Edexcel A Level Maths C4 · Core Mathematics 4 January 2010 question paper 6666

Find the binomial expansion of

in ascending powers of x up to and including the term in x³, simplifying each term.

Substitute x into the binomial expansion in part (a) and hence obtain an approximation to √23. Give your answer to 5 decimal places.

Figure 1 shows a sketch of the curve with equation y = x 1n x, x 1. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 4. The table shows corresponding values of x and y for y = x 1n x.

Complete the table with the values of y corresponding to x = 2 and x = 2.5, giving your answers to 3 decimal places.

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 2 decimal places.

Hence find the exact area of R, giving your answer in the form a ln 2 b), where a and b are integers.

The curve C has the equation

The point P lies on C where x

Find the value of y at P.

Find the equation of the tangent to C at P, giving your answer in the form ax + by + cπ = 0, where a, b and c are integers.

The line l₁ has vector equation

and the line l₂ has vector equation

where λ and μ are parameters. The lines l₁ and l₂ intersect at the point A and the acute angle between l₁ and l₂ is .

Write down the coordinates of A.

Find the value of cos .

The point X lies on l₁ where λ = 4.

Find the coordinates of X.

Hence, or otherwise, show that = 4√26.4 26.

The point Y lies on l₂. Given that the vector is perpendicular to l₁,

find the length of AY, giving your answer to 3 significant figures.

Given that y = 8 at x =1, solve the differential equation

giving your answer in the form y² = g(x).

The area A of a circle is increasing at a constant rate of 1.5 cm² s^–1. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm².

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

The curve C cuts the x-axis at the points A and B.

Find the x-coordinate at the point A and the x- coordinate at the point B.

The region R, as shown shaded in Figure 2, is enclosed by the loop of the curve.

Use integration to find the area of R.

Using the substitution x = 2 cos u, or otherwise, find the exact value of

The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = √2. The shaded region S is rotated through 2π radians about the x-axis to form a solid of revolution.

Using your answer to part (a), find the exact volume of the solid of revolution formed.