Edexcel C4 january 2013 exam answers review

1. Given
   f(x) = (2 + 3x)^-3, |x|<2/3

   find the binomial expansion of f(x), in ascending powers of x, up to and including the term
   in x³.

   Give each coefficient as a simplified fraction.

   (5)

1. Use integration to find
   \( \int \)1/x³lnx dx

   (5)

2. Hence calculate

   \(\int_{0}^{2}\)1/x³lnx dx

   (2)

Figure 1 shows a sketch of part of the curve with equation y = x/1 + √x The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis, the line with equation x = 1 and the line with equation x = 4.

1. Complete the table with the value of y corresponding to x = 3, giving your answer to 4 decimal places.(1)
   

   1	2	3	4
   0.5	0.8284		1.3333

2. Use the trapezium rule, with all the values of y in the completed table, to obtain an
   estimate of the area of the region R, giving your answer to 3 decimal places.(3)
3. Use the substitution u = 1 + √x, to find, by integrating, the exact area of R.(8)

Figure 2 shows a sketch of part of the curve C with parametric equations

The curve crosses the y-axis at the point A and crosses the x-axis at the point B.

1. Show that A has coordinates (0, 3).(2)
2. Find the x coordinate of the point B.(2)
3. Find an equation of the normal to C at the point A.(5)

The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = –1 and
the x-axis.

4. Use integration to find the exact area of R.(6)

Figure 3 shows a sketch of part of the curve with equation y = 1 – 2cos x, where x is measured in radians. The curve crosses the x-axis at the point A and at the point B.

1. Find, in terms of π, the x coordinate of the point A and the x coordinate of the point B.(3)

The finite region S enclosed by the curve and the x-axis is shown shaded in Figure 3. The region S is rotated through 2 radians about the x-axis.

2. Find, by integration, the exact value of the volume of the solid generated.(6)

With respect to a fixed origin O, the lines l1 and l2 are given by the equations

where λ and µ are scalar parameters.

1. Given that l¹ and l² meet, find the position vector of their point of intersection.(5)
2. Find the acute angle between l¹ and l², giving your answer in degrees to 1 decimal
   place.(3)

Given that the point A has position vector 4i + 16j – 3k and that the point P lies on l₁ such
that AP is perpendicular to l₁,

3. find the exact coordinates of P.(6)

A bottle of water is put into a refrigerator. The temperature inside the refrigerator remains
constant at 3 °C and t minutes after the bottle is placed in the refrigerator the temperature
of the water in the bottle is θ °C.

The rate of change of the temperature of the water in the bottle is modelled by the differential equation,

1. By solving the differential equation, show that,
   θ = e^–0.008t + 3

   where A is a constant.

   (4)

Given that the temperature of the water in the bottle when it was put in the refrigerator was 16 °C,

2. find the time taken for the temperature of the water in the bottle to fall to 10 °C, giving your answer to the nearest minute.(5)