Edexcel F3 (Further Pure Mathematics 3) 2021 January

1. Relative to a fixed origin \(O\), the points \(A\), \(B\), \(C\) and \(D\) have coordinates \(( 0,4,1 ) , ( 4,0,0 )\), \(( 3,5,2 )\) and \(( 2,2 , k )\) respectively, where \(k\) is a constant.
   1. Determine the exact area of triangle \(A B C\).
   2. Determine in terms of \(k\), the volume of the tetrahedron \(A B C D\), simplifying your answer. \(( 3,5,2 )\) and \(( 2,2 , k )\) respectively, where \(k\) is a constant.
   3. Determine the exact area of triangle \(A B C\).
   $$\text { etrahedron } A B C D \text {, simplifying }$$

2. $$y = \ln ( \tanh 2 x ) \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosech } 4 x$$ where \(p\) is a constant to be determined.
2. Hence determine, in simplest form, the exact value of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)

3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & k & 2
2 & 2 & k
1 & 2 & 2 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Determine the values of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. find \(\mathbf { A } ^ { - 1 }\), giving your answer in terms of \(k\).
   3.

4. Using the substitution \(x = 4 \cosh \theta\) show that $$\int \frac { 1 } { \left( x ^ { 2 } - 16 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { a x } { \sqrt { x ^ { 2 } - 16 } } + c \quad | x | > 4$$ where \(a\) is a constant to be determined and \(c\) is an arbitrary constant.
(6)

5. $$\mathbf { M } = \left( \begin{array} { r r r } 6 & - 2 & - 1
- 2 & 6 & - 1
- 1 & - 1 & 5 \end{array} \right)$$ Given that 8 is an eigenvalue of \(\mathbf { M }\)

1. determine an eigenvector corresponding to the eigenvalue 8
2. Determine the other two eigenvalues of \(\mathbf { M }\).
3. Hence find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { T } \mathbf { M P } = \mathbf { D }\)
   5.
   

6. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x \quad n \in \mathbb { N }$$

1. Show that $$I _ { n } = \frac { x ^ { n - 1 } } { n } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } - \frac { 3 ( n - 1 ) } { n } I _ { n - 2 } \quad n \geqslant 3$$
2. Hence show that $$\int \frac { x ^ { 5 } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x = \frac { 1 } { 5 } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } \left( x ^ { 4 } + p x ^ { 2 } + q \right) + k$$ where \(p\) and \(q\) are integers to be determined and \(k\) is an arbitrary constant.
   

1. The point \(P\) has coordinates \(( 1,2,1 )\)

The line \(l\) has Cartesian equation $$\frac { x - 3 } { 5 } = \frac { y + 1 } { 3 } = \frac { z + 5 } { - 8 }$$ The plane \(\Pi _ { 1 }\) contains the point \(P\) and the line \(l\).

1. Show that a Cartesian equation for \(\Pi _ { 1 }\) is $$6 x - 2 y + 3 z = 5$$ The point \(Q\) has coordinates \(( 2 , k , - 7 )\), where \(k\) is a constant.
2. Show that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is $$\frac { 2 } { 7 } | k + 7 |$$ The plane \(\Pi _ { 2 }\) has Cartesian equation \(8 x - 4 y + z = - 3\)
   Given that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is the same as the shortest distance between \(\Pi _ { 2 }\) and \(Q\),
3. determine the possible values of \(k\).
   

1. The curve \(C\) has equation

$$y = 2 + \ln \left( 1 - x ^ { 2 } \right) \quad \frac { 1 } { 2 } \leqslant x \leqslant \frac { 3 } { 4 }$$

1. Show that the length of the curve \(C\) is given by $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x$$
2. Hence, using algebraic integration, show that the length of the curve \(C\) is \(p + \ln q\) where \(p\) and \(q\) are rational numbers to be determined.

9. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 4 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l\) is the normal to the ellipse at the point \(P\).

1. Show that an equation for \(l\) is $$5 x \sin \theta - 4 y \cos \theta = 9 \sin \theta \cos \theta$$ The point \(F\) is the focus of \(E\) that lies on the positive \(x\)-axis.
2. Determine the coordinates of \(F\). The line \(l\) crosses the \(x\)-axis at the point \(Q\).
3. Show that $$\frac { | Q F | } { | P F | } = e$$ where \(e\) is the eccentricity of \(E\).
   

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