Edexcel C4 june 2013 exam answers review

1. Find \( \int x^2e^x dx \)(5)
2. Hence find the exact value of \( \int_{0}^{2} x^2e^x dx \)(2)

1. Use the binomial expansion to show that
   √(1 + x/1 – x) ≈ 1 + x + 1/2x², |x|<1

   (6)

2. Substitute x = 1/26 into;
   √(1 + x/1 – x) = 1 + x + 1/2x²

   to obtain an approximation to √3

   Give your answer in the form a/b where a and b are integers.

   (3)

[IMAGE]
Figure 1 shows the finite region R bounded by the x-axis, the y-axis, the line x = π/2 and the curve with equation

The table shows corresponding values of x and y for y = sec (1/2x)

x	0	π/6	π/3	π/2
y	1	1.035276		1.414214

1. Complete the table above giving the missing value of y to 6 decimal places(1).
2. Using the trapezium rule, with all of the values of y from the completed table, find an
   approximation for the area of R, giving your answer to 4 decimal places.(3)

Region R is rotated through 2 radians about the x-axis.

3. Use calculus to find the exact volume of the solid formed.(4)

A curve C has parametric equations

1. Find dy/dx at the point where t = π/6(4)
2. Find a cartesian equation for C in the form
   y = f(x), – k≤x≤k,

   stating the value of the constant k.

   (3)

3. Write down the range of f(x).(2)

Use the substitution x = u², u > 0, to show that

1. \( \int \)1/x(2√x − 1)dx = \( \int \)2/u(2u – 1)/span>du

   (3)

2. Hence show that

   \( \int_{9}{1}\) 1/x(2√x − 1)dx = 2ln(a/b)

   where a and b are integers to be determined.

   (7)

Water is being heated in a kettle. At time t seconds, the temperature of the water is θ °C.

The rate of increase of the temperature of the water at any time t is modelled by the differential equation

where λ is a positive constant.

Given that λ = 20 when t = 0,

1. solve this differential equation to show that

   λ = 120 – 100e^–λt

   (8)

When the temperature of the water reaches 100 °C, the kettle switches off.

2. Given that λ = 0.01, find the time, to the nearest second, when the kettle switches off.(3)

A curve is described by the equation

1. Find dy/dx in terms of x and y.(5)

A point Q lies on the curve.

The tangent to the curve at Q is parallel to the y-axis.

Given that the x coordinate of Q is negative,

2. use your answer to part (a) to find the coordinates of Q.(7)

With respect to a fixed origin O, the line l has equation

[IMAGE]

The point A lies on l and has coordinates (3, – 2, 6).

The point P has position vector (–p i + 2p k) relative to O, where p is a constant.

Given that vector PA is perpendicular to l,

1. find the value of p.(4)

Given also that B is a point on l such that ∠BPA = 45°,

2. find the coordinates of the two possible positions of B.(5)