also that $\lambda = 5$ is an eigenvalue of $\mathbf { M }$, and $p < 0$ and $q < 0$, find
the values of $p$ and $q$,
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
$\overrightarrow { P Q } \times \overrightarrow { P R }$
(3)
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 $\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } - 5 \mathbf { k } ) = 6$.
Find cartesian equations of the line through $P$ and $S$.
Hence show that $P S$ is parallel to $Q R$. Given that $P Q R S$ is a parallelogram and that $T$ has coordinates $( 5,2 , - 1 )$,
find the volume of the pyramid $P Q R S T$.
(3)
[0pt] [FP3 June 2008 Qn 7]

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