Imperial Aeronautical Engineering: ESAT requirements

One question per module

Imperial Aeronautical Engineering asks for Mathematics 1, Mathematics 2 and Physics. Try one original question from each.

1. Mathematics 1

   A map has a scale of $1:25\,000$. A lake has an area of $12\,\mathrm{cm^2}$ on the map. What is the actual area of the lake?
   A $0.3\,\mathrm{km^2}$B $0.75\,\mathrm{km^2}$C $3\,\mathrm{km^2}$D $7.5\,\mathrm{km^2}$E $75\,\mathrm{km^2}$

2. Mathematics 2

   For which values of $k$ does the equation $kx^2+4x+k$ $=3$ have no real solutions?
   A $-1$ $<k$ $<4$B $k$ $<-4$ or $k$ $>1$C $-4$ $<k$ $<1$D $k$ $>4$ onlyE $k$ $<-1$ or $k$ $>4$

3. Physics

   A trolley of mass $2.0\,\mathrm{kg}$ moving at $3.0\,\mathrm{m\,s^{-1}}$ collides with a stationary trolley of mass $1.0\,\mathrm{kg}$. The two trolleys stick together. How much kinetic energy is transferred to other stores in the collision?
   A $0\,\mathrm{J}$B $3.0\,\mathrm{J}$C $6.0\,\mathrm{J}$D $7.5\,\mathrm{J}$E $9.0\,\mathrm{J}$