Edexcel F3 (Further Pure Mathematics 3) 2021 June

1. (a) Using the definitions of hyperbolic functions in terms of exponentials, show that

$$1 - \tanh ^ { 2 } x \equiv \operatorname { sech } ^ { 2 } x$$ (b) Solve the equation $$2 \operatorname { sech } ^ { 2 } x + 3 \tanh x = 3$$ giving your answer as an exact logarithm.
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2. A curve has equation $$y = \sqrt { 9 - x ^ { 2 } } \quad 0 \leqslant x \leqslant 3$$

1. Using calculus, show that the length of the curve is \(\frac { 3 \pi } { 2 }\) The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Using calculus, find the exact area of the surface generated.
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3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 1 & p
1 & 1 & 2
- 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant (a) Find the exact values of \(p\) for which \(\mathbf { M }\) has no inverse. Given that \(\mathbf { M }\) does have an inverse, (b) find \(\mathbf { M } ^ { - 1 }\) in terms of \(p\).
3. \(\mathbf { M } = \left( \begin{array} { r c c } 3 & 1 & p
1 & 1 & 2
- 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant
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4. (i) $$f ( x ) = x \arccos x \quad - 1 \leqslant x \leqslant 1$$ Find the exact value of \(f ^ { \prime } ( 0.5 )\).
(ii) $$\mathrm { g } ( x ) = \arctan \left( \mathrm { e } ^ { 2 x } \right)$$ Show that $$\mathrm { g } ^ { \prime \prime } ( x ) = k \operatorname { sech } ( 2 x ) \tanh ( 2 x )$$ where \(k\) is a constant to be found.
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1. Prove that for \(n \geqslant 2\) $$( n - 1 ) I _ { n } = \tan x \sec ^ { n - 2 } x + ( n - 2 ) I _ { n - 2 }$$
2. Hence, showing each step of your working, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 6 } x d x$$ $$I _ { n } = \int \sec ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$ Prove that for \(n > 2\)

1. The line \(l _ { 1 }\) has equation

$$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathrm { r } = 2 \mathbf { i } + s \mathbf { j } + \mu ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ where \(s\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) both lie in a common plane \(\Pi _ { 1 }\)

1. show that an equation for \(\Pi _ { 1 }\) is \(3 x + y - z = 3\)
2. find the value of \(s\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 3\)
3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in degrees to 3 significant figures.

1. Using calculus, find the exact values of
   1. \(\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } - 4 x + 5 } \mathrm {~d} x\)
   2. \(\int _ { \sqrt { 3 } } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 3 } } { x ^ { 2 } } \mathrm {~d} x\)
      
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8. The hyperbola \(H\) has equation $$4 x ^ { 2 } - y ^ { 2 } = 4$$

1. Write down the equations of the asymptotes of \(H\).
2. Find the coordinates of the foci of \(H\). The point \(P ( \sec \theta , 2 \tan \theta )\) lies on \(H\).
3. Using calculus, show that the equation of the tangent to \(H\) at the point \(P\) is $$y \tan \theta = 2 x \sec \theta - 2$$ The point \(V ( - 1,0 )\) and the point \(W ( 1,0 )\) both lie on \(H\).
   The point \(Q ( \sec \theta , - 2 \tan \theta )\) also lies on \(H\).
   Given that \(P , Q , V\) and \(W\) are distinct points on \(H\) and that the lines \(V P\) and \(W Q\) intersect at the point \(S\),
4. show that, as \(\theta\) varies, \(S\) lies on an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are integers to be found.
   
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