Edexcel FP1 (Further Pure Mathematics 1) 2022 June

1. An ellipse has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1\) and eccentricity \(e _ { 1 }\) A hyperbola has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) and eccentricity \(e _ { 2 }\)

Given that \(e _ { 1 } \times e _ { 2 } = 1\)

1. show that \(a ^ { 2 } = 3 b ^ { 2 }\) Given also that the coordinates of the foci of the ellipse are the same as the coordinates of the foci of the hyperbola,
2. determine the equation of the hyperbola.

1. During 2029, the number of hours of daylight per day in London, H, is modelled by the equation

$$H = 0.3 \sin \left( \frac { x } { 60 } \right) - 4 \cos \left( \frac { x } { 60 } \right) + 11.5 \quad 0 \leqslant x < 365$$ where \(x\) is the number of days after 1st January 2029 and the angle is in radians.

1. Show that, according to the model, the number of hours of daylight in London on the 31st January 2029 will be 8.13 to 3 significant figures.
2. Use the substitution \(t = \tan \left( \frac { x } { 120 } \right)\) to show that \(H\) can be written as $$H = \frac { a t ^ { 2 } + b t + c } { 1 + t ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are constants to be determined.
3. Hence determine, according to the model, the date of the first day of 2029 when there will be at least 12 hours of daylight in London.

1. With respect to a fixed origin \(O\), the points \(A\) and \(B\) have coordinates \(( 2,2 , - 1 )\) and ( \(4,2 p , 1\) ) respectively, where \(p\) is a constant.

For each of the following, determine the possible values of \(p\) for which,

1. \(O B\) makes an angle of \(45 ^ { \circ }\) with the positive \(x\)-axis
2. \(\overrightarrow { O A } \times \overrightarrow { O B }\) is parallel to \(\left( \begin{array} { r } 4
   - p
   2 \end{array} \right)\)
3. the area of triangle \(O A B\) is \(3 \sqrt { 2 }\)

1. The velocity \(v \mathrm {~ms} ^ { - 1 }\), of a raindrop, \(t\) seconds after it falls from a cloud, is modelled by the differential equation

$$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.1 v ^ { 2 } + 10 \quad t \geqslant 0$$ Initially the raindrop is at rest.

1. 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 estimate the velocity of the raindrop 1 second after it falls from the cloud. Given that the initial acceleration of the raindrop is found to be smaller than is suggested by the current model,
2. refine the model by changing the value of one constant.

1. The rectangular hyperbola \(H\) has equation \(x y = 36\)
   1. Use calculus to show that the equation of the tangent to \(H\) at the point \(P \left( 6 t , \frac { 6 } { t } \right)\) is
   $$y t ^ { 2 } + x = 12 t$$ The point \(Q \left( 12 t , \frac { 3 } { t } \right)\) also lies on \(H\).
2. Find the equation of the tangent to \(H\) at the point \(Q\). The tangent at \(P\) and the tangent at \(Q\) meet at the point \(R\).
3. Show that as \(t\) varies the locus of \(R\) is also a rectangular hyperbola.

1. The points \(P , Q\) and \(R\) have position vectors \(\left( \begin{array} { r } 1
   - 2
   4 \end{array} \right) , \left( \begin{array} { r } 3
   1
   - 5 \end{array} \right)\) and \(\left( \begin{array} { l } 2
   0
   3 \end{array} \right)\) respectively.
   1. Determine a vector equation of the plane that passes through the points \(P , Q\) and \(R\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), where \(\lambda\) and \(\mu\) are scalar parameters.
   2. Determine the coordinates of the point of intersection of the plane with the \(x\)-axis.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \left| x ^ { 2 } - 8 \right|\) and a sketch of the straight line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. The equation $$\left| x ^ { 2 } - 8 \right| = m x + c$$ has exactly 3 roots, as shown in Figure 1.

1. Show that $$m ^ { 2 } - 4 c + 32 = 0$$ Given that \(c = 3 m\)
2. determine the value of \(m\) and the value of \(c\)
3. Hence solve $$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$

1. The Taylor series expansion of \(f ( x )\) about \(x = a\) is given by

$$f ( x ) = f ( a ) + ( x - a ) f ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } f ^ { ( r ) } ( a ) + \ldots$$

1. (a) Use differentiation to determine the Taylor series expansion of \(\ln x\), in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 2 }\)
   (b) Hence prove that $$\lim _ { x \rightarrow 1 } \left( \frac { \ln x } { x - 1 } \right) = 1$$
2. Use L'Hospital's rule to determine $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { ( x + 3 ) \tan ( 6 x ) \operatorname { cosec } ( 2 x ) } \right)$$ (Solutions relying entirely on calculator technology are not acceptable.)

1. A particle \(P\) moves along a straight line.

At time \(t\) minutes, the displacement, \(x\) metres, of \(P\) from a fixed point \(O\) on the line is modelled by the differential equation $$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x + 16 t ^ { 2 } x = 4 t ^ { 3 } \sin 2 t$$

1. Show that the transformation \(x =\) ty transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 16 y = 4 \sin 2 t$$
2. Hence find a general solution for the displacement of \(P\) from \(O\) at time \(t\) minutes.