Edexcel F3 (Further Pure Mathematics 3) 2022 June

1. (a) Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that

$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$ (b) Hence find the value of \(x\) for which $$\cosh ( x + \ln 2 ) = 5 \sinh x$$ giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is a rational number to be determined.
(5)

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Determine $$\int \frac { 1 } { \sqrt { 5 + 4 x - x ^ { 2 } } } d x$$
2. Use the substitution \(x = 3 \sec \theta\) to determine the exact value of $$\int _ { 2 \sqrt { 3 } } ^ { 6 } \frac { 18 } { \left( x ^ { 2 } - 9 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$ Give your answer in the form \(A + B \sqrt { 3 }\) where \(A\) and \(B\) are constants to be found.

3. $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 5 & 0
5 & 1 & - 3
0 & - 3 & 6 \end{array} \right)$$ Given that \(\mathbf { i } + \mathbf { j } + \mathbf { k }\) is an eigenvector of \(\mathbf { M }\),

1. determine the corresponding eigenvalue. Given that 8 is an eigenvalue of \(\mathbf { M }\),
2. determine a corresponding eigenvector.
3. Determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that $$\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }$$

4. $$y = \operatorname { artanh } \left( \frac { \cos x + a } { \cos x - a } \right)$$ where \(a\) is a non-zero constant.
Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \tan x$$ where \(k\) is a constant to be determined.

1. A curve has parametric equations

$$x = 4 \mathrm { e } ^ { \frac { 1 } { 2 } t } \quad y = \mathrm { e } ^ { t } - t \quad 0 \leqslant t \leqslant 4$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the curved surface generated is $$\pi \left( \mathrm { e } ^ { 8 } + A \mathrm { e } ^ { 4 } + B \right)$$ where \(A\) and \(B\) are constants to be determined.

6. $$\mathbf { A } = \left( \begin{array} { r r r } x & 1 & 3
2 & 4 & x
- 4 & - 2 & - 1 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular for all real values of \(x\).
2. Determine, in terms of \(x , \mathbf { A } ^ { - 1 }\)

7. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { 10 - x ^ { 2 } } } \mathrm {~d} x \quad n \in \mathbb { N } \quad | x | < \sqrt { 10 }$$

1. Show that $$n I _ { n } = 10 ( n - 1 ) I _ { n - 2 } - x ^ { n - 1 } \left( 10 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \quad n \geqslant 2$$
2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \sqrt { 10 - x ^ { 2 } } } \mathrm {~d} x$$ giving your answer in the form \(\frac { 1 } { 15 } ( p \sqrt { 10 } + q )\) where \(p\) and \(q\) are integers to be determined.

1. The plane \(\Pi\) has equation

$$3 x + 4 y - z = 17$$ The line \(l _ { 1 }\) is perpendicular to \(\Pi\) and passes through the point \(P ( - 4 , - 5,3 )\)
The line \(l _ { 1 }\) intersects \(\Pi\) at the point \(Q\)

1. Determine the coordinates of \(Q\) Given that the point \(R ( - 1,6,4 )\) lies on \(\Pi\)
2. determine a Cartesian equation of the plane containing \(P Q R\) The line \(l _ { 2 }\) passes through \(P\) and \(R\)
   The line \(l _ { 3 }\) is the reflection of \(l _ { 2 }\) in \(\Pi\)
3. Determine a vector equation for \(l _ { 3 }\)

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$ The line \(l\) has equation \(y = k x - 3\), where \(k\) is a constant.
Given that \(E\) and \(l\) meet at 2 distinct points \(P\) and \(Q\)

1. show that the \(x\) coordinates of \(P\) and \(Q\) are solutions of the equation $$\left( 9 k ^ { 2 } + 4 \right) x ^ { 2 } - 54 k x + 45 = 0$$ The point \(M\) is the midpoint of \(P Q\)
2. Determine, in simplest form in terms of \(k\), the coordinates of \(M\)
3. Hence show that, as \(k\) varies, \(M\) lies on the curve with equation $$x ^ { 2 } + p y ^ { 2 } = q y$$ where \(p\) and \(q\) are constants to be determined.