Edexcel C4 january 2012 exam answers review

The curve C has the equation 2x + 3y2 + 3×2 y = 4×2.

The point P on the curve has coordinates (–1, 1).

1. Find the gradient of the curve at P.(5)
2. Hence find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.(3)

1. Use integration by parts to find \( \int xsin 3x dx \)(3)
2. Using your answer to part (a), find \( \int x^22cos 3x dx \)(3)

1. Expand;

   1/(2 −5x)², |x| < 2/5

   in ascending powers of x, up to and including the term in x², giving each term as a simplified fraction.

   Given that the binomial expansion of;

   2 + kx/(2 – 5x)², |x| < 2/5

   is;

   1/2 + 7/4x + Ax² + …

   (5)

2. find the value of the constant k, (3)
3. find the value of the constant A. (3)

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Figure 1 shows the curve with equation

The finite region S, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 2

The region S is rotated 360° about the x-axis.

Use integration to find the exact value of the volume of the solid generated, giving your
answer in the form k ln a, where k and a are constants.

(5)

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

1. Find an expression for dy/dx in terms of t.(3)
2. Find the coordinates of all the points on C where dy/dx = 0(5)

[IMAGE]
Figure 3 shows a sketch of the curve with equation Y = 2sin3x/(1+cosx), 0≤x≤π/2

The finite region R, shown shaded in Figure 3, is bounded by the curve and the x-axis.

The table below shows corresponding values of x and y for y = 2sin3x/(1+cosx)

Sales	0	π/8	π/4	3π/8	π/2
Sales	0		1.17157	1.02280	0

1. Complete the table above giving the missing value of y to 5 decimal places. (1)
2. Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 4 decimal places.(3)

3. Using the substitution u = 1 + cos x , or otherwise, show that

   \( \int \) 2sin 2x/(1 + cosx)dx = 4ln(1 + cosx) – 4cosx + k

   …where k is a constant.
   (5)

4. Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.(3)

Relative to a fixed origin O, the point A has position vector ( 2i – j + 5k ),
the point B has position vector ( 5i + 2j + 10k ),
and the point D has position vector ( –i + j + 4k ).

The line l passes through the points A and B.

1. Find the vector AB.(2)
2. Find a vector equation for the line l.(2)
3. Show that the size of the angle BAD is 109°, to the nearest degree.(4)
4. The points A, B and D, together with a point C, are the vertices of the parallelogram
   ABCD, where AB = DC.

   Find the position vector of C.

   (4)

5. Find the area of the parallelogram ABCD, giving your answer to 3 significant figures.(3)
6. Find the shortest distance from the point D to the line l, giving your answer to
   3 significant figures.(2)

1. Express 1/P(5 − P) in partial fractions.(3)

A team of conservationists is studying the population of meerkats on a nature reserve.
The population is modelled by the differential equation

dP/dt = 1/15(5 – P), t≥0

where P, in thousands, is the population of meerkats and t is the time measured in years since the study began.

Given that when t = 0, P = 1,

2. solve the differential equation, giving your answer in the form,
   P = a/b+ce^-1/3t

   where a, b and c are integers.(8)

3. Hence show that the population cannot exceed 5000(1)