Edexcel FP3 (Further Pure Mathematics 3) 2016 June

1. $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 3
k & 1 & 3
2 & - 1 & k \end{array} \right) \text {, where } k \text { is a constant }$$ Given that the matrix \(\mathbf { A }\) is singular, find the possible values of \(k\).

1. The curve \(C\) has equation

$$y = \frac { x ^ { 2 } } { 8 } - \ln x , \quad 2 \leqslant x \leqslant 3$$ Find the length of the curve \(C\) giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational numbers to be found.

3. (a) Prove that $$\frac { \mathrm { d } ( \operatorname { arcoth } x ) } { \mathrm { d } x } = \frac { 1 } { 1 - x ^ { 2 } }$$ Given that \(y = ( \operatorname { arcoth } x ) ^ { 2 }\),
(b) show that $$\left( 1 - x ^ { 2 } \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 x \frac { d y } { d x } = \frac { k } { 1 - x ^ { 2 } }$$ where \(k\) is a constant to be determined.

4. (i) Find, without using a calculator, $$\int _ { 3 } ^ { 5 } \frac { 1 } { \sqrt { 15 + 2 x - x ^ { 2 } } } d x$$ giving your answer as a multiple of \(\pi\).
(ii)

1. Show that $$5 \cosh x - 4 \sinh x = \frac { \mathrm { e } ^ { 2 x } + 9 } { 2 \mathrm { e } ^ { x } }$$
2. Hence, using the substitution \(u = e ^ { x }\) or otherwise, find $$\int \frac { 1 } { 5 \cosh x - 4 \sinh x } d x$$

5. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The point \(P ( 4 \sec \theta , 3 \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\), lies on \(H\).

1. Show that an equation of the normal to \(H\) at the point \(P\) is $$3 y + 4 x \sin \theta = 25 \tan \theta$$ The line \(l\) is the directrix of \(H\) for which \(x > 0\)
   The normal to \(H\) at \(P\) crosses the line \(l\) at the point \(Q\). Given that \(\theta = \frac { \pi } { 4 }\)
2. find the \(y\) coordinate of \(Q\), giving your answer in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers to be found.
   

   VIIIV SIHI NI IIIUM IONOO

   VI4V SIHI NI IM IMM ION OC

   VI4V SIHI NI JIIYM IONOO

   \includegraphics[max width=\textwidth, alt={}, center]{f0c9aceb-98f3-4451-8a65-cee6d26be08c-09_2258_47_315_37}

6. $$\mathbf { M } = \left( \begin{array} { r r r } p & - 2 & 0
- 2 & 6 & - 2
0 & - 2 & q \end{array} \right)$$ where \(p\) and \(q\) are constants.
Given that \(\left( \begin{array} { r } 2
- 2
1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),

1. find the eigenvalue corresponding to this eigenvector,
2. find the value of \(p\) and the value of \(q\). Given that 6 is another eigenvalue of \(\mathbf { M }\),
3. find a corresponding eigenvector. Given that \(\left( \begin{array} { l } 1
   2
   2 \end{array} \right)\) is a third eigenvector of \(\mathbf { M }\) with eigenvalue 3
4. find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$

7. Given that $$I _ { n } = \int \frac { \sin n x } { \sin x } \mathrm {~d} x , \quad n \geqslant 1$$

1. prove that, for \(n \geqslant 3\) $$I _ { n } - I _ { n - 2 } = \int 2 \cos ( n - 1 ) x \mathrm {~d} x$$
2. Hence, showing each step of your working, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 6 } } \frac { \sin 5 x } { \sin x } d x$$ giving your answer in the form \(\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )\), where \(a\), \(b\) and \(c\) are integers to be found.

1. The plane \(\Pi _ { 1 }\) has equation

$$x - 5 y - 2 z = 3$$ The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } ) + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(\Pi _ { 1 }\) is perpendicular to \(\Pi _ { 2 }\)
2. Find a cartesian equation for \(\Pi _ { 2 }\)
3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = \mathbf { 0 }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors to be found.
   (6)