Edexcel FP3 (Further Pure Mathematics 3)

1. Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\), where \(a\) is a rational number.

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$ Calculate

1. \(\mathbf { b } \times \mathbf { c }\),
2. \(\mathbf { a . } ( \mathbf { b } \times \mathbf { c } )\),
3. the area of triangle \(O B C\),
4. the volume of the tetrahedron \(O A B C\).

4. Given that \(y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer as a simplified fraction.
2. Hence, or otherwise, find $$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$ giving your answer in the form \(\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)\), where \(a\) and \(b\) are integers.

5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$

1. Find an expression for \(\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5\).
2. Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
3. Find \(I _ { 4 }\) in the form \(k \pi\), where \(k\) is a fraction.

6. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants. The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.

1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).

7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1
- 1
2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
3
4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha
- 4
0 \end{array} \right) + \mu \left( \begin{array} { l } 0
3
2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find

1. the value of \(\alpha\),
2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
   Given that \(\alpha = 2\),
3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

8. A curve, which is part of an ellipse, has parametric equations $$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { a } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \text { where } c = \cos \theta$$ and where \(k\) and \(\alpha\) are constants to be found.
2. Using the substitution \(c = \frac { 3 } { 4 } \sinh u\), or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures. \section*{END} \section*{TOTAL FOR PAPER: 75 MARKS} Mathematical Formulae (Pink) Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP3), the paper reference (6669), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
   Full marks may be obtained for answers to ALL questions.
   There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
   You must show sufficient working to make your methods clear to the Examiner.
   Answers without working may not gain full credit. N35389RA \({ } _ { \text {This publication may only be reproduced in accordance with Edexcel Limited copyright policy. } }\) ©2010 Edexcel Limited.
   1. The line \(x = 8\) is a directrix of the ellipse with equation
   $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > 0 , \quad b > 0$$ and the point \(( 2,0 )\) is the corresponding focus.
   Find the value of \(a\) and the value of \(b\).
   2. Use calculus to find the exact value of \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\).
   3. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$
3. Solve the equation $$\cosh 2 x - 3 \sinh x = 15$$ giving your answers as exact logarithms.
   4. $$I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a \geq 0 , \quad n \geq 0$$
4. Show that, for \(n \geq 2\), $$I _ { n } = n a ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
5. Hence evaluate \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x \mathrm {~d} x\).
   5. Given that \(y = ( \operatorname { arcosh } 3 x ) ^ { 2 }\), where \(3 x > 1\), show that
6. \(\left( 9 x ^ { 2 } - 1 \right) \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 36 y\),
7. \(\left( 9 x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 18\).
   6. \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 0 & 3
   0 & - 2 & 1
   k & 0 & 1 \end{array} \right)\), where \(k\) is a constant.
   Given that \(\left( \begin{array} { l } 6
   1
   6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),
8. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6
   1
   6 \end{array} \right)\),
9. show that \(k = 3\),
10. 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 \(l _ { 2 }\).
11. Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).
    7. The plane \(\Pi\) has vector equation $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$
12. 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 \(N\).
13. Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\). The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
14. Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.
    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 \(P ( 4 \sec t , 2 \tan t )\).
15. 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 \(Q\).
16. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$

9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$

1. Show that, for \(n > 0\), $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
2. Find \(I _ { 2 }\).