Edexcel FP3 (Further Pure Mathematics 3) Specimen

1. Find the eigenvalues of the matrix \(\left( \begin{array} { l l } 7 & 6
   6 & 2 \end{array} \right)\)
2. Find the values of \(x\) for which

$$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms.
(Total 6 marks)

3. \begin{figure}[h] \end{figure} The parametric equations of the curve \(C\) shown in Figure 1 are $$x = a ( t - \sin t ) , \quad y = a ( 1 - \cos t ) , \quad 0 \leq t \leq 2 \pi$$ Find, by using integration, the length of \(C\).

4. Find \(\int \sqrt { } \left( x ^ { 2 } + 4 \right) \mathrm { d } x\).

5. Given that \(y = \arcsin x\) prove that

1. \(\quad \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { \left( 1 - x ^ { 2 } \right) } }\)
2. \(\quad \left( 1 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0\)

6. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \sin x \mathrm {~d} x$$

1. Show that for \(n \geq 2\) $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
2. Hence obtain \(I _ { 3 }\), giving your answers in terms of \(\pi\).

7. $$\mathbf { A } ( x ) = \left( \begin{array} { c c c } 1 & x & - 1
3 & 0 & 2
1 & 1 & 0 \end{array} \right) , x \neq \frac { 5 } { 2 }$$

1. Calculate the inverse of \(\mathbf { A } ( x )\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & - 1
   3 & 0 & 2
   1 & 1 & 0 \end{array} \right)$$ The image of the vector \(\left( \begin{array} { c } p
   q
   r \end{array} \right)\) when transformed by \(\mathbf { B }\) is \(\left( \begin{array} { l } 2
   3
   4 \end{array} \right)\)
2. Find the values of \(p , q\) and \(r\).

8. The points \(A , B , C\), and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } + \mathbf { k } , \mathrm { b } = \mathbf { i } + 3 \mathbf { j } , \mathbf { c } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \mathbf { d } = 4 \mathbf { j } + \mathbf { k }$$ respectively.

1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\) and hence find the area of triangle \(A B C\).
2. Find the volume of the tetrahedron \(A B C D\).
3. Find the perpendicular distance of \(D\) from the plane containing \(A , B\) and \(C\).

9. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)

1. Show that an equation of the normal to \(C\) at \(P ( a \sec \theta , b \tan \theta )\) is $$b y + a x \sin \theta = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(A B\) is \(M\).
2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies.
   (Total 13 marks)