Edexcel FP3 (Further Pure Mathematics 3) 2014 June

1. Solve the equation

$$5 \tanh x + 7 = 5 \operatorname { sech } x$$ Give each answer in the form \(\ln k\) where \(k\) is a rational number.

2. $$9 x ^ { 2 } + 6 x + 5 \equiv a ( x + b ) ^ { 2 } + c$$

1. Find the values of the constants \(a\), \(b\) and \(c\). Hence, or otherwise, find
2. \(\int \frac { 1 } { 9 x ^ { 2 } + 6 x + 5 } d x\)
3. \(\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 6 x + 5 } } \mathrm {~d} x\)

1. The curve \(C\) has equation

$$y = \frac { 1 } { 2 } \ln ( \operatorname { coth } x ) , \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosech } 2 x$$ The points \(A\) and \(B\) lie on \(C\). The \(x\) coordinates of \(A\) and \(B\) are \(\ln 2\) and \(\ln 3\) respectively.
2. Find the length of the arc \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
   (6)

4. $$I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } \left( 3 - x ^ { 2 } \right) ^ { n } \mathrm {~d} x , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 1\) $$I _ { n } = \frac { 6 n } { 2 n + 1 } I _ { n - 1 }$$
2. Hence find the exact value of \(I _ { 4 }\), giving your answer in the form \(k \sqrt { 3 }\) where \(k\) is a rational number to be found.

5. The ellipse \(E\) has equation $$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The point \(P ( a \cos \theta , b \sin \theta )\) is a general point on the ellipse \(E\).

1. Write down the value of \(a\) and the value of \(b\). The line \(L\) is a tangent to \(E\) at the point \(P\).
2. Show that an equation of the line \(L\) is given by $$3 y \sin \theta + x \cos \theta = 3$$ The line \(L\) meets the \(x\)-axis at the point \(Q\) and meets the \(y\)-axis at the point \(R\).
3. Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is given by $$k \operatorname { cosec } 2 \theta$$ where \(k\) is a constant to be found. The point \(M\) is the midpoint of \(Q R\).
4. Find a cartesian equation of the locus of \(M\), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

6. The symmetric matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2
2
1 \end{array} \right) , \left( \begin{array} { r } - 2
1
2 \end{array} \right)\) and \(\left( \begin{array} { r } 1
- 2
2 \end{array} \right)\) with eigenvalues 5, 2 and - 1 respectively.

1. Find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$ Given that \(\mathbf { P } ^ { - 1 } = \mathbf { P } ^ { \mathrm { T } }\)
2. show that $$\mathbf { M } = \mathbf { P D P } ^ { - 1 }$$
3. Hence find the matrix \(\mathbf { M }\).

7. The curve \(C\) has equation $$y = \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$ The part of the curve \(C\) between \(x = 0\) and \(x = \ln 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the area \(S\) of the curved surface generated is given by $$S = 2 \pi \int _ { 0 } ^ { \ln 3 } \mathrm { e } ^ { - x } \sqrt { 1 + \mathrm { e } ^ { - 2 x } } \mathrm {~d} x$$
2. Use the substitution \(\mathrm { e } ^ { - x } = \sinh u\) to show that $$S = 2 \pi \int _ { \operatorname { arsinh } \alpha } ^ { \operatorname { arsinh } \beta } \cosh ^ { 2 } u \mathrm {~d} u$$ where \(\alpha\) and \(\beta\) are constants to be determined.
3. Show that $$2 \int \cosh ^ { 2 } u \mathrm {~d} u = \frac { 1 } { 2 } \sinh 2 u + u + k$$ where \(k\) is an arbitrary constant.
4. Hence find the value of \(S\), giving your answer to 3 decimal places.

8. The plane \(\Pi _ { 1 }\) has vector equation \(\mathbf { r }\). \(\left( \begin{array} { l } 2
1
3 \end{array} \right) = 5\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } - 1
2
4 \end{array} \right) = 7\)

1. Find a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The plane \(\Pi _ { 3 }\) has cartesian equation $$x - y + 2 z = 31$$
2. Using your answer to part (a), or otherwise, find the coordinates of the point of intersection of the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\)
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