6. A car was purchased for £18 000 on 1st January.
On 1st January each following year, the value of the car is 80% of its value on 1st January
in the previous year.
(a) Show that the value of the car exactly 3 years after it was purchased is £9216. The value of the car falls below £1000 for the first time n years after it was purchased.
(b) Find the value of n.
An insurance company has a scheme to cover the maintenance of the car. The cost is £200 for the first year, and for every following year the cost increases by 12% so that for the 3rd year the cost of the scheme is £250.88
(c) Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
(d) Find the total cost of the insurance scheme for the first 15 years.
The curve C has equation y = x2 - 5x + 4. It cuts the x-axis at the points L and M as shown
in Figure 2.
(a) Find the coordinates of the point L and the point M.
(b) Show that the point N (5, 4) lies on C.
(c) Find ∫x2 - 5x + 4 dx
The finite region R is bounded by LN, LM and the curve C as shown in Figure 2.
(d) Use your answer to part (c) to find the exact value of the area of R.
Figure 3 shows a sketch of the circle C with centre N and equation
(x - 2)2 + (y + 1)2 = 169/4
(a) Write down the coordinates of N.
(b) Find the radius of C.
The chord AB of C is parallel to the x-axis, lies below the x-axis and is of length 12 units as shown in Figure 3.
(c) Find the coordinates of A and the coordinates of B.
(d) Show that angle ANB = 134.8°, to the nearest 0.1 of a degree.
The tangents to C at the points A and B meet at the point P.
(e) Find the length AP, giving your answer to 3 significant figures.
9. The curve C has equation y = 12√(x) - x3/2 - 10, x > 0
(a) Use calculus to find the coordinates of the turning point on C.
(b) Find d2y/dx2
(c) State the nature of the turning point.