Edexcel F3 (Further Pure Mathematics 3) 2023 January

1. Given that

$$y = 3 x \arcsin 2 x \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$

1. determine an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence determine the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 4 }\), giving your answer in the form \(a \pi + b\) where \(a\) and \(b\) are fully simplified constants to be found.

1. A hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 5 } = 1 \quad \text { where } a \text { is a positive constant }$$ The line with equation \(x = \frac { 4 } { 3 }\) is a directrix of \(H\)

1. Write down an equation of the other directrix.
2. Determine
   1. the value of \(a\)
   2. the coordinates of each of the foci of \(H\)

1. Solve the equation

$$4 \tanh x - \operatorname { sech } x = 1$$ giving your answer in the form \(x = \ln k\) where \(k\) is a fully simplified rational number.
(6)

1. (a) Determine

$$\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x$$ (b) Hence determine the exact value of $$\int _ { - 2 } ^ { 2 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } d x$$ Give your answer in the form \(a \ln ( b + c \sqrt { 13 } )\), where \(a , b\) and \(c\) are rational numbers.

5. $$\mathbf { A } = \left( \begin{array} { r r r } a & a & 1 \\ - a & 4 & 0 \\ 4 & a & 5 \end{array} \right) \quad \text { where } a \text { is a positive constant }$$

1. Determine the exact value of \(a\) for which the matrix \(\mathbf { A }\) is singular. Given that 2 is an eigenvalue of \(\mathbf { A }\)
2. determine
   1. the value of \(a\)
   2. the other two eigenvalues of \(\mathbf { A }\) A normalised eigenvector for the eigenvalue 2 is \(\left( \begin{array} { c } \frac { 1 } { \sqrt { 6 } } \\ \frac { 1 } { \sqrt { 6 } } \\ - \frac { 2 } { \sqrt { 6 } } \end{array} \right)\)
3. Determine a normalised eigenvector for each of the other eigenvalues of \(\mathbf { A }\)

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1. A curve has parametric equations
   where \(a\) is a positive constant.

$$\begin{aligned} & x = a ( \theta - \sin \theta ) \\ & y = a ( 1 - \cos \theta ) \end{aligned}$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = k a ^ { 2 } \sin ^ { 2 } \frac { \theta } { 2 }$$ where \(k\) is a constant to be determined. The part of the curve from \(\theta = 0\) to \(\theta = 2 \pi\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Determine the area of the surface generated, giving your answer in terms of \(\pi\) and \(a\).
   [0pt] [Solutions relying on calculator technology are not acceptable.]

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

$$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Determine a vector perpendicular to \(\Pi\) The line \(l\) meets \(\Pi\) at the point ( \(1,2,3\) ) and passes through the point ( \(1,0,1\) )
2. Determine the size of the acute angle between \(\Pi\) and \(l\) Give your answer to the nearest degree.
3. Determine the shortest distance between \(\Pi\) and the point \(( 6 , - 3 , - 6 )\)

8. $$I _ { n } = \int \cos ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$

1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x + \frac { n - 1 } { n } I _ { n - 2 }$$
2. Show that for positive even integers \(n\) $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { n } x d x = \frac { ( n - 1 ) ( n - 3 ) \ldots 5 \times 3 \times 1 } { n ( n - 2 ) ( n - 4 ) \ldots 6 \times 4 \times 2 } \times \overline { 2 }$$
3. Hence determine the exact value of $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { 6 } x \sin ^ { 2 } x d x$$

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1. The ellipse \(E\) has equation

$$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The foci of \(E\) are \(F _ { 1 }\) and \(F _ { 2 }\)

1. 1. Determine the coordinates of \(F _ { 1 }\) and the coordinates of \(F _ { 2 }\)
   2. Write down the equation of each of the directrices of \(E\) The point \(P\) lies on the ellipse.
2. Show that \(\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 6\) The straight line through \(P\) with equation \(y = 2 x + c\) meets \(E\) again at the point \(Q\) The point \(M\) is the midpoint of \(P Q\)
3. Show that as \(P\) varies the locus of \(M\) is a straight line passing through the origin.