Edexcel FP3 (Further Pure Mathematics 3) 2014 June

1. The line \(l\) passes through the point \(P ( 2,1,3 )\) and is perpendicular to the plane \(\Pi\) whose vector equation is

$$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) = 3$$ Find

1. a vector equation of the line \(l\),
2. the position vector of the point where \(l\) meets \(\Pi\).
3. Hence find the perpendicular distance of \(P\) from \(\Pi\).

2. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 0 & 2
0 & 4 & 1
0 & 5 & 0 \end{array} \right)$$

1. Show that matrix \(\mathbf { M }\) is not orthogonal.
2. Using algebra, show that 1 is an eigenvalue of \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
3. Find an eigenvector of \(\mathbf { M }\) which corresponds to the eigenvalue 1 The transformation \(M : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\).
4. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac { y } { 2 } = \frac { z } { - 1 }$$

1. Using calculus, find the exact value of
   1. \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { \left( x ^ { 2 } - 2 x + 3 \right) } } \mathrm { d } x\)
   2. \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 x } \sinh x \mathrm {~d} x\)
      

1. Using the definitions of hyperbolic functions in terms of exponentials,
   1. show that
   $$\operatorname { sech } ^ { 2 } x = 1 - \tanh ^ { 2 } x$$
2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$

1. Given that \(y = \operatorname { artanh } \frac { x } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
   show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
   
2. \hspace{0pt} [In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.]

The points \(P ( 3 \cos \alpha , 2 \sin \alpha )\) and \(Q ( 3 \cos \beta , 2 \sin \beta )\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Show the equation of the chord \(P Q\) is $$\frac { x } { 3 } \cos \frac { ( \alpha + \beta ) } { 2 } + \frac { y } { 2 } \sin \frac { ( \alpha + \beta ) } { 2 } = \cos \frac { ( \alpha - \beta ) } { 2 }$$
2. Write down the coordinates of the mid-point of \(P Q\). Given that the gradient, \(m\), of the chord \(P Q\) is a constant,
3. show that the centre of the chord lies on a line $$y = - k x$$ expressing \(k\) in terms of \(m\).

7. A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x ^ { 2 } + y ^ { 2 } = r ^ { 2 }\) where \(r\) is a constant.

1. Show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \frac { r ^ { 2 } } { r ^ { 2 } - x ^ { 2 } }\)
2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4 \pi r ^ { 2 }\).
3. Write down the length of the arc of the curve \(y = \sqrt { } \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = 1\)

8. The position vectors of the points \(A , B\) and \(C\) from a fixed origin \(O\) are $$\mathbf { a } = \mathbf { i } - \mathbf { j } , \quad \mathbf { b } = \mathbf { i } + \mathbf { j } + \mathbf { k } , \quad \mathbf { c } = 2 \mathbf { j } + \mathbf { k }$$ respectively.

1. Using vector products, find the area of the triangle \(A B C\).
2. Show that \(\frac { 1 } { 6 } \mathbf { a } . ( \mathbf { b } \times \mathbf { c } ) = 0\)
3. Hence or otherwise, state what can be deduced about the vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).

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 \(I _ { 2 }\)