Edexcel GCE Core Mathematics C2 may/june 2013 exam answers review

Using calculus, find the coordinates of the stationary point on the curve with equation

1. Complete the table below with the value of y corresponding to x = 1.3, giving your
   answer to 4 decimal places.

   x	1	1.1	1.2	1.3	1.4	1.5
   y	0.7071	0.7591	0.8090	0.9037	0.9487

   (1)

2. Use the trapezium rule, with all the values of y in the completed table, to obtain an approximate value for
   \( \int_{1}{1.5}\)x/√(1 + x)

   giving your answer to 3 decimal places.

   You must show clearly each stage of your working.

   (4)

Find the first 4 terms, in ascending powers of x, of the binomial expansion of

giving each term in its simplest form.

(4)

f(x) = ax³ – 11x² + bx + 4, where a and b are constants.

When f(x) is divided by (x – 3) the remainder is 55

When f(x) is divided by (x + 1) the remainder is –9

1. Find the value of a and the value of b.(5)

Given that (3x + 2) is a factor of f(x),

2. factorise f(x) completely.(4)

The first three terms of a geometric series are 4p, (3p + 15) and (5p + 20) respectively, where p is a positive constant.

1. Show that 11p² – 10p – 225 = 0(4)
2. Hence show that p = 5 (2)
3. Find the common ratio of this series.(2)
4. Find the sum of the first ten terms of the series, giving your answer to the nearest integer.(3)

Given that log3 x = a, find in terms of a,

1. log₃ (9x)(2)
2. log₃(x⁵/81)(3)

giving each answer in its simplest form.

3. Solve, for x,

   log₃(9x) + log₃(x⁵/81) = 3

   giving your answer to 4 significant figures.

   (4)

The line with equation y = 10 cuts the curve with equation y = x² + 2x + 2 at the points A and B as shown in Figure 1. The figure is not drawn to scale.

1. Find by calculation the x-coordinate of A and the x-coordinate of B.(2)

The shaded region R is bounded by the line with equation y = 10 and the curve as shown in Figure 1.

2. Use calculus to find the exact area of R.(7)

Figure 2 shows the design for a triangular garden ABC where AB = 7 m, AC = 13 m and BC = 10 m.

Given that angle BAC = θ radians,

1. show that, to 3 decimal places, θ = 0.865

The point D lies on AC such that BD is an arc of the circle centre A, radius 7 m.

The shaded region S is bounded by the arc BD and the lines BC and DC. The shaded region S will be sown with grass seed, to make a lawned area.

Given that 50 g of grass seed are needed for each square metre of lawn,

2. find the amount of grass seed needed, giving your answer to the nearest 10 g.(7)

1. Solve, for 0≤θ≤180°

   sin (2θ – 30°) + 1 = 0.4

   giving your answers to 1 decimal place.

   (5)

2. Find all the values of x, in the interval 0≤x≤360°, for which
   9cos² x – 11cos x + 3sin² x = 0

   giving your answers to 1 decimal place.

   You must show clearly how you obtained your answers.

   (7)