Edexcel FP1 AS (Further Pure 1 AS) 2019 June

1. (a) Write down the \(t\)-formula for \(\sin x\).
   (b) Use the answer to part (a)
   1. to find the exact value of \(\sin x\) when
   $$\tan \left( \frac { x } { 2 } \right) = \sqrt { 2 }$$
2. to show that $$\cos x = \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } }$$ (c) Use the \(t\)-formulae to solve for \(0 < \theta \leqslant 360 ^ { \circ }\) $$7 \sin \theta + 9 \cos \theta + 3 = 0$$ giving your answers to one decimal place.

1. A student was set the following problem.

Use algebra to find the set of values of \(x\) for which $$\frac { x } { x - 24 } > \frac { 1 } { x + 11 }$$ The student's attempt at a solution is written below. $$\begin{gathered} x ( x - 24 ) ( x + 11 ) ^ { 2 } > ( x + 11 ) ( x - 24 ) ^ { 2 }
x ( x - 24 ) ( x + 11 ) ^ { 2 } - ( x + 11 ) ( x - 24 ) ^ { 2 } > 0
( x - 24 ) ( x + 11 ) [ x ( x + 11 ) - x - 24 ] > 0
( x - 24 ) ( x + 11 ) \left[ x ^ { 2 } + 10 x - 24 \right] > 0
( x - 24 ) ( x + 11 ) ( x + 12 ) ( x - 2 ) > 0
x = 24 , x = - 11 , x = - 12 , x = 2
\{ x \in \mathbb { R } : - 12 < x < - 11 \} \cup \{ x \in \mathbb { R } : 2 < x < 24 \} \end{gathered}$$ Line 3 There are errors in the student's solution.

1. Identify the error made
   1. in line 3
   2. in line 7
2. Find a correct solution to this problem.

1. Julie decides to start a business breeding rabbits to sell as pets.

Initially she buys 20 rabbits. After \(t\) years the number of rabbits, \(R\), is modelled by the differential equation $$\frac { \mathrm { d } R } { \mathrm {~d} t } = 2 R + 4 \sin t \quad t > 0$$ Julie needs to have at least 40 rabbits before she can start to sell them.
Use two iterations of the approximation formula $$\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }$$ to find out if, according to the model, Julie will be able to start selling rabbits after 4 months.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a solid doorstop made of wood. The doorstop is modelled as a tetrahedron. Relative to a fixed origin \(O\), the vertices of the tetrahedron are \(A ( 2,1,4 )\), \(B ( 6,1,2 ) , C ( 4,10,3 )\) and \(D ( 5,8 , d )\), where \(d\) is a positive constant and the units are in centimetres.

1. Find the area of the triangle \(A B C\). Given that the volume of the doorstop is \(21 \mathrm {~cm} ^ { 3 }\)
2. find the value of the constant \(d\).

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the rectangular hyperbola \(H\) with equation $$x y = c ^ { 2 } \quad x > 0$$ where \(c\) is a positive constant.
The point \(P \left( c t , \frac { c } { t } \right)\) lies on \(H\).
The line \(l\) is the tangent to \(H\) at the point \(P\).
The line \(l\) crosses the \(x\)-axis at the point \(A\) and crosses the \(y\)-axis at the point \(B\).
The region \(R\), shown shaded in Figure 2, is bounded by the \(x\)-axis, the \(y\)-axis and the line \(l\). Given that the length \(O B\) is twice the length of \(O A\), where \(O\) is the origin, and that the area of \(R\) is 32 , find the exact coordinates of the point \(P\).