1. (a) Find the binomial expansion of
√(1 - 8x), |x| < 1/8
in ascending powers of x up to and including the term in x3
(b) Show that, when x = 1/100, the exact value of √(1 - 8x) is √23/5
(c) Substitute x = 1/100 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.
(a) Complete the table with the values of y corresponding to x = 2 and x = 2.5, giving your answers to 3 decimal places.
(b) 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.
(c) (i) Use integration by parts to find ∫ x lnx dx
(ii) Hence find the exact area of R, giving your answer in the form ¼(a ln2 + b) where a and b are integers.
3. The curve C has the equation
cos 2x + cos 3y = 1, -π/4 ≤ x ≤ π/4, 0 ≤ y ≤ π/6
(a) Find dy/dx in terms of x and y.
The point P lies on C where x = π/6
(b) Find the value of y at P.
(c) 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 lines ll and l2 intersect at the point A and the acute angle between ll and l2
intersect at the point A and the acute angle between ll and l2 is θ
(a) Write down the coordinates of A.
(b) Find the value of cos θ.
The point X lies on ll where λ = 4.
(c) Find the coordinates of X.
(d) Find the vector AX.
(e) Hence, or otherwise, show that |AX| = 4√26.
The point Y lies on l2. Given that the vector YX is perpendicular to ll
(f) find the length of AY, giving your answer to 3 significant figures.