Edexcel FP1 AS (Further Pure 1 AS) 2018 June

1. (a) Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that the equation

$$5 \sin x + 12 \cos x = 2$$ can be written in the form $$7 t ^ { 2 } - 5 t - 5 = 0$$ (b) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$5 \sin x + 12 \cos x = 2$$ giving your answers to one decimal place.

1. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of coffee in a cup, \(t\) minutes after the cup of coffee is put in a room, is modelled by the differential equation

$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$ where \(k\) is a constant.
The coffee has an initial temperature of \(80 ^ { \circ } \mathrm { C }\)
Using \(k = 0.1\)

1. use two iterations of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { 0 } = \frac { y _ { 1 } - y _ { 0 } } { h }\) to estimate the temperature of the coffee 3 minutes after it was put in the room. The coffee in a different cup, which also had an initial temperature of \(80 ^ { \circ } \mathrm { C }\) when it was put in the room, cools more slowly.
2. Use this information to suggest how the value of \(k\) would need to be changed in the model.

   V349 SIHI NI IMIMM ION OC

   VJYV SIHIL NI LIIIM ION OO

   VJYV SIHIL NI JIIYM ION OC

1. Use algebra to find the values of \(x\) for which

$$\frac { x } { x ^ { 2 } - 2 x - 3 } \leqslant \frac { 1 } { x + 3 }$$

1. A scientist is investigating the properties of a crystal. The crystal is modelled as a tetrahedron whose vertices are \(A ( 12,4 , - 1 ) , B ( 10,15 , - 3 ) , C ( 5,8,5 )\) and \(D ( 2,2 , - 6 )\), where the length of unit is the millimetre. The mass of the crystal is 0.5 grams.
   1. Show that, to one decimal place, the area of the triangular face \(A B C\) is \(52.2 \mathrm {~mm} ^ { 2 }\)
   2. Find the density of the crystal, giving your answer in \(\mathrm { g } \mathrm { cm } ^ { - 3 }\)

   V349 SIHI NI IMIMM ION OC

   VJYV SIHIL NI LIIIM ION OO

   VJYV SIHIL NI JIIYM ION OC

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant.

The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$p ^ { 3 } x - p y + c \left( 1 - p ^ { 4 } \right) = 0$$ The normal to \(H\) at the point \(P\) meets \(H\) again at the point \(Q\).
2. Find the coordinates of the midpoint of \(P Q\) in terms of \(c\) and \(p\), simplifying your answer where possible.