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
$$\frac { 7 } { 4 } \ln ( 2 + \sqrt { 3 } ) - \frac { \sqrt { 3 } } { 2 }$$
- (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.
- Given that \(\mathbf { A B } = \mathbf { A C }\), and that \(\mathbf { A }\) is not singular, prove that \(\mathbf { B } = \mathbf { C }\).
- 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]